Chapter 5 Local and Nonlocal Robin Laplacian

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Stochastic and Analytic Aspects in the Situation of Robin Boundary Value Prolems Akhlil Khalid December 2013

` ma m`ere... A

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Contents Introduction

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Related Papers

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1 Preliminaries 10 1.1 Dirichlet Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Hunt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Potential Theory 2.1 Relative Capacity . . . 2.2 Smooth Measures . . . 2.3 Revuz Correspondence 2.4 Application . . . . . . 3

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Stochastic Approach to General Robin Boundary Value Problems 41 3.1 Review of Analytic Approach . . . . . . . . . . . . . . . . . . 41 3.2 Positive Smooth Measures Case . . . . . . . . . . . . . . . . . 45 3.2.1 General Reflecting Brownian Motion . . . . . . . . . . 45 3.2.2 Probabilistic Representation . . . . . . . . . . . . . . . 48 3.2.3 Properties of the Semigroup . . . . . . . . . . . . . . . 52

4 Signed Smooth Measure Case 4.1 A Special Kato Class of Measures 4.1.1 Definition and Properties . 4.1.2 Analytic Definition . . . . 4.2 Signed Measures Case . . . . . . 4.3 Domination Results . . . . . . . . 5

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55 55 56 58 59 61

Local and Nonlocal Robin Laplacian 68 5.1 Dirichlet and Neumann Boundary Conditions: What is in all between? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 iii

5.2

5.3

5.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 5.1.2 New Method for the Sandwiched Property . . . . . . Generalized Nonlocal Robin Laplacian on Arbitrary Domains 5.2.1 Closability . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Domination . . . . . . . . . . . . . . . . . . . . . . . Nonlocal Robin Laplacian involving Bounded Operators . . . 5.3.1 The Nonlocal Robin Laplacian . . . . . . . . . . . . . 5.3.2 Positivity . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Domination . . . . . . . . . . . . . . . . . . . . . . .

Bibliographie

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69 71 75 76 79 81 81 84 87 91

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Acknowledgements Je tiens `a remercier tout particuli`erement mon directeur de th`ese, Omar El-mennaoui ; pour sa disponibilit´e tout au long de ces longues ann´ees de th`ese. Je n’oublierais jamais tous les conseils qu’il m’a prodigu´e et la patience avec laquelle il a fait le suivi de ma progression. Je lui dois ´egalement l’opportunit´e que j’ai eu pour mon s´ejour de recherche en Allemagne. Si je devais ˆetre reconnaissant envers quelqu’un en dehors de ma m`ere, il serait bien lui. Encore merci. J’aimerais ´egalement exprimer toute ma gratitude `a Wolfgang Arendt de l’universit´e de Ulm pour son acceuil chaleureux, me permettant de m’int´egrer facilement dans son ´equipe de recherche. Par ses conseils, j’ai pu m’ouvrir sur d’autres horizons math´ematiques. J’´etais tr`es touch´e par l’amiti´e avec laquelle j’´etais re¸cu par tous les membres de l’institut d’analyse appliqu´ee de l’universit´e de Ulm en Allemagne. Je mentionne en particulier, Daniel Hauer, James Kennedy, Markus Kunze, Stephan Fackler et Moritz Gerlach. Je les remercie donc de m’avoir “adopter” d´es les premiers jours de mon s´ejour. Je remercie aussi le DAAD (der Deutsche Akademische Austauschdienst) pour avoir financer mon s´ejour en Allemagne, et de m’avoir donner l’occasion de m’enrichir de la culture locale. Je remercie enfin tout ceux qui ont particip´e de pr`es ou de loin `a ce modeste travail, par leurs suggestions, les discussions qu’on a eu ensemble ou tout simplement par leurs encouragements.

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Abstract Using a capacity approach, and the theory of measure’s perturbation of Dirichlet forms, we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain Ω. In the first time we study the case of positive smooth measures, and then we focus on signed smooth measures case by defining a special Kato class of measures. Some other topics related to the first one are also studied. In particular, we aim to characterize all semigroups sandwiched between Dirichlet and Neumann ones. In addition we characterize the closability of a general nonlocal Robin boundary value problem by defining a new notion of admissibility and we study the positivity of the nonlocal Robin Laplacian involving bounded operators.

Introduction The theory of Dirichlet forms is a well developed subject and plays a prominent role in various field of mathematics like potential theory, differential geometry, calculus of variations, stochatstic processes and partial differential equations. This mainly due to the fact that they allow to develop highly nontrivial extentions of classical theories under minimal regularity hypothesis, as for perturbation theory where Dirichlet form methods allows a high singularities on the potentials. The other reason is that they constitute an effective link between different mathematical theories and towards applications. The theory of Dirichlet forms originates in the pioneering work of Beurling and Deny [25] on functional analytic approach to potential theory. Fukushima and Silverstein [80, 45] developed the beautiful and fruitful corespondence between Dirichlet forms and symmetric Markov processes. Since that time, Dirichlet forms has turned out to be the most useful tool in the interplay of stochastics and analysis. In the last 40 years, this theory has turned out to be particulary suited for application in quantum theory. In this field, but also in other contexts, there is the necessary of studying certain generalized functionals of the process, coresponding to singular perturbation of a given Dirichlet form( see [6], [7], [8], [9], [29], [31], [36], [38], [48], [50], [83], [84], [88] and [89]). If we consider for example, the evolution associated with the heat equation with absorption ut = 12 Δu − V u, then the impact of the absorption-excitation effect “measured” with means of the rate V , can be seen by looking at the Feynman-Kac formula [61, 87] 1

et( 2 Δ−V ) f (x) = Ex [f (Bt )e−

t 0

V (Bs )ds

]

describing the solution of the absorbed heat equation with initial data f ∈ L2 (Rd ), and (Bt )t≥0 is a Standard Brownian motion on Rd . If we try to replace V with a certain positive measure μ, a need of a more sophisticated methods appears, and it is hier exactly where the theory of Dirichlet forms and the associated stochastic analysis step in. 1

To illustrate the special charm and the power of this theory, we consider 2 2 perturbations aμ := a + qμ of Dirichlet  forms a on some L space L (X, m) given by quadratic forms qμ (u, v) = X uvdμ with μ a positive Borel measure. The Borel measures that makes the Dirichlet forms aμ closed in L2 (X, m) are those which are ”absolutely continous” with respect to the capacity generated by a, which means that μ charges no set of zero capacity. In [7], it is proved that the set S of smooth measures is the appopriate one where the forms aμ are closed. In the same way as the Feynman Kac formula, one can write explicitly the strongly continous symmetric semigroup Ptμ associated with the closed form aμ as follow: μ

Ptμ f (x) = Ex [f (Xt )e−At ],

f ∈ L2 (X, m)

where (Xt )t≥0 is the Hunt process associated with the Dirichlet form a, and (Aμt )t≥0 is the positive continous additive functional (PCAF) of (Xt )t≥0 in Revuz correspondence with μ. The Feynman-Kac formula is a simple example of the above discussion. In fact, it suffices to take  t μ = V m which corresponds to the positive continous additive functional 0 V (Bs )ds with mean of Revuz correspondence. In the current thesis, we will be concerned with Robin boundary conditions on arbitrary domains. The classical Robin boundary conditions on a smooth domain Ω of Rd (d ≥ 1), is given by : ∂u + βu = 0 on ∂Ω, (0.1) ∂ν where ν is the outward normal vector field on the boundary ∂Ω, and β a positive bounded Borel measurable function defined on ∂Ω. Such boundary conditions appears in the modelisation of some physical, chimical or biological processes governed by the Laplacian equation ( stationary diffusion). Such processes are called Laplacian transport phenomena or diffusive transport phenomena. More in details, the diffusive transport phenomena describes the transport of species between two distinct “regions” separated by an interface. In biology, it describes the process when water and minerals are pumped by roots from earths, or when ions and biological species penetrate through cellular membranes, or also when oxygen molecules diffuse towards and pass through alveolar ducts. Transport processes are relvant for many other scientific domains, for example, heterogeneous catalysis and electrochemistry (see for example [51]). The theoretical analysis of these phenomena is in general complicated by an irregular geometry of the interface. In this situation the existence of the process associated to the transport phenomena, which is called partially reflecting Brownian motion, is compromised. 2

The goal is then to settle a theory permetting to deal with Robin boundary value problem in a general framework. The treatment of such problem as one can guess is situated in the confluence between Dirichlet forms, potential theory and stochastic analysis. One can then use the same approach as for perturbation theory of Dirichlet forms by measures. In fact, the Robin boundary conditions (0.1) are nothing but a perturba∂ , which represent Neumann boundary conditions, by the measure tion of ∂ν μ = β.σ, where σ is the surface measure. Consequently, the associated diffusion process is the reflecting Brownian motion killed by a certain additive functional, and the semigroup generated by the Laplacian with classical Robin boundary conditions can be giving as: Ptμ f (x) = Ex [f (Xt )e−

t 0

β(Xs )dLs

]

(0.2)

where (Xt )t≥0 is a reflecting Brownian motion (RBM), and Lt is the boundary local time, which corresponds to σ by Revuz correspondence. It is clear that the smoothness of the domain Ω in classical Robin boundary value problem, follows the smoothness of the domains where RBM is constructed( see [20, 21, 30, 23, 35, 54, 55] and references therein for more details about RBM). The probabilistic treatment of Robin boundary value problems has been considered by many authors [78, 71, 73, 76]. The first two authors considered bounded C 3 −domains since the third considered bounded domains with Lipschitz boundary, and [76] was concerned with C 3 − domains but with smooth measures instead of β. If one want to generalize the probabilistic treatment to a general domains, a difficulty arise when we try to get a diffusion process representing Neumann boundary conditions. In [21], the RBM is defined to be the Hunt process associated with the form (a, D(a)) defined on L2 (Ω) by:  a(u, v) = ∇u∇vdx , ∀u, v ∈ D(a) = H 1 (Ω) Ω

where Ω is assumed to be bounded with Lipschitz boundary so that the Dirichlet form (a, D(a)) is regular. If Ω is an arbitrary domain, then the Dirichlet form need not to be Regular, and to not loose the generality we  1 (Ω), the closure of H 1 (Ω) ∩ Cc (Ω) in H 1 (Ω). The domain consider D(a) = H 1  (Ω) is so defined to insure the Dirichlet form (a, D(a)) to be regular. H Now, if we perturb the Neumann boundary conditions by Borel positive measure concentrated on the boundary ∂Ω [15, 16, 90], we get the Dirichlet

3

form (aμ , D(aμ ) defined on L2 (Ω) by:    1 (Ω) ∩ L2 (∂Ω, dμ) ∇u∇vdx + u vdμ , ∀u, v ∈ D(aμ ) = H aμ (u, v) = Ω

∂Ω

(0.3) In the case of μ = β.σ (Ω bounded with Lipschitz boundary), (0.3) is the form associated with Laplacian with classical Robin boundary conditions and (0.2) gives the associated semigroup. In the case of an arbitrary domain Ω we make use of the theory of measure’s perturbation of Dirichlet forms (see e.g. [6, 7, 29, 46, 62, 81, 83, 84, 88, 89]). This thesis is organized as follows: In chapter 2, we will introduce the notion of Revuz correspondence adapted to our context. More specifically, we adapt the potential theory, and associated stochastic analysis to the situation where the regular Dirichlet form associated with general Robin boundary value problem is studied. The Revuz correspendance gives a link between smooth measures and additive functionals. It is in fact this notion who permits to build a bridge between the Dirichlet form describing general Robin boundary value problem and the associated probabilistic representation. First of all, we define the notion of relative capacity as a central tool in potential theory permitting to describe the phenomena occuring on the boundary of a general Euclidean domain. In the second section, we focus on the notion of smooth measures. All families of measures on ∂Ω in this section was originally defined on X, a locally compact separable metric space. We reproduce then the same definitions and the most of their properties on ∂Ω as we deal with measures concentrated on the boundary. There is three famillies of measures S0 , S00 and S. We put ∂Ω between brackets to recall our context, and we keep in mind that the same things are valid if we put Ω or Ω instead of ∂Ω. The classical reference in this chapiter is [46] and [15]. In chapter 3 and 4, we will concentrate on the main subject of this thesis: The probabilistic treatment of the general Robin boundary value problem. After passing in review the analytic approach develloped by W. Arend and M. Warma [15, 16, 90], we will focus on the probabilistc aspects. We first define a general reflecting Brownian motion, that is the diffusion process (Xt )t≥0 associated with the regular Dirichlet form (aμ , D(aμ )), we then apply a decomposition theorem of additive functionals to write Xt in the form Xt = x + Bt + Nt , we prove that the additive functional Nt is supported by ∂Ω, and we investigate when it is of bounded variations. We get then the probabilistic representation of the semigroup associated with (aμ , D(aμ )), 4

and we prove that it is sandwiched between the semigroup generated by the Laplacian with Dirichlet boundary conditions, and that of Neumann ones. In addition, we prove some convergence theorems, and we give a probabilistic interpretation of the phenomena occurring on the boundary. In the litterature, there is no study of the Robin boundary value problems in an arbitrary domain involving smooth measures on the boundary. There is two reasons for this: First, one need the reflecting Brownian motion X on Ω, which is defined to be the Hunt process associated with the Dirichlet form  ∇u∇vdx, ∀u, v ∈ H 1 (Ω) a(u, v) = Ω

1

The dirichlet form (a, H (Ω)) need not to be regular, and then nothing insure the existence of X. Moreover, one can not define, for a “bad” Ω, the capacity induced by (a, H 1 (Ω)), and then to be able to reproduce the theory of perturbation of regular Dirichlet forms [6, 7, 29, 46, 83, 84] in our special case. Throught [46] for example,the form (a, D(a)) is a regular Dirichlet form on L2 (X, m) , where X is a locally compact separable metric space, and m a positive Radon measure on X with supp[m] = X. For our purposes we take as in [16] X = Ω, where Ω is an open subset of Rd , and the measure m on the σ−algebra B(X) is given by m(A) = λ(A ∩ Ω) for all A ∈ B(X) with λ the Lebesgue measure, it follows that L2 (Ω) = L2 (X, B(X), m), and we define a regular Dirichlet form (a, D(a)) on L2 (Ω) by:  ∇u∇vdx , ∀u, v ∈ D(a) a(u, v) = Ω

 1 (Ω) is the closure of H 1 (Ω) ∩ Cc (Ω) in H 1 (Ω). The domain where D(a) = H 1  H (Ω) is so defined to insure the Dirichlet form (a, D(a)) to be regular. In the special case where Ω is bounded with Lipschitz boundary, we have  1 (Ω) = H 1 (Ω). H To deal with signed smooth measures case, one should define a specific Kato class adapted to the treatment of the Robin Laplacian in the same way as for Kato class defined in the context of perturbation of Dirichlet forms. One should keep always in mind that Robin boundary condition is in fact a perturbation of the Neumann boundary condition by a certain measure. In the litterature, the first who had defined such class of measures was V. G. Papanicolau [71]. His aim was to give the probabilistic solution of ∂ + β on the Schr¨odinger operator −Δ + V with Robin boundary condition ∂ν the boundary ∂Ω, where Ω is bounded domain with C 3 boundary, and ν the 5

outward unit normal vector on ∂Ω. The Borel function β belong to a specific Kato class Σ(∂Ω), which means that,  t  lim sup Ex (0.4) |β(Xs )|dLs = 0 t↓0

x∈Ω

0

where L is the boundary local time of the standard reflecting Brownian motion X on Ω. With the same smootness assymption on the domain as above, R. Song [76] worked with a generalized Kato class of measure on Ω to study in a probabilistic point of view, the third boundary value problems, semilinear and generalized mixed boundary value problems. Ramasubramanian in [73] remarks that, one can generalize the treatment in [71] to bounded Lipschitz domains. We consider then a perturbation on the boundary by signed smooth measure, we define then, for μ ∈ S(∂Ω) − S(∂Ω)   ∇u∇vdx + uvdμ, ∀u, v ∈ D(aμ ) aμ (u, v) = Ω

∂Ω

 1 (Ω) ∩ L2 (∂Ω, |μ|). where D(aμ ) = H More precisely, we define a particular Kato class of measures, adapted to our context, we give also some of its properties and its analytic description. We consider then, the Robin problem involving signed smooth measures. We will see that when μ ∈ S(∂Ω) − SK (∂Ω), the Dirichlet form (aμ , D(aμ )) is closed and the associated selfadjoint operator Δμ is a realization of the Laplacian on L2 (Ω). In the special case where |μ| is locally infinite on ∂Ω, then Δμ is the Laplacian with Dirichlet boundary conditions. Moreover, (aμ , D(aμ )) is regular if and only if |μ| is a Radon measure. We will also prove a domination theorem. It says that the semigroup (e−tΔμ )t≥0 is sandwitched between (e−tΔμ+ )t≥0 and (e−tΔ−μ− )t≥0 . We will see that the converse is also true. That means that if one have a semigroup (T (t))t≥0 sandwitched between (e−tΔμ+ )t≥0 and (e−tΔ−μ− )t≥0 , then T (t) = e−tΔν−μ− , where ν is a Radon measure charging no set of zero relative capacity. In chapter 5, we focus on three different topics. In the first topic, we want to characterize all semigroups sandwiched between Dirichlet and Neumann ones. In [15], the answer was given in the case where the form associated with the semigroup is local, but here, not only we give another method but also we prove that the locality is automatic for the form of any sandwiched semigroup. The result is obtained by using a general representation formula called Beurling-Deny and Lejan formula. It gives a decomposition of any 6

regular Dirichlet form on the sum of a strongly local form, local part and a nonlocal part. The second topic is about a general nonlocal Robin boundary value problem. More precisely, we let Ω be an open set of Rd , and we consider μ to be a Borel measure on ∂Ω, and θ a symmetric Radon measure on ∂Ω × ∂Ω\d where d indicates the diagonal of ∂Ω × ∂Ω. We define   1 E = {u ∈ H 1 (Ω) ∩ Cc (Ω) : |u|2 dμ + (u(x) − u(y))2 dθ < ∞} 2 ∂Ω×∂Ω\d ∂Ω and the bilinear symmetric form aμ,θ on L2 (Ω) by  aμ,θ (u, v) =

 Ω

∇u∇vdx +

uvdμ + ∂Ω

1 2

 (u(x) − u(y))(v(x) − v(y))dθ ∂Ω×∂Ω\d

We want to characterize the closability of the above form. To do this, we give a new notion of admissibility concerning the pair of measures (μ, θ), and then we prove that the form (aμ,θ , E) is closable if and only if the pair (μ, θ) is admissible. The operator Δμ,θ associated with the closure of the form (aμ,θ , E) is a realization of the Laplacian on L2 (Ω). Moreover, the semigroup (etΔμ,θ )t≥0 is not sandwiched between Dirichlet and Neumann semigroups. The third topic concerns another problem of the nonlocal Robin Laplacian involving bounded operators. Let B be a bounded operator on L2 (Γ), where Γ is the boundary of a bounded domain Ω of Rd with Lipschitz boundary. We consider the following bilinear form on L2 (Ω) with domain H 1 (Ω)   aB (u, v) = ∇u∇vdx + Buvdσ, u, v ∈ H 1 (Ω) Ω

Γ

1

We will prove that the form (aB , H (Ω)) is continuous and elliptic. The associated operator ΔB is called the nonlocal Robin Laplacian. We give a characterization of the positivity of the associated semigroup e−tΔB and we see what happens in the case of dimension one. In addition, we give a domination result. It says that the semigroup associated with the bilinear form (aB , H 1 (Ω)) is not always sandwiched between the semigroup associated with the Laplacian with Dirichlet boundary condition and the one with Neumann ones. In fact when B ≥ 0 then by a Zaanen’s characterization Theorem we obtain that B = mI, where 0 ≤ m ∈ L∞ (∂Ω) and then the sandwiched property is valid, but when B ≤ 0 then one have that e−tΔD ≤ etΔB but etΔB  e−tΔN except when B = 0.

7

Related Papers This thesis is based on the following papers: [1] K. Akhlil: Probabilistic Solution of the General Robin Boundary Problem on Arbitrary Domains; International Journal of Stochastic Analysis, Vol. 2012 (2012), [2] K. Akhlil: General Robin Boundary Value problem on arbitrary Domain: Stochastic Approach, Ulmer Seminare, Heft 18(2013), [4] K. Akhlil : The Laplacian with General Robin Boundary conditions involving Signed Measures. Submitted to Acta Mathematicae Applicatae Sinica, [3] K. Akhlil: Dirichlet and Neumann boundary conditions: What is in all between? (in press in Ulmer Seminare), [5] K. Akhlil: Generalized Nonlocal Robin Boundary Conditions on Arbitrary Domains.(In preparation)

8

9

Chapter 1 Preliminaries In this chapter, we focus on two subjects related by a theorem of Fukushima. It concerns Dirichlet forms and Hunt processes, and we know by the Fukushima theorem that for any regular Dirichlet form, there is a Hunt processes associated with it.This Chapter is based on the famous book [46]. For further developements on functional analysis, Dirichlet forms and Hunt processes we refer to [4], [12], [13], [17], [24]. [25], [30], [32], [53], [64], [70], [80] and [91].

1.1

Dirichlet Forms

Let H be a real Hilbert space with inner product (·, ·). A non negative definite symmetric bilinear form densely defined on H is henceforth called simply a symmetric form on H. To be precise, a is called a symmetric form on H if the following conditions are satisfied: 1. a is defined on D(a) × D(a) with values in R, D(a) being a dense linear subspace of H, 2. a(u, v) = a(v, u), a(u + αv, w) = a(u, w) + αa(v, w) and a(u, u) ≥ 0 where u, v, w ∈ D(a), α ∈ R. We call D(a) the domain of a. The inner product (·, ·) on H is a specific symmetric form defined on the whole space H. Given a symmetric form a on H  aα (u, v) = a(u, v) + α(u, v) u, v ∈ D(a) D(aα ) = D(a) defines a new symmetric form on H for each α > 0. Note that the space D(a) is then a pre-Hilbert space with inner product aα . Furthermore aα and 10

aβ determine equivalent metrics on D(a) for different α, β > 0. If D(a) is complete with respect to this metric, then a is said to be closed. In other words, a symmetric form a is said to be closed if un ∈ D(a), a1 (un − um , un − um ) → 0, n, m → ∞ ⇒ ∃u ∈ D(a) a1 (un − u, un − u) → 0, n → ∞

(1.1)

Cleary D(a) is then a real Hilbert space with inner product aα for each α > 0. We say that a symmetric form a is closable if the following condition is fulfilled : For un ∈ D(a), satisfaying a(un − um , un − um ) → 0 (un , un ) → 0 ⇒ a(un , un ) → 0

(1.2)

Given two symmetric form a1 and a2 , a2 is said to be an extension of a1 if D(a1 ) ⊂ D(a2 ) and a2 = a1 on D(a1 ) × D(a1 ). A necessary and sufficient condition for a symmetric form possess a closed extension is that the symmetric form is closable. We consider a σ−finite measure space (X, B, m) such that  X is locally compact separable metric space (1.3) m is a positive Radon measure on X such that supp[m] = X i.e. m is a non-negative Borel measure on X finite on compact sets and strictly positive on non-empty open sets. We take as a real Hilbert space H the L2 −space L2 (X, m) which is endowed with the inner product  u(x)v(x)m(dx), u, v ∈ L2 (X, m) (u, v) = X

We call a symmetric form on L2 (X, m) Markovian symmetric form if the following property hold u ∈ D(a), v = (0 ∨ u) ∧ 1 ⇒ v ∈ D(a), a(v, v) ≤ (u, u) A Dirichlet form is by definition a symmetric Markovian closed form on L2 (X, m). A core of a symmetric form is by definition a subset C of D(a) ∩ Cc (X) such that C is dense in D(a) with a1 −norm and dense in Cc (X) with uniform norm. a is called regular if a possess a core. It is clear that a is regular if and only if the space D(a) ∩ Cc (X) is a core of a. For a m−measurable function u, the support supp[u] of the measure u.dm is simply denoted supp[u]. If u ∈ C(X), then supp[u] is just the closure of 11

{x ∈ X : u(x) = 0}. We say that a symmetric form a posses the local property or simply a is local if for u, v ∈ D(a), such that supp[u] and supp[v] are disjoint compact sets then a(u, v) = 0. a will be said to be strong local if for u, v ∈ D(a), such that supp[u] and supp[v] are compact and v is constant on a neighbourhood of supp[u] then a(u, v) = 0. Let us consider an abstract real Hilbert space H with inner product (·, ·). Consider a family {T (t), t > 0} of linear operators on H satisfaying the following conditions: 1. Each T (t) is a symmetric operators with domain D(T (t)) = H, 2. Semigroup property: T (t)T (s) = T (t + s), t, s > 0, 3. Contraction property: (T (t)u, T (t)u) ≤ (u, u) t > 0, u ∈ H. Then {T (t), t > 0} is called a semigroup (of symmetric operators) on H. It is called strongly continuous if in addition : (T (t)u − u, T (t)u − u) → 0, t  0 with u ∈ H A resolvent on H is by definition a family {Gα , α > 0} of linear operators on H satisfaying the following conditions, 1. Each Gα is symmetric operator with domain D(Gα ) = H, 2. Resolvent equation : Gα − Gβ + (α − β)Gα Gβ , α, β > 0, 3. Contraction property: (αGα u, αGα u) ≤ (u, u), α > 0, u ∈ H, 4. (αGα u − u, αGα u − u) → 0 α → ∞ where u ∈ H. Given a strongly continous semigroup {T (t), t > 0} on H, the following Bochner integral  ∞ Gα u = T (t)udt 0

determines a strongly continous resolvent {Gα , α > 0} on H. This is called the resolvent of the given semigroup. The generator A of a strongly continous semigroup {T (t), t > 0} on H is defined by  Au = limt0 T (t)u−u t D(A) = {u ∈ H : Au exists as a strong limit}

12

Theorem 1.1.1. There is a one to one corespondence between the family of closed symmetric forms a on H, and the family of nonpositive definite selfadjoint operators A on H. The corespondence is determined by √  D(a) = D(√ −A) √ a(u, v) = ( −Au, −Au) This corespondence can be characterized also by  D(A) ⊂ D(a) a(u, v) = (−Au, v), u ∈ D(A), v ∈ D(a) From the discussion above, we find the operator A on H associated with a by the following procedure  D(A) = {u ∈ D(a) : ∃v ∈ H, a(u, ϕ) = (v, ϕ)H ∀ϕ ∈ D(a)} (1.4) Au =v Then A is selfadjoint and −A generates a contraction C0 −semigroup T = (T (t))t≥0 of symmetric operators on H. We also write e−tA = T (t) and call T the semigroup associated with a. Example 1.1.2. Let Ω be an Euclidean domain of Rd . Let a be the symmetric billinar form defined on L2 (Ω) by  ∇u(x)∇v(x)dx a(u, v) = Ω

1. Dirichlet boundary conditions: If we take D(a) = H01 (Ω) as the domain D(a) of a, then (a, D(a)) is a Regular Dirichlet form on L2 (Ω) and is associated with the Laplacian with Dirichlet boundary conditions noted ΔD , 2. Neumann boundary conditions: If we take D(a) = H 1 (Ω) as the domain D(a) of a, then (a, D(a)) is not, in general, a regular Dirichlet form on L2 (Ω). One should then suppose some smoothness assumptions on the domain Ω. In fact, if we suppose that Ω is a bounded domain with Lipschitz boundary, then the form (a, D(a)) is a regular Dirichlet form on L2 (Ω) and is associated with the Laplacian with Neumann boundary conditions noted ΔN . To preserve the generality on the domain Ω, one can “regularize” the  1 (Ω), the cloform (a, D(a)) in a certain way. We consider D(a) = H 1 1 1  sure of H (Ω) ∩ Cc (Ω) in H (Ω). The domain H (Ω) is so defined to insure the Dirichlet form (a, D(a)) to be regular. In this situation we say always that the form (a, D(a)) is associated with Laplacian with Neumann boundary conditions, noted also ΔN 13

3. Robin boundary conditions: Let Ω be a bounded domain on Rd with Lipschitz boundary. Define the symmetric bilinear form (aβ , D(aβ )) on L2 (Ω) by   aβ (u, v) = ∇u(x)∇v(x)dx + β(x)u(x)v(x)dσ Ω

∂Ω

where 0 ≤ β ∈ L∞ (∂Ω). The domain of aβ is D(aβ ) = H 1 (Ω) ∩ L2 (∂Ω, μ), where dμ = β.dσ (σ is the surface measure on ∂Ω). Considering the boundedness of β and the fact that each function u ∈ H 1 (Ω) 2(d−1) has a trace which is in L d−2 (∂Ω), which implies that D(aβ ) = H 1 (Ω). The operator associated with (aβ , H 1 (Ω)) is the Laplacian with Robin boundary condition, which is given by (0.1) with help of (1.4). To end this section we define the approximation form determined by a resolvent. Let (a, D(a)) be a symmetric closed form on L2 (X, m), and let {Gα , α > 0} the resolvent associated with a. To prove the association between a resolvent and a form, we use in general the following characterization: Gα (L2 (X, m)) ⊂ D(a) and if u ∈ L2 (X, m) and v ∈ D(a) aα (Gα u, v) = (u, v),

(1.5)

We define a symmetric form a(α) on L2 (X, m) by; for u, v ∈ L2 (X, m) we let a(α) (u, v) = α(u − αGα u, v) For any u ∈ L2 (X, m), a(α) (u, u) is decreasing as α  ∞ and  D(a) = {u ∈ L2 (X, m) : lim a(α) (u, u) < ∞} α→∞

a(u, v) = lim a(α) (u, v), u, v ∈ D(a)

(1.6)

(1.7)

α→∞

which justifies the saying that a(α) is approximating form determined by Gα . This approximating form will be used to construct a cetain measure in Chapter 4, with help of the following Lemma contained in [46, lemma 1.4.1] Lemma 1.1.3. If S is a positive symmetric linear operator on L2 (X, m), then there exists a unique positive Radon measure ν on the product space X × X satisfying the following property; for all u, v ∈ L2 (X, m),  (u, Sv) = u(x)v(x)dν (1.8) X×X

14

This Lemma is central in many situations: First of all, it is used to prove the sandwiched property in [16], we will use the same lemma and then the same procedure as in [16], to prove a sandwiched property in the signed measure case. In the other hand, this Lemma gives a link between our approach of sandwiched property in Section 5.1, and the one in [16]. In fact, the above Lemma is used also to prove the representaion formula of regular Dirichlet forms called Beurling-Deny and Lejan formula (see Section 5.1 for more details).

1.2

Hunt Process

Let (S, B) be a measurable space, and denote by B the smallest σ−field making things inside the brackets measurable and N the set of negligible sets. Let (Ω, M, Xt , P ) be a stochastic process with state space (S, B), i.e (S, B) is a measurable space, (Ω, M, P ) is a probability space, and Xt is a measurable map from Ω to S for each t. The last condition of measurability is explicitly indicated by Xt ∈ MB. We set 0 F∞ = σ{Xs , s < ∞},

Ft0 = σ{Xs , s ≤ t} We say that a family {Mt }t≥0 of sub−σ−fields of M is an admissible filtration if Mt is inceasing in t and Xt ∈ Mt B for each t ≥ 0. We may then call {Ft0 }t≥0 the minimum admissible filtration. An admissible filtration {Mt } is called right continous if Mt = Mt+ (= Mt ), ∀t ≥ 0 t >t

Adjoining an extra point Δ to a measurable space (S, B), we set SΔ = S ∪ Δ and BΔ = B ∩ {B ∪ Δ : B ∈ B}

 A quadruple M = Ω, M, {Xt }t∈[0,∞] , {Px }x∈SΔ is said to be a Markov process on (S, B) if the following conditions are satisfied: 

1. For each x ∈ SΔ , Ω, M, {Xt }t∈[0,∞] , {Px }x∈SΔ is a stochastic process with state space (S, B). X∞ (ω) = Δ ∀ω ∈ Ω. 2. Px (Xt ∈ E) is B−measurable in x ∈ S for each t ≥ 0 and E ∈ B. 15

3. There exists an admissible filtration {Mt }t≥0 such that Px (Xt+s ∈ E|Mt ) = PXt (Xs ∈ E),

Px − a.s.

(1.9)

for any x ∈ S, t, s ≥ 0 and E ∈ B. 4. PΔ (Xt = Δ) = 1, ∀t ≥ 0. The above condition requires the point Δ to play the role of the “cemetery”. If the additional condition 5. Px (X0 = x) = 1, x ∈ S is satisfied, then the Markov process M is called normal. The condition (1.9) is called the Markov property with respect to the admissible filtration {Mt }. As an immediate consequence of this property the function pt (x, E) defined by pt (x, E) = Px (Xt ∈ E),

x ∈ S, t ≥ 0, E ∈ B

is a Markovian transition function. We call this the transition function of the Markov process M . Its Laplace transform  ∞ Rα (x, E) = e−αt pt (x, E)dt 0

gives a markovian resolvent kernel, which is called the resolvent of the Markov process M . Given an admissible filtration {Mt }t≥0 , a [0, ∞]−valued function σ on Ω is called an {Mt }− stopping time if {σ ≤ t} ∈ Mt ,

∀t ≥ 0.

If {Mt }t≥0 is right continuous, then σ is an {Mt }−stopping time if and only if {σ < t} ∈ Mt , ∀t ≥ 0 because {σ ≤ t} = n≥0 {σ < 1 + n1 } ∈ Mt+ . We define a sub−σ−field Mσ by Mσ = {Λ ∈ M : Λ ∩ {σ ≤ t} ∈ Mt .∀t ≥ 0}. Let us consider a normal Markov process on (S, B) and assume the following additional conditions (H) concerning the pair (Ω, Xt ): (H.i) X∞ (ω) = Δ, ∀ω ∈ Ω, (H.ii) Xt (ω) = Δ, ∀t ≥ ξ(ω), where ξ(ω) = inf{t ≥ 0 : Xt (ω) = Δ}, (H.iii)for each t ∈ [0, ∞], there exists a map θt from Ω to Ω such that Xs ◦ θt = Xs+t , s ≥ 0 16

(H.iv) for each ω ∈ Ω, the sample path t → Xt (ω) is right continuous on [0, ∞[ and has the left limit on ]0, ∞[ (inside SΔ ). The θt and ξ in the preceding paragraph are called the translation operator and the life time, respectively.  Given a Markov process M we denote Pμ (Λ) = SΔ Px (Λ)μ(dx), where 0 0 μ ∈ P(SΔ ) and Λ ∈ F∞ , which defines a probability measure Pμ on (Ω, F∞ ). For an admissible filtration {Mt }, we can say that M is strong Markov with respect to {Mt } if {Mt } is right continuous and Pμ (Xσ+s ∈ E|Mσ ) = PXσ (Xs ∈ E), Pμ − a.s.,

(1.10)

μ ∈ P(SΔ ) the set of probability measures on SΔ ,E ∈ BΔ , s ≥ 0, for any {Mt }− stopping time σ. We say that M is quasi-left-continuous if for any {Mt }−stopping time σn increasing to σ Pμ ( lim Xσn = Xσ , σ < ∞) = Pμ (σ < ∞), n→∞

μ ∈ P(SΔ ),

A normal Markov process M = (Ω, M, Xt , Px ) on (S, B) satisfying the condition (H) is called a Hunt process if there exists an admissible filtration {Mt } such that M is strong Markov and quasi-left-continuous with respect to {Mt }. Notice that the right continuity of {Mt } is implied in the above definition of the Hunt process. The following Theorem is important in the way that it associates the world of Dirichlet forms and the one of stochastic processes(see [46] for more details) Theorem 1.2.1. There is a Hunt process associated which each regular Dirichlet form. We call a Hunt process a diffusion if Px (Xt is continous in t ∈ [0, ξ[) = 1 for every x ∈ X. The following theorem give the association betweet the local property of the Dirichlet form and the continuity of the sample paths of its associated Hunt process, the proof is contained in [46, Theorem 5.5.1]. Theorem 1.2.2. The following conditions are equivalent to each other: 1. The Dirichlet form (a, D(a)) pocesses the local property, 2. The associated Hunt process M is of continous sample paths for q.e. starting point, 17

3. there is a symmetric diffusion on X which is equivalent to M . Example 1.2.3. We give four classical Hunt processes: The Brownian motion on Rd and three of its classical subprocesses defined by their regular Dirichlet forms on L2 (Ω), where Ω is a domain on Rd which is bounded with Lipschitz boundary. 1. Brownian motion: The Brownian motion (Bt )t≥0 is the Hunt process on Rd associated with the regular Dirichlet form on L2 (Rd ) defined by  a(u, v) = ∇u(x)∇v(x)dx, D(a) = H 1 (Rd ) Rd

In addition, the form is local, then the associated Hunt process is in fact a diffusion process(i.e. a strong Markov process with continuous sample paths). The operator associated with the above form is the Laplacian in the whole Rd , and the associated semigroup is given by e−tΔ f (x) = E x [f (Bt )] ,

∀f ∈ L2 (Rd )

One can see easily some simple facts about this semigroup. For example, it is positive and Markovian. 2. Killed Brownian motion: Let (aD , D(aD ) be the regular Dirichlet form on L2 (Ω) defined in the first example in Example 1.1.2, then there is a Hunt process associated with it. In addition, the form is local, then the associated Hunt process is in fact a diffusion process, we call it “killed Brownian motion”. The word “killed” comes from the fact that the kernel of the process is submarkovian and then one should adjoint a “cemetery” state ∂ to Ω. The semigroup associated with the form (aD , D(aD ) is given by e−tΔD f (x) = E x [f (Bt )1t n, it follows that P2 u = Pp u ∈ C(Rd ) whenever u = ϕ|Ω for some ϕ ∈ D(Rd ).  1 (Ω) ∩ Cc (Ω) is dense in H  1 (Ω). Proposition 2.1.2. The space H Proof. Let ξ ∈ D(Rd ) such that ξ(x) = 1 for some |x| ≤ 1. Let ξn (x) = ξ( nx ). If u ∈ H 1 (Ω) ∩ C(Ω), then ξn u ∈ H 1 (Ω) ∩ Cc (Ω) and ξn u → u(n → ∞) in  1 (Ω) is the closure of H 1 (Ω) ∩ C(Ω) the claim follows. H 1 (Ω). Since H It follows that the Dirichlet form (a.D(a)) defined by (2.1) is regular ; i.e., D(a) ∩ Cc (Ω) is dense in (D(a), .a ), and also in (Cc (Ω), .∞ ) by the Stone-Weierstrass theorem. We define the relative capacity CapΩ of a subsets A of Ω by  1 (Ω), ∃O ⊂ Rd open CapΩ (A) := inf{||u||2H 1 (Ω) : u ∈ H such that A ⊂ O and u(x) ≥ 1 a.e on Ω ∩ O} (2.2) Here and further on, the word “relative” means with respect to the fixed Ω. The notion of relative capacity was introduced for the first time in [16], as a special case of the capacity associated with a regular Dirichlet form (a, D(a)) defined by (2.1)(see [46]). The relative capacity is an efficient tool to study the phenomenon occuring on the boundary ∂Ω of Ω. We say that a set N ⊂ Ω is relatively polar if CapΩ (N ) = 0. Proposition 2.1.3. We have the following: 1. CapΩ (∅) = 0, 23

2. If An is an arbitrary sequence of subsets of Ω, then     CapΩ An ≤ CapΩ (An ) n≥1

n≥1

However, CapΩ is not a Borel measure. Theorem 2.1.4. The relative capacity is a Choquet capacity; i.e. it has the following properties. 1. A ⊂ B ⇒ CapΩ (A) ≤ CapΩ (B), 2. If Kn is a decreasing sequence of compact subset of Ω, then   = inf CapΩ (Kn ), CapΩ n

n≥1

3. if An is an increasing sequence of arbitrary subsets of Ω, then :    An = supn CapΩ (An ) CapΩ n≥1

A statement depending on x ∈ A ⊂ Ω is said to hold relatively quasieverywhere (r.q.e.) on A, if there exist a relatively polar set N ⊂ A such that the statement is true for every x ∈ A \ N .  1 (Ω) as defined on Ω. We call a Now we may consider functions in H function u : Ω → R relatively quasi-continuous (r.q.c.) if for every > 0 there exists a relatively open set G ⊂ Ω such that CapΩ (G) < and u|Ω\G is continuous.  1 (Ω) there It follows form [30, 9, I, Proposition 8.2.1] that for each u ∈ H (x) = u(x) exists a relatively quasi-continous function u  : Ω → R such that u m−a.e. This function is unique relatively quasi-everywhere. We call u  the relatively quasi-continous representative of u. It can be seen form the proof of [30, 9, I, Proposition 8.2.1] that u  can be chosen Borel measurable. The following two results can be found in [46, Lemma 2.1.4] and [46, Theorem 2.1.4], Proposition 2.1.5. Let G be a relatively open set of Ω and u be relatively quasi-continous on G. If u ≥ 0 a.e. on G, then u ≥ 0 r.q.e. on G. Theorem 2.1.6. We have the following: 24

  1 (Ω)(the classe of relatively quasi-continous representation of 1. If un ∈ H  1 (Ω)) constitutes an a1 −Cauchy sequence, then there are elements of H   1 (Ω) such that un → u r.q.e. and (un ) a subsequence nk and a u ∈ H k is a1 −convergent to u,  1 (Ω) constitutes an a1 −Cauchy sequence and if some mod2. If (un ) ∈ H   1 (Ω) of un converges to a function u ifications u n ∈ H  r.q.e., then   1 (Ω) and (un ) is a1 −convergent to u u ∈H . For B ⊂ Ω, we let  1 (Ω) : u  ≥ 1 r.q.e on B} LB := {u ∈ H where u  denote a relatively quasi-continuous version of u. If B is relatively open, then :  1 (Ω) : u  ≥ 1 a.e on B} LB = {u ∈ H Proposition 2.1.7. Fix an arbitrary set B ⊂ Ω 1. If LB = ∅, then there exist a unique element eB ∈ LB minimizing the  1 (Ω) and eB satisfies Cap (B) = ||eB ||2 1 , norm of H Ω H (Ω)  1 (Ω) satisfying : 0 ≤ eB ≤ 1 a.e., eB = 1 2. eB is the unique element of H  1 (Ω) such that v ≥ 0 r.q.e. r.q.e on B and a1 (eB , v) ≥ 0 for all v ∈ H on B. It follows from the preceding Proposition that for every B ⊂ Ω, CapΩ (B) = inf{u2H 1 (Ω) : u ∈ LB } Proposition 2.1.8. Let K ⊂ Ω be compact set. Then CapΩ (K) = inf{u2H 1 (Ω) : u ∈ H 1 (Ω) ∩ Cc (Ω) : u(x) ≥ 1, ∀x ∈ K} We end this section with an important result contained in [15], it characterizes the space H01 (Ω) with help of relative capacity, Theorem 2.1.9. One has  1 (Ω) : u H01 (Ω) = {u ∈ H (x) = 0 r.q.e. on ∂Ω}

(2.3)

For further properties of relative capacity we refer to [16], [26] and [90]. 25

2.2

Smooth Measures

All families of measures on ∂Ω defined in this section, was originally defined on X [46], and then in our settings on X = Ω, as a special case. We reproduce the same definitions, and most of their properties on ∂Ω, as we deal with measures concentrated on the boundary of Ω for our approach to Robin boundary conditions involving measures. There is three families of measures, as we will see in the sequel, the family S0 , S00 and S. We put ∂Ω between brackets to recall our context, and we keep in mind that the same things are valid if we put Ω or Ω instead of ∂Ω. Let Ω ⊂ Rd be open and (a, D(a)) the regular Dirichlet form as defined in (2.1). A positive Radon measure μ on ∂Ω is said to be of finite energy integral if    1 (Ω) ∩ Cc (Ω) |v(x)|μ(dx) ≤ C a1 (v, v) , v ∈ H (2.4) ∂Ω

for some positive constant C. A positive Radon measure on ∂Ω is of finite energy integral if and only if there exists, for each α > 0, a unique function  1 (Ω) such that Uα μ ∈ H  aα (Uα μ, v) = v(x)μ(dx) (2.5) ∂Ω

We call Uα μ an α−potential. Let {Pt , t > 0} be the Markovian semi-group associated with the Dirichlet form (a, D(a)), then u ∈ L2 (Ω) is called α−excessive(with respect to {Pt , t > 0}) if u ≥ 0, e−αt Pt u ≤ u a.e., ∀t > 0 Theorem 2.2.1. The following conditions are equivalent to each other for  1 (Ω) and α > 0: u∈H 1. u is an α−potential, 2. u is α− excessive, 3. u ≥ 0, βGβ+α u ≤ u m-a.e., ∀β > 0, where Gβ is the resolvent of Pt ,  1 (Ω), v ≥ 0 a.e., 4. aα (u, v) ≥ 0, ∀v ∈ H  1 (Ω) ∩ Cc (Ω), v ≥ 0 m-a.e., 5. aα (u, v) ≥ 0, ∀v ∈ H 26

Proof. See [46] for the proof.  1 (Ω) are α−potentials, then so are u1 ∧ u2 Corollary 2.2.2. If u1 , u2 ∈ H and u1 ∧ 1. We denote by S0 (∂Ω), the family of all positive Radon measures of finite energy integral. Lemma 2.2.3. For μ ∈ S0 (∂Ω) and α > 0, let gn = n(Uα μ − nGn+α (Uα μ)),

n = 1, 2, ...

Then gn .m converges vaguely to μ and Gα gn converges aα −weakly to Uα μ. Proof. Applying Theorem 2.2.1 to u = Uα μ, we see that gn ≥ 0 a.e. Moreover, aα (Gα gn , v) = (gn , v) = n(u − nGn+α u, v)

(2.6)

 1 (Ω) when n → ∞, v ∈ H

−→ aα (u, v) In particular,  lim (gn , v) =

n→∞

v(x)μ(dx),

 1 (Ω) ∩ Cc (Ω). ∀v ∈ H

∂Ω

Lemma 2.2.4. Each measure in S0 (∂Ω) charges no set of zero relative capacity. Proof. It suffices to prove for μ ∈ S0 (∂Ω)   μ(G) ≤ a1 (U1 μ, U1 μ) CapΩ (G),

G⊂Ω

(2.7)

This follows from the preceding lemma with α = 1:  μ(G) ≤ lim inf G gn (x)dx n→∞

≤ lim (gn , eG ) n→∞

(2.8)

= a1 (U1 μ, eG )   ≤ a1 (U1 μ, U1 μ) a1 (eG , eG ), which equals the right-hand side of (2.7) in view of Proposition 2.1.7. 27

Theorem 2.2.5. For any μ ∈ S0 (∂Ω),   1 (Ω) ⊂ L1 (∂Ω, μ) H   1 (Ω) aα (Uα μ, v) = v˜μ(dx), α > 0, v ∈ H

(2.9) (2.10)

∂Ω

 1 (Ω). Here v˜ denotes any quasi-relative continuous modification of v ∈ H  1 (Ω) ∩ Cc (Ω) which  1 (Ω), we choose a sequence vn ∈ H Proof. For any v ∈ H is a1 −convergent to v. By Theorem 2.1.6 and Lemma 2.1.5, a subsequence nk exists such that vnk converges r.q.e on Ω to v˜, v˜ being any fixed relatively quasi continous modification of v. Since the inequality (2.4) holds for every vn , we have from the preceding lemma and Fatou’s lemma that   |˜ v (x) − vn (x)|μ(dx) = ∂Ω lim inf |vnk (x) − vn (x)|μ(dx) ∂Ω nk →∞

≤ C · lim inf nk →∞

 a(vnk (x) − vn (x), vnk (x) − vn (x))

(2.11) which implies that v˜ ∈ L1 (∂Ω, μ) and vn is L1 (∂Ω, μ)− convergent to v˜. By letting n tend to infinity in equation in (2.5) with v being replaced by vn , we arrive at (2.10). For μ ∈ S0 (∂Ω) we set aα (μ) = aα (Uα μ, Uα μ). We may now call this the α−energy integral of μ because we have from Theorem 2.2.5 

α μ(dx) aα (μ) = U ∂Ω

We now present several other consequences of Theorem 2.2.5. We shall first formulate the following maximum principle. Lemma 2.2.6. Consider the two α−potentials u1 = Uα μ and u2 = Uα ν, α > 0, μ, ν ∈ S0 (∂Ω). 1. If u˜1 ≤ u˜1 μ−a.e., then u1 ≤ u2 m−a.e. 2. If u˜1 ≤ c μ−a.e. for some constant c, then u1 ≤ c m−a.e.

28

Proof. (i) Set u = u1 ∧ u2 . Then by (2.10)  aα (u1 , u) = ∂Ω u˜(x)μ(dx)  = ∂Ω u1 (x)μ(dx)

(2.12)

= aα (u1 , u1 ) Since u is also an α−potential and u ≤ u2 m − a.e., we have aα (u − u1 , u − u1 ) = aα (u, u − u1 ) − aα (u1 , u − u1 ) ≤0

(2.13)

which proves that u1 = u ≤ u2 m − a.e. (ii) It suffices to set u2 = c. Let us consider a subset S00 (∂Ω) of S0 (∂Ω) defined by: S00 (∂Ω) = {μ ∈ S0 (∂Ω) : μ(∂Ω) < ∞, ||U1 μ||∞ < ∞} where  · ∞ denote the norm in L∞ (Ω, m). Lemma 2.2.6 leads us to Lemma 2.2.7. For any μ ∈ S0 (∂Ω), there exists an increasing sequence (Fn )n≥0 of compact sets of ∂Ω such that: 1Fn .μ ∈ S00 (∂Ω)

, n = 1, 2, ...

CapΩ (K \ Fn ) −→ 0, n → +∞ for any compact set K ⊂ ∂Ω  Proof. For μ ∈ S0 (∂Ω), take a relatively quasi continous modification U 1 μ(x) of U1 μ and consider an associated nest {Fn0 }. Let {En } be an increasing sequence of relatively compact open sets of ∂Ω such that E n ⊂ En+1 , En ↑ ∂Ω. If we let  Fn = {x ∈ Fn0 ∩ E n ; U 1 μ(x) ≤ n},

n = 1, 2...

then {Fn } is an increasing sequence of compact sets and, for any compact K,  lim CapΩ (K \ Fn ) ≤ lim [CapΩ (∂Ω \ Fn0 ) + CapΩ ({U 1 μ > n})],

n→∞

n→∞

which vanishes.

1 (1Fn ·μ ) ≤ U  Since U 1 μ ≤ n r.q.e. on Fn then U1 (1Fn · μ) ≤ n m-a.e. by vertue of Lemma 2.2.6.

29

We note that μ ∈ S0 (∂Ω) vanishes on ∂Ω \ ∪n Fn for the sets Fn of the Lemma 2.2.7, because of the Lemma 2.2.4.  1 (Ω) and Lemma 2.2.8. The following conditions are equivalent for u ∈ H a closed set F ⊂ ∂Ω. 1. u = Uα μ, μ ∈ S0 (∂Ω), with supp[μ] ⊂ F .  1 (Ω), v˜ ≥ 0 r.q.e. on F . 2. aα (u, v) ≥ 0, ∀v ∈ H  1 (Ω) ∩ Cc (Ω), v ≥ 0 r.q.e. on F . 3. aα (u, v) ≥ 0, ∀v ∈ H In the previous section, we have defined for any B ⊂ Ω witch CapΩ (B) <  1 (Ω) ∞ the 1−equilibrium potential eB characterized as a unique element of H satisfaying r.q.e. on B (2.14) e B = 1 a1 (eB , v) ≥ 0,

 1 (Ω), v˜ ≥ 0 r.q.e. on B ∀v ∈ H

(2.15)

We consider such B as subsets of ∂Ω. In view of the previous Lemma, eB = U1 νB with a unique νB ∈ S0 (∂Ω). We call νB the 1-equilibrium measure of B and supp[νB ] is contained in B ⊂ ∂Ω. In particular, we have for any compact set K ⊂ ∂Ω CapΩ (K) = a1 (eK , eK ) = νK (K)

(2.16)

Theorem 2.2.9. The following conditions are equivalent for a Borel set B ⊂ ∂Ω 1. CapΩ (B) = 0, 2. μ(B) = 0,

∀μ ∈ S0 ,

3. μ(B) = 0,

∀μ ∈ S00 .

Proof. See Theorem 2.2.3 of [46]. We now turn to a class of measures S(∂Ω) larger than S0 (∂Ω). Let us call a (positive) Borel measure μ on ∂Ω smooth if it satisfies the following conditions: - μ charges no set of zero relative capacity. - There exist an increasing sequence (Fn )n≥0 of closed sets of ∂Ω such that: (2.17) μ(Fn ) < ∞ , n = 1, 2, ... 30

lim CapΩ (K \ Fn ) = 0 for any compact K ⊂ ∂Ω

n→+∞

(2.18)

Let us note that μ then satisfies μ(∂Ω \ ∪n Fn ) = 0

(2.19)

An increasing sequence (Fn ) of closed sets satisfying condition (2.18) will be called a generalized nest, if further each Fn is compact, we call it a generalized compact nest. Given a smooth measure μ, we call (Fn ) satisfaying (2.17) and (2.18) a generalized nest associated with μ. We denote by S(∂Ω) the family of all smooth measures. The class S(∂Ω) is quiet large and it contains all positive Radon measure on ∂Ω charging no set of zero relative capacity. In particular S0 (∂Ω) ⊂ S(∂Ω). There exist also, by Theorem 1.1 [7] a smooth measure μ on ∂Ω ( hence singular with respect to m) ”nowhere Radon” in the sense that μ(G) = ∞ for all non-empty relatively open subset G of ∂Ω (See Example 1.6[7]). We claim that any measure in S(∂Ω) can be approximated by measures in S0 (∂Ω) and in S00 (∂Ω) as well. Lemma 2.2.10. Let ν be a bounded positive Borel measure on ∂Ω. If ν(A) ≤ c.CapΩ (A) for any Borel set A and for a positive constant c, then ν ∈ S0 (∂Ω)  1 (Ω) ∩ Cc (Ω) with a1 (v, v) = 1, we have Proof. For any non-negative v ∈ H   k+1 v(x)ν(dx) ≤ ν(∂Ω) + ∞ ν({x : 2k ≤ v(x) ≤ 2k+1 }) k=0 2 ∂Ω ∞ k+1 ≤ ν(∂Ω) + c k=0 2 CapΩ ({x : v(x) ≥ 2k }) (2.20)  k+1 −2k 2 ≤ ν(∂Ω) + c ∞ k=0 2 ≤ ν(∂Ω) + 4c which shows that ν ∈ S0 (∂Ω) Lemma 2.2.11. Let ν be a bounded positive Borel measure on ∂Ω charging no set of zero relative capacity. Then there exists a decreasing sequence {Gn } of open sets of ∂Ω such that CapΩ (Gn ) → 0,

ν(Gn ) → 0,

n→∞

ν(A) ≤ 2n CapΩ (A) for any Borel set A ⊂ ∂Ω \ Gn

31

Proof. Fix n and let α = inf{2n CapΩ (A) − ν(A); A is a Borel set of ∂Ω} Clearly −ν(∂Ω) ≤ α. If α < 0, shoose an open set B1 such that 2n CapΩ (B1 ) − ν(B1 ) ≤

α 2

Then α1 = inf{2n CapΩ (A) − ν(A); A ⊂ ∂Ω \ B1 } is not less than

α 2

because

α ≤ 2n CapΩ (A ∪ B1 ) − ν(A ∪ B1 ) ≤ {2n CapΩ (A) − ν(A)} + {2n CapΩ (B1 ) − ν(B1 )}

(2.21)

If α1 < 0, choose a relatively open set B2 ⊂ ∂Ω \ B1 such that 2n CapΩ (B2 ) − ν(B2 ) ≤

α1 2

Continuimg in this fashion we find open sets {B1 ∪ B2 ∪ · · · ∪ Bk } such that  n 2 CapΩ (B1 ∪ B2 ∪ · · · ∪ Bk ) − ν(B1 ∪ B2 ∪ · · · ∪ Bk ) ≤ 0 (2.22) 2n CapΩ (A) − ν(A) ≥ 2−k α, ∀A ⊂ ∂Ω \ (B1 ∪ B2 ∪ · · · ∪ Bk )  Let us put Gn = ∞ k=1 Bk . Then by (2.22), 2n CapΩ (A) ≥ ν(A) for any Borel set A of ∂Ω \ Gn Furthermore, 2n CapΩ (Gn ) ≤ ν(Gn ) ≤ ν(∂Ω). Then it suffices to set ∞  Gn = m=n Gm , because {Gn } is a decreasing sequence of open sets with CapΩ (Gn ) ≤ 2−n+1 ν(∂Ω) → 0,

n → ∞,

Since ν charges no set of zero relative capacity, 0 = ν(∩n Gn ) = lim ν(Gn ). n→∞

Theorem 2.2.12. The following conditions are equivalent for a positive Borel measure μ on ∂Ω: 1. μ ∈ S(∂Ω), 2. There exists a generalized nest (Fn ) satisfying (2.19) and 1Fn .μ ∈ S0 (∂Ω) for each n. 3. There exists a generalized compact nest (Fn ) satisfying (2.19) and 1Fn .μ ∈ S00 (∂Ω) for each n. 32

Proof. (3.)⇒(1.) is clear. The implication (1.)⇒(2.) follows from the preceding two lemmas. In fact, when μ is a bounded positive Borel measure charging no set of zero relative capacity, it is sufficient to adopt as Fn the complementary set of Gn of Lemma 2.2.11. For a general measure μ ∈ S(∂Ω), let {El } be an associated generalized nest. Since μl = 1El · μ is bounded for each l, μl admits a sequence {Fnl } with the properties of the statement of Theorem 2.2.12 (2.). We set, Fn =

n 

{El ∩ Fn(l) },

n = 1, 2, . . .

l=1

The {Fn } are then increasing closed sets such that 1Fn · μ ∈ S0 (∂Ω) for each n. (2.17) and (2.18) for {Fn } follow from the inclusion K \ Fn ⊂ (l) (K \ El ) ∪ (K \ Fn ) for any compact set K. We can prove the implication (2.)⇒(iii) by using Lemma 2.2.7 in a manner as in the preceding paragraph.

2.3

Revuz Correspondence

Now we turn our attention to the correspondence between smooth measures and additive functionals, known as Revuz correspondence. As the support of an additive functional is the quasi-support of its Revuz measure, we restrict our attention, as for smooth measures, to additive functionals supported by ∂Ω. Recall that as the Dirichlet form (a, D(a)) is regular, then there exists a Hunt process M = (Ω, Xt , ξ, Px ) on Ω which is symmetric and associated with it. Moreover, (a, D(a) is local, then M is in fact, a diffusion process on Ω. Definition 2.3.1. A function A : [0, +∞[×Ω → [−∞, +∞] is said to be an Additive functional (AF) if: 1. At is Ft −measurable, 2. There exist a defining set Λ ∈ F∞ and an exceptional set N ⊂ Ω with CapΩ (N ) = 0 such that Px (Λ) = 1, ∀x ∈ Ω \ N , θt Λ ⊂ Λ, ∀t > 0; ∀ω ∈ Λ, A0 (ω) = 0; |At (ω)| < ∞ for t < ξ, 3. A. (ω) is right continuous and has left limit, and At+s (ω) = At (ω) + As (θt ω) s, t ≥ 0

33

An additive functional is called positive continuous (PCAF) if, in addition, At (ω) is nonnegative and continuous for each ω ∈ Λ. The set of all PCAF’s on Ω is denoted A+ c (Ω). In this thesis we need just PCAFs supported by ∂Ω. In fact, we know from [46, Theorem 5.1.5] that the support of a PCAF is a quasi support of its Revuz measure, we restrict then ouerselves to Additive functionals supported by ∂Ω as our measures are also supported by ∂Ω. We note then, A+ c (∂Ω) the set of all PCAF’s on ∂Ω. Two additive functionals A1 and A2 are said to be equivalent if for each t > 0, Px (A1t = A2t ) = 1 r.q.e x ∈ Ω. We say that A ∈ A+ c (∂Ω) and μ ∈ S(∂Ω) are in the Revuz correspondence, if they satisfy, for all γ−excessive function h, and f ∈ B+ (Ω), the relation:  t   1 f (Xs )dAs = h(x)(f.μ)(dx) lim Eh.m t0 t ∂Ω 0 The family of all equivalence classes of A+ c (∂Ω) and the family S(∂Ω) are in one to one correspondence under the Revuz correspondence. In this case, μ ∈ S(∂Ω) is called the Revuz measure of A. Theorem 2.3.1. The family of all equivalent classes of A+ c (∂Ω) and the family S(∂Ω) are in one to one correspondence under the Revuz correspondence. Proof. For the proof see [46, Theorem 5.1.4] We set:  UAα f (x) = Ex [

0 ∞

 RαA f (x) = Ex [

∞

0



Rα f (x) = Ex [

e−αt f (Xt )dAt ]

e−αt e−At f (Xt )dt] ∞

0

e−αt f (Xt )dt]

The following Theorem give equivalent characterizations of the Revuz correspondence( for more details see [46, Theorem 5.1.3]) Theorem 2.3.2. The following conditions are equivalent to each other 1. A ∈ A+ c (∂Ω) and μ ∈ S(∂Ω) are in the Revuz correspondence,

34

2. for any γ−excessive function h (γ ≥ 0) and f ∈ B + α(h, UAα+γ f )  f.μ, h,

α ∞

3. for any t > 0, f, h ∈ B +  Eh.m [(f A)t ] =

t 0

f.μ, ps hds

4. for α > 0, f, h ∈ B + (h, UAα f ) = f.μ, RαA h Example 2.3.3. We suppose Ω to be bounded with Lipschitz boundary. We have [71]:  t   1 1 lim Eh.m f (Xs )dLs = h(x)f (x)σ(dx) t0 t 2 ∂Ω 0 where Lt is the boundary local time of the reflecting Brownian motion on Ω. It follows that 21 σ is the Revuz measure of Lt . In the following we give some facts useful in the proofs of our results in Chapter 4, Proposition 2.3.4. Let μ ∈ S0 (∂Ω) and A ∈ A+ c (∂Ω) the corresponding PCAF. For α > 0, f ∈ Bb+ , UAα is a relatively quasi-continuous version of Uα (f.μ). + A Proposition 2.3.5. Let A ∈ A+ c (∂Ω), and f ∈ Bb , then Rα is relatively quasi-continuous and

RαA f − Rα f + UAα RαA f = 0 In general, the support of an AF A is defined by supp[A] = {x ∈ X \ N : Px (R = 0) = 1} where R(ω) = inf{t > 0 : At (ω) = 0} Theorem 2.3.6. The support of A ∈ A+ c (∂Ω) is the relative quasi-support of its Revuz measure.

35

In the following we give a well known theorem of decomposition of additive functionals of finite energy. We will apply it to get a decomposition of the diffusion process associated with (a, D(a)).  1 (Ω), the AF A[u] = u (Xt ) − u (X0 ) can be Theorem 2.3.7. For any u ∈ H expressed uniquely as (X0 ) = M [u] + N [u] u (Xt ) − u [u]

(2.23) [u]

where Mt is a martingale additive functional of finite energy and Nt continuous additive functional of zero energy.

is a

 1 (Ω), if σ(u) is the comA set σ(u) is called the (0)−spectrum of u ∈ H plement of the largest open set G ⊂ Ω such that a(u, v) vanishes for any  1 (Ω) ∩ Cc (X) with supp[v] ⊂ G. The following Theorem means that : v∈H  1 (Ω), supp[N [u] ] ⊂ σ(u), ∀u ∈ H  1 (Ω), the CAF N [u] vanishes on the comTheorem 2.3.8. For any u ∈ H plement of the spectrum F = σ(u) of u in the following sense: [u]

Px (Nt = 0 : ∀t < σF ) = 1 r.q.e x ∈ X The decomposition theorem will be used in two situations: • In the next section, to decompose the reflecting Brownian motion associated with the Laplacian with Neumann boundary condition on bounded with Lipschitz boundary domains. This decomposition is well known as Skorohod decomposition, and established by the above decomposition. This was done by R.F. Bass and E.P. Hsu in [21]. • In Chapter 3, and in a more general framework of arbitrary domains we will use the same decomposition with the same procedure as in [21] to prove a type of Skorohod decomposition of the form Xt = x + Bt + Nt , but in this case Xt is not semimartingal in general and then we can not write Nt as in the Skorohod decompostion, but it is proved that when the domain Ω is a Caccioppoli set we refind the same decomposition as the Skorohod one.

2.4

Application

Here we give an application of the decomposition Theorem used by Bass and Hsu to construct the RBM in a bounded Lipschitz domain Ω ⊂ Rn . 36

Theorem 2.4.1. Suppose that Ω is bounded Lipschitz domain. Then reflecting Brownian motion X is a continuous Ω−valued semimartingale, and the Skorohod equation  1 t X(t) = X(0) + W (t) + ν(X(s))dLs (2.24) 2 0 holds, where W is a standard d−dimensional Brownian motion, L is the boundary local time (continuous additive functional) associated with the surface measure σ on ∂Ω, and ν is the inward unite normal vector field on the boundary The inward pointing normal vector is only defined a.e. (with respect to surface measure). However, the continuous additive functional L is associated with σ and so does not charge the null set. Hence, the integral in the statement of the Theorem is unamniguously defined. Proof. A reflecting Brownian motion is the Hunt process associated with the following regular Dirichlet form.  1 a(u, v) = ∇u(x) · ∇v(x)dx, D(a) = H 1 (D) (2.25) 2 Ω Suppose f ∈ H 1 (Ω) ∩ C(Ω). According to [] the continuous additive functional f (Xt ) − f (X0 ) can be decomposed as follows f (Xt ) − f (X0 ) = Mtf + Ntf

(2.26)

where M f is a martingale additive functional of finite energy and N f is a continuous additive functional of zero energy. Since (a, F) is local then X has a continuous sample paths, in addition f is assumed to be continuous on Ω, then M f is a continuous martingale whose quadratic variation process is given by  < M f , M f >t =

t

0

|∇f |2 (Xs )ds

(2.27)

If we further assume that f ∈ C 2 (Ω), then ... N f is of bounded variation and its associated measure μ is uniquely characerized by the relation   1 ∇f (x) · ∇v(x)dx = v˜(x)μf (dx), ∀v ∈ H 1 (Ω) (2.28) 2 Ω Ω v˜ is a quasi-continuous modification of v. Since Ω is Lipschitz, we can use Green’s identity in the equation (2.28). This allows us to identify the associated measure of the boundary Ω. 37

Now we apply the above discussion to the coordinate functions fi (x) = xi We have 1 (2.29) X t = x + M t + Nt 2 where M = (M f1 , M f3 , ..., M fd ) and N = (N f1 , N f3 , ..., N fd ). It remains to show that M is a Brownian motion, we use L´evy’s criterion. Namly, we need to verify that i, j = 1, ..., d (2.30) < M fi , M fj >= δij , This follow immediatly from (2.27): Therefore M is a Brownian motion. Let ν(x) = (ν 1 (x), ν 2 (x), ..., ν d (x)) be the components of the normal vector ν. From (2.28), the measure associated with the continuous additive functional N fi is ν i (x)σ(x). Let Lt =

d  

t 0

i=1

ν i (x)dNsfi

(2.31)

d i2 It follows that the measure associated with L is i=1 ν σ(dx) = σ(dx). This show that L is just the boundary local time with respect to the surface measure. Since the measure for N fi is ν i σ(dx), we have  t ν i (Xs )dLs i = 1, 2, ..., d Ntfi = 0



Hence we obtain Nt =

t 0

ν(Xs )dLs

and the proof of the Skorohod equation is complete. For general domains, the reflecting Brownian motion may not be a continuous process on the Euclidian closure of the domain Ω. It is continuous on special compactification of Ω, the so called Kuramochi compactification. In [21], it is shown that if Ω is bounded Liptschitz domain, then the Kuramoshi compactification of Ω is the same as the Euclidian compactification. Thus for such domain, the RBM does live on the set Ω. This chapter will be the central tool to prove our main results in Chapter 3. More precisely the adaptation of the potential theory and the Revuz corespondence to measures and additive functionals supported by the boundary of Ω, will be the central point to deal with general Robin boundary value problems on arbitrary domains and involving smooth measures. The advantage of the Stochastic approach is that it gives a probabilistic representation of the semigroup and a generalisation to a smooth measures instead 38

of only positive Radon measures charging no set of zero relative capacity. This treatement have as result two papers. The first [1] (see also [2]) deals with positive smooth measures, and the second [3] deals with signed smooth measures on the boundary.

39

40

Chapter 3 Stochastic Approach to General Robin Boundary Value Problems In this Chapter, we will deal with our aim results concerning a probabilistic approach to the Laplacian with general Robin boundary condition on arbitrary domains and involving measures. After exposing a review of the analytic approach, we will define a general reflecting Brownian motion associated with the form (a, D(a)). This diffusion process will be decomposed in the same way as in Section 2.4 on the sum of the Brownian motion and a process located on the boundary, which is sufficient for our approach. In the seconde step, we will develop our stochastic approach inspired by the one of [6] and [7], see also [46], [8], [88] and [89].

3.1

Review of Analytic Approach

Let Ω be an open set of Rd and μ a positive Borel measure on ∂Ω. Let  |u|2 dμ < ∞} E := {u ∈ H 1 (Ω) ∩ Cc (Ω) : ∂Ω

Define the billinear symmetric form aμ with domain E on L2 (Ω) by   aμ (u, v) = ∇u∇vdx + uvdμ Ω

∂Ω

The question asked in [90, 15] is when (aμ , E) is closable? For example, if μ is locally infinite everywhere on ∂Ω, then (aμ , E) is always closable, but if μ is locally finite on ∂Ω, or locally finite only on a part Γ ⊂ ∂Ω, 41

then (aμ , E) can be not closable. However, and since (aμ , E) is a symmetric bilinear positive form, there exists( By a Reed-Simon theorem [75]) a largest closable form ((aμ )r , E) which is smaller than (aμ , E). Note its domain by V . Since this closed form ((aμ )r , V ) is symmetric and densely defined, we can then, associate with it a self-adjoint operator Δμ on L2 (Ω) with domain 1

D(Δμ2 ) = V . We can see that we always have the following situations: 1. if μ is locally infinite everywhere, then (aμ , E) is closable always on L2 (Ω), 2. if μ is Radon measure, then (aμ , E) is closable on L2 (Ω) if and only if, μ does not charges relatively polar Borel subsets. It follows form (1) and (2) that if μ is locally finite on a part of ∂Ω, then (aμ , E) is closable on L2 (Ω) if and only if the restriction of μ to the part on which it is locally finite does not charges relatively polar Borel subsets of this part. Firstly, we give the following propositions which say that the Dirichlet boundary condition is included in Robin boundary condition by considering infinite valued measures, Proposition 3.1.1. Let μ be a Borel measure on ∂Ω and assume that μ is locally infinite everywhere on ∂Ω; i.e., ∀x ∈ ∂Ω, and, r > 0 : μ(B(x, r) ∩ ∂Ω) = ∞ Then the form aμ is closable and its closure which we denote by a∞ is given by:  a∞ (u, v) = ∇u∇vdx Ω

with domain H01 (Ω). Next, let the bilinear form which we denote by ((aμ )r , V ) be defied by   ∇u∇vdx + uvdμ (aμ )r (u, v) = Ω

S

Following Daners approach [41] for the more general case involving Borel measures, Arendt and Warma proved that ((aμ )r , V ) is closed on L2 (Ω). We will denote by Δμ the selfadjoint operator on L2 (Ω) associated with ((aμ )r , V ); i.e.,  D(Δμ ) := {u ∈ V : ∃v ∈ L2 (Ω) : (aμ )r (u, φ) = (v, φ)∀φ ∈ V } Δμ u := −v. 42

Since for each u ∈ D(Δμ ) we have    uφdμ = vφdx ∇u∇φdx = Ω

Ω

S

for φ ∈ V , if we choose φ ∈ D(Ω), the above equality can be written < −Δu, φ >=< v, φ > where denotes the duality between D(Ω) and D(Ω). Since φ ∈ D(Ω) is arbitrary, it follows that −Δu = v

in D(Ω)

Thus Δμ is a realization of the Laplacian on L2 (Ω). Next one can set, M0 = {μ : Borel measures on ∂Ω : CapΩ (N ) = 0 ⇒ μ(N ) = 0, ∀N ∈ B(∂Ω)} Theorem 3.1.2. Let μ be a Radon measure on ∂Ω. Then the following assertions are equivalent. 1. The form (aμ , E) is closable on L2 (Ω), 2. μ ∈ M0 Corollary 3.1.3. Let μ be a Borel measure on ∂Ω and let Γ := {z ∈ ∂Ω : ∃r > 0/μ(B(z, r) ∩ ∂Ω < ∞} be a relatively open subset of ∂Ω on which μ is locally finite. Then the following assertions are equivalent. 1. The form (aμ , E) is closable on L2 (Ω), 2. The measure μ does not charge relatively polar Borel subsets of Γ. It follows from the preceding corollary that the class M0 can be defined M0 = {μ : Borel measures on ∂Ω : CapΩ (N ) = 0 ⇒ μ(N ) = 0, ∀N ∈ B(Γ)} where Γ denotes the relatively open set of ∂Ω on which μ is locally finite. Let μ be a Borel measure on ∂Ω in M0 . By definition, the domain V of the closure of the form (aμ , E) is the completion of E with respect to the aμ −norm. The following result gives a characterization of V. 43

Proposition 3.1.4. Let μ be a Borel measure on ∂Ω in M0 . Then  1 (Ω) : V = {u ∈ H

u  ∈ L2 (∂Ω, μ)}

 aμ (u, v) =

 Ω

∇u∇vdx +

u vdμ ∂Ω

where u  is the relatively quasi-continuous version of u If μ is a Radon measure on ∂Ω, we can use the Riesz decomposition of measures to reconstruct the closure part of (aμ , E). So, there exist a unique pair (μr , μs ) of measures on (∂Ω, B(Ω)) such that: 1. μ = μr + μs , 2. μr (A) = 0 ∀A ∈ B(∂Ω) with CapΩ (A) = 0, 3. μs = χN μ for some N ∈ B(∂Ω), with CapΩ (N ) = 0. Definition 3.1.1. We call the measure μr the regular part of μ with respect to the relative capacity. Remark 3.1.5. If μ is not a Radon measure on ∂Ω, since its restriction to the part Γ on which it is locally finite is a Radon measure, we can also decompose μ|Γ = μr + μs . For simplicity, we assume Γ = ∂Ω. Proposition 3.1.6. Let μ be a Radon measure on ∂Ω and μ = μr + μs be its Riez decompostion. Then the closure (aμr , V ) is the closable part of (aμ , E) is given by   aμr (u, v) = ∇u∇vdx + u vdμr Ω

∂Ω

 =

Ω

with domain



u vdμ

∇u∇vdx + S

1 (Ω) : V = {u ∈ H

u  ∈ L2 (∂Ω, μr )}

where S = ∂Ω \ N Let μ ∈ M0 , and Δμ the self-adjoint operator on L2 (Ω) associated with the closure of (aμ , E). Δμ generates a holomorphic C0 −semigroup Tμ = (e−tΔμ )t≥0 on L2 (Ω).

44

3.2

Positive Smooth Measures Case

The aim of this section is to give the probabilistic representation of the semigroup generated by the Lapalacian with general Robin boundary conditions, which is, actually, a measure perturbation on the boundary of the derivative on the inward normal vector. We start with the regular Dirichlet form (a, D(a)) on L2 (Ω) 

1 (Ω) a(u, v) = ∇u∇vdx , D(a) = H (3.1) Ω

which we call always as the Dirichlet form associated to Laplacian with Neumann boundary conditions. The notation (a, D(a)) will denote always the same Dirichlet form in what remains of this thesis.

3.2.1

General Reflecting Brownian Motion

We concentrate then our attention on the process associated with the regular Dirichlet form (a, D(a)) on L2 (Ω) defined by (3.1). Due to the Theorem of Fukushima [45], there is a Hunt process (Xt )t≥0 associated with it. In addition, (a, D(a)) is local, thus the Hunt process is in fact a diffusion process (i.e. A strong Markov process with continuous sample paths). The diffusion process M = (Xt , Px ) on Ω is associated with the the form a in the sense that the transition semigroup pt f (x) = Ex [f (Xt )], x ∈ Ω is a version of the L2 −semigroup Pt f generated by a for any nonnegative L2 −function f . M is unique up to set of zero relative capacity. Definition 3.2.1. We call the diffusion process on Ω associated with (a, D(a)) the General reflecting Brownian motion. The process Xt is so named to recall the standard reflecting Brownian motion in the case of bounded smooth Ω, as the process associated with (a, H 1 (Ω)). Indeed, when Ω is bounded with Lipschitz boundary we have  1 (Ω) = H 1 (Ω), and by [21] the reflecting Brownian motion Xt admits that H the following Skorohod representation:  1 t X t = x + Wt + ν(Xs )dLs , (3.2) 2 0 where W is a standard N −dimensional Brownian motion, L is the boundary local(continuous additive functional) associated with surface measure σ on ∂Ω, and ν is the inward unit normal vector field on the boundary. For a general domains, the form (a, H 1 (Ω)) need not to be regular. 45

Fukuchima in [44] constructed the reflecting brownian motion on a special compactification of Ω, the so called Kuramuchi compactification. In [21] it is shown that if Ω is a bounded Lipschitz domain, then the Kuramochi compactification of Ω is the same as Euclidean Compactification. Thus for such domains, the reflecting Brownian motion is a continuous process who does live on the set Ω. Now,we apply a general decomposition theorem of additive functionals to our process M , in the same way as in [21]. The continuous additive functional (X0 ) can then be decomposed as follows: u (Xt ) − u [u]

[u]

u (Xt ) − u (X0 ) = Mt + Nt [u]

[u]

where Mt is a martingale additive functional of finite energy and Nt is a continuous additive functional of zero energy. [u] Since (Xt )t≥0 has continuous sample paths, Mt is a continuous martingale whose quadratic variation process is:  t < M [u] , M [u] >t = |∇u|2 (Xs )ds (3.3) 0

Instead of u we take coordinate function φi (x) = xi . We have X t = X 0 + M t + Nt We claim that Mt is a Brownian motion with respect to the filtration of Xt . To see that, we use L´evys criterion. This follows immediately from (3.3), which became in the case of coordinate function: < M [φi ] , M [φi ] >= δij t Now we turn our attention to the additive functional Nt . Two natural questions need to be answered. The first is, where is the support of Nt located, and the second concern the boundedness of its total variation. For the first question we claim the following: Proposition 3.2.1. The additive functional Nt is supported by ∂Ω. Proof. Following Theorem 2.3.8, we have that supp[Nt ] ⊂ σ(φ), where σ(φ) is the (0)−spectrum of φ, which means the complement of the largest open set G such that a(φi , v) = 0 for all v ∈ D(a) ∩ Cc (Ω) with supp[v] ⊂ G. First case: If Ω is smooth( Bounded with Lipschitz boundary, for example), then we have:  v.ni dσ a(φi , v) = − ∂Ω

46

Then, a(φi , v) = 0 for all v ∈ D(a) ∩ Cc (Ω) with supp[v] ⊂ Ω. We can then see that the largest G is Ω. Consequently σ(φ) = Ω \ Ω, and then σ(φ) = ∂Ω. Second case: If Ω isarbitrary, then we take an increasing sequence of subset of Ω such that ∞ n=0 Ωn = Ω. Define the family of Dirichlet forms (aΩn , D(a)Ωn ) to be the parts of the form (a, D(a)) on each Ωn as defined in section 4.4 of [46]. By Theorem 4.4.5 in the same section, we have that D(a)Ωn ⊂ D(a) and aΩn = a on D(a)Ωn × D(a)Ωn . We have that Ωn is the largest open set such that aΩn (φi , v) = 0 for all v ∈ D(a)Ωn ∩ Cc (Ωn ). By limit, we get the result. The interest of the question of boundedness of total variation of Nt appears when one need to study the semimartingale property and the Skorohod equation of the process Xt . Let |N | be the total variation of Nt , i.e., |N |t = supp

n−1 

|Nti − Nti−1 |.

i=1

where the supremum is taken over all finite partition 0 = t0 < t1 < ... < tn = t, and |.| denote the Euclidian distance. If |N | is bounded, then we have the following expression:  t νs d|N |s Nt = 0

where ν is a process such that |ν|s = 1 for |N |−almost all s. According to §5.4. in [46], we get the following result: Theorem 3.2.2. Assume that Ω is bounded, and that the following inequality is satisfied:    ∂v  1  dx (3.4)  ∂xi  ≤ C||v||∞ , ∀v ∈ H (Ω) ∩ Cb (Ω) Ω for some constant C. Then, Nt is of bounded variation. A bounded set verifying (3.4) is called strong Caccioppoli set. This notion is introduced in [37], and is a purely measure theoretic notion. An example of this type of sets are bounded sets with Lipschitz boundary. Theorem 3.2.3. If Ω is a Caccioppoli set, then there exist a finite signed smooth measure ν such that:   ∂v  1 (Ω) ∩ Cb (Ω). dx = − vdμ , ∀v ∈ H (3.5) Ω ∂xi ∂Ω 47

and ν = ν 1 − ν 2 is associated with the CAF −Nt = −A1t + A2t with the Revuz correspondence. Consequently ν charges no set of zero relative capacity. To get a Skorohod type representation, we set: ν=

N 

|μi | (3.6)

i=1

dμi φi = dν

i = 1, ..., N

We define the measure σ on ∂Ω by:  σ(dx) = 2

N 

 12 |φi (x)|

2

ν(dx)

(3.7)

i=1

and the vector of length 1 at x ∈ ∂Ω by: ⎧ ⎨  φi (x) if 1 ( Ni=1 |φi (x)|2 ) 2 ni (x) = ⎩ 0 if

N i=1

|φi (x)|2 > 0;

i=1

|φi (x)|2 = 0

N

Thus, μi (dx) = 12 ni (x)σ(dx) , i = 1, .., N . Then  t Nt = n(Xs )dLs 0

where L is the PCAF associated with 12 σ. Theorem 3.2.4. If Ω is a Caccioppoli set, then for r.q.e x ∈ Ω, we have:  t Xt = x + B t + n(Xs )dLs . 0

where B is an N −dimensional Brownian motion, and L is a PCAF associated by the Revuz correspondence to the measure 21 σ.

3.2.2

Probabilistic Representation

This section is concerned with the probabilistic representation of the semigroup generated by the Lapalacian with general Robin boundary conditions, which is, actually, obtained by perturbing the Neumann boundary conditions by a measure. We start with the Regular Dirichlet form defined by (3.1), which we call always as the Dirichlet form associated with the Laplacian with Neumann boundary conditions. 48

Let μ be a positive Radon measure on ∂Ω charging no set of zeo relative capacity. Consider the perturbed Dirichlet form (aμ , Fμ ) on L2 (Ω) defined by: Fμ = D(a) ∩ L2 (∂Ω, μ)  aμ (u, v) = a(u, v) + uvdμ u, v ∈ Fμ ∂Ω

We shall see in the following theorem that the transition function: μ

Ptμ f (x) = Ex [f (Xt )e−At ] is associated with (aμ , Fμ ), where Aμt is the positive additive functional whose Revuz measure is μ, note that the support of the AF is the same as the relative quasi-support of its Revuz measure. Proposition 3.2.5. Ptμ is a strongly continuous semigroup on L2 (Ω). In addition it is sub-markovian i.e. Ptμ ≥ 0 for all t ≥ 0, and Ptμ f ∞ ≤ f ∞

(t ≥ 0)

Proof. We have μ Pt+s f (x) = Ex [f (Xs+t )exp(−Aμt+s )]

= Ex [f (Xs+t )exp(−Aμt − Aμs .θt )] = Ex [f (Xs+t )exp(−Aμs .θt )exp(−Aμt )] = Ex [EXt [f (Xs )exp(−Aμs )]exp(−Aμt )]

(3.8)

= Ex [Psμ f (Xt )exp(−Aμt )] = Ptμ (Psμ f ) (x) Also P0μ f = f , it remains to prove that Ptμ is strongly continunous on L2 (Ω) We have   |Ptμ f (x)|2 ≤ Ex |f |2 (Xt ) .Ex [exp(−2Aμt )] (3.9)   ≤ Ex |f |2 (Xt ) And then Ptμ 2,2 ≤ 1. In addition we have by (3.9): lim sup Ptμ 2,2 ≤ f 2 t0

49

And by Dominated Convergence Theorem, limt0 < Ptμ f, f >= f 22 . Then, the limit of the relation, Ptμ f − f 22 = Ptμ 22 + f 22 − 2 < Ptμ f, f > gives limt0 Ptμ f − f 22 ≤ 0. Thus, (Ptμ )t≥0 ) is a strongly continuous semigroup on L2 (Ω). It is clear that if f ∈ L2 (Ω)+ , then Ptμ f ≥ 0 for all t ≥ 0. In addition we have: |Ptμ f (x)| ≤ Ex [|f |(Xt )], and then Ptμ f ∞ ≤ f ∞ (t ≥ 0) Now we give our central theorem, which traduces the fact that Ptμ is the strongly continuous semigroup associated with the Dirichlet form (aμ , Fμ ) on L2 (Ω). Theorem 3.2.6. Let μ be a positive Radon measure on ∂Ω charging no set of zero relative capacity and (Aμt )t≥0 be its associated PCAF of (Xt )t≥0 . Then Ptμ is the strongly continuous semigroup associated with the Dirichlet form (aμ , Fμ ) on L2 (Ω). Proof. To prove that Ptμ is associated with the Dirichlet form (aμ , Fμ ) on L2 (Ω) it suffices to prove the assertion RαA f ∈ Fμ

, (aμ )α (RαA , u) = (f, u) , f ∈ L2 (Ω, m), u ∈ Fμ

(3.10)

Since

1 f L2 (Ω) , α we need to prove (3.10) only for bounded f ∈ L2 (Ω). We first prove that (3.10) is valid when μ ∈ S00 (∂Ω). According to a results in the preceding Chapter we have RαA f L2 (Ω) ≤ Rα f L2 (Ω) ≤

RαA f − Rα f + UAα RαA f = 0 , α > 0, f ∈ B + (Ω) If μ ∈ S00 (∂Ω), and if f is bounded function in L2 (Ω), then Rα f  < ∞, and UAα RαA f is a relative quasi continuous version of the α−potential Uα (RαA f.μ) ∈  1 (Ω). Since Uα (RA f.μ)∞ ≤ RA f ∞ Uα μ∞ < ∞ and μ(∂Ω) < ∞, we H α α have that A Rα f = Rα f − UAα RαA f ∈ Fμ and that aα (RαA f, u) = aα (Rα f, u) − aα (UAα RαA f, u) = (f, u) − (RαA f, u)μ 50

, u ∈ Fμ

(3.10) follows. For general positive measure μ charging no set of zero relative capacity, we can take an increasing sequence (Fn ) of generalized nest of ∂Ω, and μn = 1Fn .μ ∈ S00 (∂Ω). Since μ charges no set of zero relative capacity, μn (B) increases to μ(B) for any B ∈ B(∂Ω). Let An = 1Fn .A. Then An is a PCAF of Xt with Revuz measure μn . Since μn ∈ S00 (∂Ω) we have for f ∈ L2 (Ω): RαAn f ∈ Fμn

, (aμn )α (RαAn , u) = (f, u) , f ∈ L2 (Ω), u ∈ Fμn

|RαAn f |

Clearly ≤ Rα |f | < ∞ r.q.e, and hence for r.q.e x ∈ Ω. For n < m, we get from (3.11):

(3.11)

limn→+∞ RαAn f (x)

= RαA f (x)

(aμn )α (RαAn f − RαAm f, RαAn f − RαAm f ) ≤ (f, RαAn f − RαAm f )

(3.12)

which converges to zero as n, m → +∞. Therefore (RαAn f )n is a1 −convergent   1 (Ω) and the limit function RA f is in H  1 (Ω). On the other hand we also in H α get from (3.11): RαAn f L2 (∂Ω,μ) ≤ (f, RαAn f )L2 (Ω) ≤ α1 f L2 (Ω) . And by Fatou’s Lemma: RαA f L2 (Ω) ≤ √1α f ||L2 (Ω) , getting RαA f ∈ Fμ . Finally, observe the estimate: |(RαAn f, u)μn −(RαA f, u)μ | ≤ RαAn f −RαA f L2 (∂Ω,μn ) uL2 (∂Ω,μ) +|(Rα f, u)μ−μn | holding for u ∈ L2 (∂Ω, μ). The second term of the right-hand side tends to zero as n → +∞. The first term also tends to zero because we have from (4.3): RαAn f − RαAm f L2 (∂Ω,μn ) ≤ (f, RαAn f − RαAm f ), and it suffices to let first m → +∞ and then n → +∞. By letting n → +∞ in (3.11) we arrive to desired equation (3.10). The proof of the Theorem 3.2.6 is similar to the Theorem 6.1.1 [46] which was formulated in the first time by S. Albeverio and Z. M. Ma [6] for general smooth measures in the context of general (X, m). In the case of X = Ω, and working just with measures on S0 (∂Ω) the proof still the same, and works also for any smooth measure concentrated on ∂Ω. Consequently, the theorem still verified for smooth measures ”nowhere Radon” i.e. measures locally infinite on ∂Ω. Example 3.2.7. We give some particular examples of Ptμ : (1) If μ = 0, then Pt0 f (x) = Ex [f (Xt )] the semigroup generated by Laplacian with Neumann boundary conditions. 51

(2) If μ is locally infinite (nowhere Radon) on ∂Ω, then Pt∞ f (x) = Ex [f (Bt )1{t 0 by, (4.2) Φ(μ, α) : Cc (Ω)+ → [0, ∞]    ∞ Φ(μ, α)f := e−αt pt f dt dμ (4.3) ∂Ω

0

where (pt )t≥0 is the markovian transition function of the general reflected brownian motion: The diffusion process associated with (a, F). Recall that (pt )t≥0 is a continuous version of the semigroup (Pt )t≥0 assciated with (a, F). Theorem 4.1.4. Let μ ∈ S(∂Ω). Then the following assertions are equivalent to each other: 1. μ ∈ SK (∂Ω), 2. Φ(μ, α) extends to a bounded linear functional on L1 (Ω) and lim Φ(μ, α)1 =0

α→∞

In view of the KLMN theorem, the more suitable class of measures is when the limit in the above theoem is less strictly than 1. It is exactly, what is done in [84], where an extended Kato class of measures was defined. In our context, and to deal just with measures concentrated in ∂Ω, the definition can be wrote as follow: SˆK (∂Ω) = {μ ∈ S(∂Ω) : ∃α > 0 s.t. Φ(μ, α) extends to a bounded linear functional on L1 (Ω) and c(μ) < 1} (4.4) where c(μ) := lim cα (μ) and cα (μ) := Φ(μ, α)∞ = Φ(μ, α)L1 (Ω) . α→∞

In the same manner as Theorem 3.3. in [84], one can deduce the following theorem: Theorem 4.1.5. Let μ ∈ SˆK (∂Ω). Then μ is a−bounded. More precisely,  |u|2 dμ ≤ c(μ)a(u, u) + u22 , ∀u ∈ F (4.5) ∂Ω

58

4.2

Signed Measures Case

Let μ = μ+ − μ− be a signed Broel measure on ∂Ω. If |μ| is a smooth measure, then μ is called a signed smooth measure, and we shall write μ ∈ S(∂Ω) − S(∂Ω). It is evident that μ ∈ S(∂Ω) − S(∂Ω) if μ+ and μ− are both + − in S(∂Ω). For μ ∈ S(∂Ω) − S(∂Ω) we shal set Aμt := Aμt − Aμt . We shal μ call μ the Revuz measure of At . We define for μ ∈ S(∂Ω) − S(∂Ω)   ∇u∇vdx + uvdμ, ∀u, v ∈ F μ aμ (u, v) = Ω

∂Ω

 1 (Ω) ∩ L2 (∂Ω, |μ|) where F μ = H First, we begin with the following result Theorem 4.2.1. Let μ be a signed Borel measure on ∂Ω and assume that |μ| is locally infnite everywhere on ∂Ω; i.e., ∀x ∈ ∂Ω and r > 0

|μ|(B(x, r)) = ∞.

Then the form aμ , which we denote by a∞ , is closed and is given by  ∇u∇vdx , D(a∞ ) = H01 (Ω) a∞ (u, v) = Ω

 its relatively continuous version, it follow from the Proof. Let u ∈ F and u u|2 d|μ| < ∞ that u  = 0 r.q.e on ∂Ω. Thus fact that ∂Ω | μ

 1 (Ω) : u F ∞ := F μ = {u ∈ H  = 0 r.q.e. on ∂Ω} One obtain that for all u, v ∈ F ∞ ,



a∞ (u, v) := aμ (u, v) =

Ω

∇u∇vdx.

Following a characterization of H01 (Ω) with relative capacity [15, Theorem 2.3.], one conclude that F ∞ = H01 (Ω). Proposition 4.2.2. Let μ ∈ S(∂Ω) − SK (∂Ω). Then 1. (aμ , F μ ) is lower semibounded closed quadratic form, 2. (Ptμ )t≥0 is the unique strongly continuous semigroup corresponding to (aμ , F μ ), 59

3. F μ = F ∩ L2 (∂Ω, μ+ ). Proof. The proof is the same as Proposition 3.1. in [7] with minor change. Theorem 4.2.3. Let μ ∈ S(∂Ω) − S(∂Ω). Then the following assertions are equivalent to each other: 1. (aμ , F μ ) is lower semibounded, 2. (Ptμ )t≥0 is a strongly continuous semigroup on L2 (Ω), 3. there exist constants c and β such that Ptμ f 2 ≤ ceβt f 2

, ∀f ∈ L2 (Ω) +

4. the form Qμ− is relatively form bound with respect to (aμ+ , F μ ) with bound less than 1, where qμ− is defined by qμ− (u, v) = ∂Ω uvdμ− . If one of the above assertion holds, the closed form coresponding to (Ptμ )t≥0 is the largest closable form that is smaller that (aμ , F μ ). Proof. The proof is tha same as Theorem 4.1. in [7] with minor change. Let μ ∈ S(∂Ω) − SK (∂Ω). We will denote by Δμ the selfadjoint operator in L2 (Ω) associated with (aμ , F μ ); i.e., 

D(Δμ ) := {u ∈ F μ : ∃v ∈ L2 (Ω) : aμ (u, ϕ) = (v, ϕ)∀ϕ ∈ F μ } Δμ u := −v

Since for each u ∈ D(Δμ ) we have    ∇u∇ϕdx + uϕdμ = vϕdx Ω

(4.6)

Ω

∂Ω

for all ϕ ∈ aμ , if we choose ϕ ∈ D(Ω), the equality (4.6) can be written −Δu, ϕ = v, ϕ where ,  denotes the duality between D(Ω) and D(Ω). Since ϕ ∈ D(Ω) is arbitrary, it follows that in D(Ω)

−Δu = v

Thus Δμ is a realization of the Laplacian on L2 (Ω). 60

Proposition 4.2.4. Let μ ∈ S(∂Ω) − SK (∂Ω). Then the following assertion are equivalent to each other: 1. (aμ , F μ ) is regular on Ω. 2. |μ| is a Radon measure. Proposition 4.2.5. Let μ ∈ S(∂Ω) − SK (∂Ω). Then there exists a relatively open set X0 satisfying Ω ⊂ X0 ⊂ Ω such that the form (aμ , F μ ) is regular on X0 For the proof of the above two propositions, one can follow the proof of Theorem 5.8. in [6]. The relatively open set can be explicitly written as follow X0 = Ω \ {x ∈ ∂Ω : |μ| (B(x, r)) = ∞, ∀r > 0} Now define the following subset of ∂Ω, Γ∞ := {x ∈ ∂Ω : |μ| (B(x, r)) = ∞, ∀r > 0} Note that Γ∞ is a relatively closed subset of ∂Ω. Since Γ|μ| := ∂Ω \ Γ∞ is a locally compact metric space, it follows from [75, Theorem 2.18. p.48] that |μ||Γ|μ| is a regular Borel measure. Therefore |μ| is a Radon measure on Γ|μ| . +

As for Theorem 4.2.1, we have u |Γ∞ = 0 r.q.e. for each function u ∈ FΓμ∞ , where + +  = 0 r.q.e. on Γ∞ } FΓμ∞ := {u ∈ F μ : u It follows that Δμ is the Laplacian with Dirichlet boundary conditions on Γ∞ , and with Robin boundary conditions on Γ|μ| .

4.3

Domination Results

In this section, we will prove a domination theorem. It says that the semigroup (e−tΔμ )t≥0 is sandwitched between (e−tΔμ+ )t≥0 and (e−tΔ−μ− )t≥0 . A very natural question arise: Is the converse also true? The answer is yes under a locality assymption. Definition 4.3.1. Let E be an ordered vector space 1. E is a vector lattice if any two elements f, g ∈ E have a supremum, which is denoted by f ∨ g, and an infinimum, denoted by f ∧ g. 2. Let E be a Banach lattice. A linear subspace I of E is called an ideal if f ∈ I and g ∈ E such that |g| ≤ |f | imply g ∈ I. 61

Theorem 4.3.1. Let μ ∈ S(∂Ω) − SK (∂Ω), and Δμ the closed operator on the L2 (Ω) associated with the closed form (aμ , F μ ). Then 0 ≤ e−tΔμ+ ≤ e−tΔμ ≤ e−tΔ−μ− for all t ≥ 0 in the sens of positive operators. One can see easily from the probabilistic representation of (e−tΔμ )t≥0 that the Propostion is true, but here we will prove it using the following result characterizing domination of positive semigroups due to Ouhabaz and contained in [69, Th´eor`eme 3.1.7.], Theorem 4.3.2. (Ouhabaz) Let T and S be two positive symmetric C0 −semigroups on L2 (Ω). Let (a, D(a)) be the closed form associated with T and (b, D(b)) the closed form associated with S. Then the following assertions are equivalent. 1. T (t) ≤ S(t) for all t ≥ 0 in the sense of positive operators. 2. D(a) is an ideal of D(b) and b(u, v) ≤ a(u, v) for all u, v ∈ D(a)+ . Proof. (of Theorem 4.3.1) Recall that the forms associated with e−tΔμ+ , e−tΔ−μ− and e−tΔμ are given repectively by   +  1 (Ω) ∩ L2 (∂Ω, μ+ ) ∇u∇vdx + uvdμ+ , ∀u, v ∈ F μ = H aμ+ (u, v) = Ω

aμ (u, v) = and

∂Ω

 Ω

 a−μ− (u, v) =



+

∇u∇vdx +

uvdμ, ∀u, v ∈ F μ = F μ ∂Ω

 Ω

−  1 (Ω) uvdμ− , ∀u, v ∈ F μ = H

∇u∇vdx − ∂Ω

+

It is clear that aμ (u, v) ≤ aμ+ (u, v) for all u, v ∈ F+μ = F+μ . Then by Theorem 4.3.2 we have e−tΔμ+ ≤ e−tΔμ for all t ≥ 0 in the sense of positive operator. On the other hand, one have a−μ− (u, v) ≤ aμ (u, v) for all u, v ∈ + +  1 (Ω). Let u ∈ F μ+ and F+μ . It still to prove that F μ is an ideal of H 1  v ∈ H (Ω) such that 0 ≤ |v| ≤ |u|.We may assume that u and v are r.q.c., it follows that 0 ≤ |v| ≤ |u| r.q.e. and therefore μ+ −a.e.(since μ charges no set of zero relative capacity) . It follows that   |v|2 dμ+ ≤ |u|2 dμ+ < ∞ ∂Ω

∂Ω

 1 (Ω) ∩ L2 (∂Ω, μ+ ). and then v ∈ L (∂Ω, μ ), which implies that v ∈ H 2

+

62

For two positive Borel measures μ and ν on ∂Ω we write ν ≤ μ if ν(A) ≤ μ(A) for all A ∈ B(∂Ω), and for two signed Borel measures μ and ν on ∂Ω we write ν ≤ μ if ν + ≤ μ+ and ν − ≥ μ− . Proposition 4.3.3. Let μ, ν ∈ S(∂Ω) − SK (∂Ω) such that ν ≤ μ. Let Δμ and Δν denote the selfadjoint operators on L2 (Ω) associated respectively with the closed forms (aμ , F μ ) and (aν , F ν ). Then 0 ≤ e−tΔμ+ ≤ e−tΔμ ≤ e−tΔν ≤ e−Δ−ν − for all t ≥ 0 in the sense of positive operators. Proof. It suffices to show that e−tΔμ ≤ e−tΔν for all t ≥ 0 in the sense of positive operators.We have ν + ≤ μ+ , then L2 (∂Ω, μ+ ) ⊂ L2 (∂Ω, ν + ). Thus + + F μ is continuously embedded into F ν . + + + μ+ We claim that F is an ideal of F ν . In fact, let u ∈ F μ and v ∈ F ν μ+ be such that 0 ≤ |v| ≤ |u|. We have to show that v ∈ F . We may assume  1 (Ω). Since 0 ≤ |v| ≤ |u| that u and v are r.q.c. It is clear that v ∈ H a.e., it follows that 0 ≤ |v| ≤ |u| r.q.e and therefore μ+ and ν + a.e. (since μ+ , ν + ∈ S(∂Ω)). It then follow   |v|2 dν + ≤ |u|2 dμ+ < ∞ ∂Ω 2

∂Ω +

and therefore v ∈ L (∂Ω, μ ) which proves the claim. + Now, let u, v ∈ F+μ . It follows that u and v are positive r.q.e. on Ω and thus μ a.e. on ∂Ω. We have ν + ≤ μ+ and ν − ≥ μ− , therefore aμ (u, v) ≤ aν (u, v), which completes the proof. +

Proposition 4.3.4. Let μ ∈ S(∂Ω) − SK (∂Ω). Then (aμ , F μ ) is local. Proof. The proof is similar to Proposition 3.4.20. in [90]. The main result of this section is the converse of Theorem 4.3.1. More precisely, if (T (t))t≥0 is a C0 −semigroup on L2 (Ω) satisfaying e−tΔμ+ ≤ T (t) ≤ e−tΔ−μ− for all t ≥ 0 in the sense of positive operators, under which conditions T (t) is given by a signed measure ν on ∂Ω? We suppose that Γμ = ∂Ω, we have then the following theorem:

63

Theorem 4.3.5. Let Ω ⊂ Rd be an open set and T = (T (t))t≥0 be a symmetric C0 −semigroup on L2 (Ω) satisfaying e−tΔμ+ ≤ T (t) ≤ e−tΔ−μ− for all t ≥ 0 in the sense of positive operators, where μ+ ∈ S(∂Ω) and μ− ∈ SK (∂Ω). Let (a, D(a)) be the closed form on L2 (Ω) associated with T . Suppose in addition that (a, D(a)) is regular. Then the following assertions are equivalent to each other: 1. T (t) = e−tΔν−μ− for a unique positive Radon measure ν charging no set of zero relative capacity on ∂Ω. 2. (a, D(a)) is local. Proof. (1)⇒(2) This part follows from Proposition 4.3.4.  1 (Ω) and for all u, v ∈ D(a)+ we (2)⇒(1) We have D(a) is an ideal of H have,   Ω

uvdμ− ≤ a(u, v)

∇u∇vdx − ∂Ω

For u, v ∈ D(a) ∩ Cc (Ω) we let



uvdμ− −

b(u, v) = a(u, v) + ∂Ω

 Ω

∇u∇vdx

Let {Gaβ : β > 0} be the resolvent of the operator associated with the closed −

: β > 0} be the resolvent of Δ−μ− . Let a(β) and form (a, D(a)) and {G−μ β (β)

a−μ− be the approximation forms of a and a−μ− and let (β)

b(β) (u, v) := a(β) (u, v) − a−μ− (u, v)   

− u, v = β u − βGaβ u, v − β u − βG−μ β −μ−

= β(β(Gβ

(4.7)

− Gaβ )u, v)

Since by domination criterion, b(β) (u, v) ≥ 0 for all positive u, v ∈ D(a) ∩ − − Gaβ ) is a positive symmetric operator on L2 (ω) Cc (Ω), we have that β(G−μ β and it then follows from [46](Lemma 1.4.1.) that there exists a unique positive Radon measure νβ on Ω × Ω such that for all u, v ∈ D(a) ∩ Cc (Ω) we have  − − Gaβ )u, v) = β u(x)v(y)dνβ b(β) (u, v) = β(β(G−μ β Ω

64

It is clear that b(β) (u, v) → b(u, v) as β ∞ for all u, v ∈ D(a) ∩ Cc (Ω). Since for each β > 0 and u ∈ D(a) ∩ Cc (Ω) b(β) (u, v) ≤ a(u, v) it follows that the sequence (βνβ ) is uniformly bounded on each compact subsets of Ω × Ω and hence a subsequence converges as βn → ∞ vaguely on Ω × Ω to a positive Radon measure ν. The form (a, D(a)) is regular and then ν is unique and therefore for all u, v ∈ D(a) ∩ Cc (Ω)  b(u, v) = u(x)v(y)dν Ω

 1 (Ω)) are local, it follows that b(u, v) = 0 for all Since (a, D(a)) and (a−μ− , H u, v ∈ D(a) ∩ Cc (Ω) with supp[u] ∩ supp[v] = ∅. This implies that supp[ν] ⊂ {(x, x) : x ∈ Ω}, and therefore  u(x)v(x)dν b(u, v) = Ω

Since b(u, v) = 0 for all u, v ∈ D(Ω) ⊂ D(a), we have supp[ν] ⊂ Ω \ Ω = ∂Ω and thus  u(x)v(x)dν b(u, v) = ∂Ω

Consequently, for all u, v ∈ D(a) ∩ Cc (Ω) we have    a(u, v) = ∇u∇vdx + uvdν − uvdμ− Ω

∂Ω

∂Ω

The positive Radon measure ν charges no set of zero relative capacity. In + fact, we have a(u, u) ≤ aμ+ (u, u) for all u ∈ F μ ⊂ D(a), which implies   |u|2 dν ≤ |u|2 d|μ| ∂Ω

∂Ω

With a particular choice of the function u, we have for all Borel subsets O ⊂ ∂Ω ν(O) ≤ |μ|(O) If O is of zero relative capacity then ν(O) = 0, thus ν also charges no set of zero relative capacity. To finish , it still to prove that (a, D(a)) = (aν−μ− , F ν ). 65

It is clear that F ν is a closed subspace of D(a). We show that D(a) is a  1 (Ω) subspace of F ν . Let u ∈ D(a). For n ∈ N we let un = u∧n. Then un ∈ H is relatively quasi-continuous. Since 0 ≤ un ≤ n and ν(∂Ω) < ∞, it follows that un ∈ L2 (∂Ω, ν) and therefore un ∈ F ν . It is also clear that un → u  1 (Ω) and thus after taking a subsequence if necessary, we may assume that H un → u r.q.e.(see proposition 2.1. [16]). since ν charges no set of zero relative capacity, it follows that un → u ν−a.e. on ∂Ω. Finally, since 0 ≤ un ≤ k, the Lebesgue Dominated Convergence Theorem implies that un → u in L2 (∂Ω, ν) and thus un → u in F ν and therefore u ∈ F ν . We can drop out the condition that (aμ , F μ ) is regular, but in this case we should add with the locality assymption the fact that D(a) ∩ Cc (Ω) is dense in D(a). One can then follow the proof of Theorem 4.1 in [16] and the technics in Theorem 4.3.5 to prove the existence of such measure ν. The inconvenient in this case is that ν is not necessary unique, see [3].

66

67

Chapter 5 Local and Nonlocal Robin Laplacian This Chapter is concerned with three different topics. The first topic aim to explicit all semigroups sandwiched between the semigroups generated by the Laplacian with Dirichlet boundary conditions and the one with Neumann boundary conditions. The second topic is an introduction to a general nonlocal Robin boundary value problem on arbitrary domains and involving Radon measures and the third topic concerns another notion of nonlocal Robin boundary condition involving bounded operators on the boundary.

5.1

Dirichlet and Neumann Boundary Conditions: What is in all between?

Let Ω be an open set of Rd . We want to characterize all semigroups (T (t))t≥0 on L2 (Ω) sandwiched between Dirichlet and Neumann ones, i.e.: etΔD ≤ T (t) ≤ etΔN

, for all t ≥ 0

(5.1)

in the positive operators sense. In [16], it is proved that when the (regular) Dirichlet form (a, D(a)) associated with such semigroups is local then there exists a unique positive measure on the boundary Γ := ∂Ω which charges no set of zero relative capacity such that the form is given explicitly by  

1 (Ω) ∩ L2 (Γ, dμ) ∇u∇vdx + u vdμ, D(aμ ) = H (5.2) a(u, v) = Ω

Γ

The regularity of a is needed to have the uniqueness of the measure μ, otherwise the form will be regular just on a subset X = Ω \ Γ∞ , where Γ∞ 68

is the subset of ∂Ω where μ is locally infinite everywhere, and then μ is not unique because obtained by extension to the whole boundary. In this section we focus on the hypothesis of locality, and to simplify we consider a as regular Dirichlet form, if not we do the same as in [16] or [90]. The motivation of our main results was an intuitive remark that one don’t need to suppose the locality of a, and that the locality is contained in the fact that a is sandwiched between two local forms.

5.1.1

Preliminaries

The central tool to prove our results is the following fascinating BeurlingDeny and Lejan formula. The proof can be found for example in [46] or [25], another proof based on contraction operators can be found in [10] or [11]. Theorem 5.1.1. (The Beurling-Deny and Lejan formula) Any regular Dirichlet form (a, D(a)) on L2 (X; m) can be expressed for u, v ∈ D(a) ∩ Cc (X) as follow:  u(x)v(x)k(dx) a(u, v) = a(c) (u, v) + X  + (u(x) − u(y))(v(x) − v(y))J(dx, dy) X×X\d

where a(c) is a strongly local symmetric form with domain D(a(c) ) = D(a) ∩ Cc (X), J is a symmetric positive Radon measure on X × X\d (d =diagonal), and k is a positive Radon measure on X. In addition such a(c) , J and k are uniquely determined by a. The Beurling-Deny and Lejan formula had catched a lot of attention last years and many generalization in other directions was explored. For example in the case of semi-regular Dirichlet forms, or Regular but non-symmetric Dirichlet forms. For more information we refer to [57], [58], [59] and [64] and references therein. Here, we will use the above “conventional” Beurling-Deny and Lejan formula that is the one for Regular Dirichlet forms, but there is no reason that the same arguments do not work also for the other cases. First of all, we recall the situation treated in [16]. Let μ : B → [0, ∞] be a measure. We consider the symmetric form aμ on L2 (Ω) given by   ∇u∇vdx + u(x)v(x)dμ aμ (u, v) = Γ

Ω

with domain D(aμ ) = {u ∈ H 1 (Ω) ∩ Cc (Ω) : 69

 Γ

|u|2 dμ < ∞}

Let Γμ = {z ∈ Γ : ∃r > 0 such that μ(Γ ∩ B(z, r)) < ∞} be the part of Γ on which μ is locally finite. Assume now that Γμ = ∅, thus Γμ is a locally compact space and μ is a regular Borel measure on Γμ . We say that μ is admissible if for each Borel set A ⊂ Γμ one has CapΩ (A) = 0 ⇒ μ(A) = 0 It is proved in [16] that the form aμ is closable if and only if μ is admissible, and the closure of aμ is given by   1 (Ω) : u˜ = 0 r.q.e. on Γ \ Γμ , |˜ u|2 dμ < ∞} D(aμ ) = {u ∈ H Γμ

 aμ (u, v) =

 Ω

∇u∇vdx +

u˜v˜dμ Γμ

Here u˜ is the relatively quasi-continuous representative of u. Note that the form (aμ , D(aμ ) is regular if and only if μ is a radon measure on Γ, which means that Γμ = Γ. Denote by Δμ the operator associated with aμ . Then it follows that Δμ is a realization of the Laplacian in L2 (Ω)( see [15] and [90] for more details and other properties). Next we give two trivial examples of measure μ:   1 (Ω) and aμ (u, v) = ∇u∇vdx. Example 5.1.2. (i) If μ = 0 then D(aμ ) = H Ω Let ΔN be the operator associated with (a0 , D(a0 )). It is the Neumann Laplacian, and it coincides with the usual Neumann Laplacian when Ω is bounded with Lipschitz boundary.  (ii) If Γμ = ∅, then D(aμ ) = H01 (Ω) and aμ (u, v) = Ω ∇u∇vdx. Let ΔD be the operator associated with (a∞ , D(a∞ )). It is the Dirichlet Laplacian. We have the following domination result Theorem 5.1.3. For each admissible measure μ, the semigroup (etΔμ )t≥0 satisfies (5.3) etΔD ≤ etΔμ ≤ etΔN , for all t ≥ 0 for all t ≥ 0 in the sense of positive operators. The proof is a simple application of the Ouhabaz criterion and is given in [15, Theorem 3.1.]. For a probabilistic proof see Section 3.2.3. In [15, Section 4], W. Arendt and M. Warma explored the converse of the above Theorem. More precisely, they answered the following question: 70

Having a sandwiched semigroup between Dirichlet and Neumann semigroup, can one write the associated form with help of an admissible measure? The answer was affirmative under a central hypothesis of locality of the associated Dirichlet form. The first Theorem says ([15, Theorem 4.1]) Theorem 5.1.4. Let Ω be an open subset of Rd with boundary Γ. Let T be a symmetric C0 −semigroup on L2 (Ω) associated with a positive closed form (a, D(a)). Then the following assertions are equivalent (i) There exist an admissible measure μ such that a = aμ (ii) (a) One has etΔD ≤ T (t) ≤ etΔN , (t ≥ 0). (b) a is local. (c) D(a) ∩ Cc (Ω) is dense in (D(a), .a ). In order to characterize those sandwiched semigroups which come from bounded measure we have the following Theorem ([15, Theorem 4.2.]) Theorem 5.1.5. Let Ω be a bounded open set of Rd . Let T be a symmetric C0 −semigroup on L2 (Ω) associated with a positive closed form (a, D(a)). Then the following assertions are equivalent (i) There exist a bounded admissible measure μ on Γ such that a = aμ (ii) (a) One has etΔD ≤ T (t) ≤ etΔN , (t ≥ 0). (b) a is local. (c) 1 ∈ D(a).

5.1.2

New Method for the Sandwiched Property

In the next, we will prove that one can drop the assumption of locality, this can be made by using the Beurling-Deny and Lejan decomposition formula of regular Dirichlet forms. Theorem 5.1.6. Let Ω be a bounded open set of Rd and (T (t))t≥0 be a C0 −semigroup on L2 (Ω) associated with a regular Dirichlet form (a, D(a)). Then the following assertions are equivalent: 1. a = aμ for a unique Radon measure μ on ∂Ω which charges no set of zero relative capacity. 2. etΔD ≤ T (t) ≤ etΔN for all t ≥ 0. 71

Proof. Let (a, D(a) a regular Dirichlet form associated with (T (t))t≥0 . From the formula of Beurling-Deny and Lejan we get for all u, v ∈ D(a) ∩ Cc (Ω)  a(u, v) = a(c) (u, v) + u(x)v(x)k(dx) Ω  (u(x) − u(y))(v(x) − v(y))J(dx, dy) + Ω×Ω\d

(c)

where a is a strongly local form, k a positive radon measure on Ω and J a symmetric positive radon measure on Ω × Ω\d . To simplify calculations, we note for all u, v ∈ D(a) ∩ Cc (Ω)  ak (u, v) = a(c) (u, v) + u(x)v(x)k(dx), Ω

and b(u, v) = a(u, v) − ak (u, v), We have then for all u, v ∈ D(a) ∩ Cc (Ω) such that supp[u] ∩ supp[v] = ∅  (u(x) − u(y))(v(x) − v(y))J(dx, dy) b(u, v) = Ω×Ω\d





= 2 Ω

u(x)v(x)J(dx, dy) − 2



= −2

u(x)v(y)J(dx, dy) (5.4) Ω×Ω\d

u(x)v(y)J(dx, dy)

(5.5)

Ω×Ω\d

From the fact that ak is local, we get for all u, v ∈ D(a) ∩ Cc (Ω) such that supp[u] ∩ supp[v] = ∅  u(x)v(y)J(dx, dy) a(u, v) = −2 Ω×Ω\d

From (5.1), and the Ouhabaz’s domination criterion, we have that aN (u, v) ≤ a(u, v) for all u, v ∈ D(a)+ , and then for all u, v ∈ D(a)+ ∩ Cc (Ω) such that supp[u] ∩ supp[v] = ∅ we have  u(x)v(y)J(dx, dy), 0 ≤ −2 Ω×Ω\d

 which means that −2 Ω×Ω\d u(x)v(y)J(dx, dy) = 0. Thus supp[J] ⊂ d and then for all u, v ∈ D(a) ∩ Cc (Ω)   u(x)v(y)J(dx, dy) = u(x)v(x)J(dx, dy) Ω×Ω\d

Ω

72

Then, we deduce that b(u, v) = 0 for all u, v ∈ D(a) ∩ Cc (Ω) und thus the form a is immediatly local and is reduced to the following  a(u, v) = a(c) (u, v) + u(x)v(x)k(dx), for all u, v ∈ D(a) ∩ Cc (Ω) Ω

We have

Cc∞ (Ω)



Ω

⊂ D(a), and then for all u, v ∈ Cc∞ (Ω) we have from (5.1)   ∇u∇vdx ≤ a(c) (u, v) + u(x)v(x)k(dx) ≤ ∇u∇vdx Ω

Ω

It follows that for all u, v ∈ Cc∞ (Ω), we get  a(u, v) = a(c) (u, v) + u(x)v(x)k(dx)

(5.6)

Ω

 = Ω

∇u∇vdx

(5.7)

Thus supp[k] ⊂ ∂Ω and then by putting μ = k|∂Ω , we have for all u, v ∈ D(a) ∩ Cc (Ω)   a(u, v) = ∇u∇vdx + u(x)v(x)μ(dx) Ω

∂Ω

For the rest of the proof, one can follow exactly the end of the proof of [90, Corollary 3.4.23] One can see easily that the locality property was automatic in the above proof and that just the right hand side domination property was needed for it. We give then the following Corollary. Corollary 5.1.7. Let Ω be a bounded open set of Rd and (T (t))t≥0 be a C0 −semigroup on L2 (Ω) associated with a regular Dirichlet form (a, D(a)) such that T (t) ≤ etΔN for all t ≥ 0. Then the regular Dirichlet form (a, D(a)) is local. Proof. The corollary come directly from the proof of Theorem 5.1.6. We deduce from the above discussions the following general theorem Theorem 5.1.8. Let X be locally compact separable metric space and m a positive Radon measure on X such that supp[m] = X. Let (T (t))t≥0 (resp. (S(t))t≥0 ) be a C0 −semigroup on L2 (X; m) associated with a regular Dirichlet form (a, D(a)) (resp. with a Dirichlet form (b, D(b))). Assume in addition that T (t) is subordinated by S(t) that is T (t) ≤ S(t) for all t ≥ 0 in the positive operators sense. Then b is local implies a is local. 73

Proof. The proof is based on the same argumets as for Corollary 5.1.7. In fact, we have that b ≤ a and b is local then for all u, v ∈ D(a)+ ∩ Cc (X) with disjoint support we have a(u, v) ≥ 0. Thus, by using Beurling-Deny and Lejan formula and the positivity of u and v one obtain  u(x)v(y)J(dx, dy) = 0 X×X\d

for all u, v ∈ D(a) ∩ Cc (X) with disjoint support. Thus supp[J] ⊂ d, and then the jump integral in the Beurling-Deny and Lejan decomposition of the form (a, D(a)) is null, which achieve the proof. Corollary 5.1.9. Let Ω be a bounded open set of Rd and (T (t))t≥0 be a C0 −semigroup on L2 (Ω) associated with a Dirichlet form (a, D(a)). Then the following assertions are equivalent: (i) a = aμ for some positive measure μ on ∂Ω which charges no set of zero relative capacity on which it is locally finite. (ii) (a) etΔD ≤ T (t) ≤ etΔN for all t ≥ 0 (b) D(a) ∩ Cc (Ω) is dense in D(a). Proof. Let Γ0 = {z ∈ Γ : ∃u ∈ D(a) ∩ Cc (Ω), u(z) = 0} From the Stone-Weierstrass theorem, one can see that the form (a, D(a)) is regular on L2 (X) where X = Ω ∪ Γ0 , one can see it exactly from the proof in [15, Theorem 4.1] or the one in [90, Theorem 3.4.21]. Following the same procedure as in Theorem 5.1.6, we obtain that there exist a unique positive Radon measure k on X such that  u(x)v(x)k(x) a(u, v) = a(c) (u, v) + X

for all u, v ∈ D(a) ∩ Cc (X). Now one can just again follow the same steps as in the proof of [15, Theorem 4.1] or the one in [90, Theorem 3.4.21] Proposition 5.1.10. Let Ω ⊂ Rd be a bounded open set of class C 1 with boundary Γ. Let T be a symmetric C0 −semigroup associated with a regular Dirichlet form (a, D(a)). Denote by A the generator of T . Assume that a) etΔD ≤ T (t) ≤ etΔN for all t ≥ 0, and that b) there exists u ∈ D(A) ∩ C 2 (Ω) such that u(z) > 0 for all z ∈ Γ. Then there exists a function β ∈ C(Γ)+ such that A = −Δβ . 74

Proof. The proof is exactly the same as [15, Proposition 5.2], we have just omitted the locality hypothesis in the proposition. In what follow we give some complements and remarks about our approach and the one of W. Arendt and M. Warma. • Apparently the both methods, ours and the one in [15] seem to be different, but in fact there is some connection between them. Seeing things more in details in the proofs of [15, Theorem 4.1], and the proof of the formula of Beurling-Deny and Lejan, one can see that they make use of [46, Lemma 1.4.1] as a central tool to find a measure. • The ideal property was not used in the sufficient implication of Theorem 5.1.6. • One can expect to generalize our results without difficulty to the semiregular Dirichlet forms and to regular but non-symmetric regular Dirichlet forms. • In the context of p−Laplacian operator R. Chill and M. Warma proved a version of the [15, Theorem 4.1. and Theorem 4.2] see [39]. Unfortunately, there is no version of the formula of Beurling-Deny and Lejan in Lp (X, m), see [28] as a beginning of interest. • It is proved in an unpublished note of W. Arendt and M. Warma that a locality of the Dirichlet form implies the locality of the associated operator, but the converse is not true. We can see easily from our results that the operator is local if and only if J|Ω×Ω\d = 0, which means that there is an equivalence between the locality of the operator and the one of the form when there is no jump inside the domain.

5.2

Generalized Nonlocal Robin Laplacian on Arbitrary Domains

Let Ω be an open set of Rd , and we consider μ to be a Borel measure on ∂Ω, and θ a symmetric Radon measure on ∂Ω × ∂Ω\d where d indicates the diagonal of ∂Ω × ∂Ω. Let   1 (u(x) − u(y))2 dθ < ∞} |u|2 dμ + E = {u ∈ H 1 (Ω) ∩ Cc (Ω) : 2 ∂Ω×∂Ω\d ∂Ω We define the bilinear symmetric form aμ,θ on L2 (Ω) by 75

 aμ,θ (u, v) =

 Ω

∇u∇vdx +

uvdμ + ∂Ω

1 2

 (u(x) − u(y))(v(x) − v(y))dθ ∂Ω×∂Ω\d

With means of a Reed-Simon Theorem [75, Theorem S15, p:373], there exist a closable positive form (aμ,θ )r ≤ aμ,θ such that b ≤ (aμ,θ )r whenever b is a closable form such that b ≤ aμ,θ . Thus (aμ,θ )r is the largest closable form smaller or equal than aμ,θ . Cleary, aμ,θ is closable if and only if aμ,θ = (aμ,θ )r .

5.2.1

Closability

ˆ Define the Borel measure θ(dx) := θ(dx × ∂Ω \ d). We want to determine (aμ,θ )r and in particular characterize when aμ,θ is closable. One can see that the closability of the form aμ,θ is inherent to the measure θˆ rather than to the measure θ. Let ˆ ∩ B(z, r)) < ∞} Γμ,θ := {z ∈ ∂Ω : ∃r > 0 such that μ(∂Ω ∩ B(z, r)) + θ(∂Ω be the part of ∂Ω on which μ and θˆ are locally finite. It is easy to verify that u = 0 on ∂Ω \ Γμ,θ for all u ∈ E. We introduce for the first time the notion of admissible pair of measures. We say that the pair (μ, θ) is admissible if for each Borel set A ⊂ Γμ,θ one has ˆ =0 CapΩ (A) = 0 ⇒ (μ + θ)(A) where CapΩ (A) is the relative capacity of the subset A. The following Lemma is an adapation of [31, Lemma 4.1.1], and will be used to prove Theorem 5.2.2. ˆ Lemma 5.2.1. Let B be a Borel set of ∂Ω such that θ(B) > 0. Then there exists disjoint compact sets K and C such that K ⊂ B and θ(K × C) > 0 Proof. By the the inner regularity of the positive Radon measure θ we choose a set K0 ⊂ ∂Ω × B \ d which is compact as a subset of ∂Ω × ∂Ω \ d and therefore compact as a subset of ∂Ω × ∂Ω such that θ(K0 ) > 0. We choose a metric ρ0 which is compatible with the topology of ∂Ω. Then the metric ρ on ∂Ω × ∂Ω, given by ρ((x, y); (x , y  )) := ρ0 (x, x ) + ρ0 (y, y  ),

∀x, x , y, y  ∈ ∂Ω

is compatible with the product topology on ∂Ω × ∂Ω. Since K0 is compact and the diagonal d is closed we can choose a real number a such that 0 < 76

2a < dist(K0 , d). Since ∂Ω is relativly compact each x ∈ ∂Ω has a compact neighborhood Kx with diameter less that a. Since K0 can be covered by finitly many sets of the form Kx × Ky and θ(K0 ) > 0 we can choose (x0 , y0 ) ∈ ∂Ω × ∂Ω such that θ(K0 ∩ Kx0 × Ky0 ) > 0. Suppose that there exist z ∈ Kx0 ∩ Ky0 . Choose (x , y  ) ∈ K0 ∩ Kx0 × Ky0 . Then ρ((z, z); (x , y  )) ≤ 2a, contradicting the fact that dist(d, K0 ) > 2a. Thus Kx0 ∩ Ky0 = ∅. Now set C := Kx0 and K := Ky0 ∩ {y ∈ ∂Ω : ∃x∂Ω with (x, y) ∈ K0 }. Then C and K are disjoint compact sets and KsubsetK. Moreover θ(C × K) > 0 since K × C ⊃ Kx0 × Ky0 ∩ K0 . Theorem 5.2.2. The form aμ,θ is closable if and only if the pair (μ, θ) is admissible Proof. (⇐) The condition that μ and θˆ are locally finite is not necessary for this part, we can also omit the assymption that θ is Radon . Let uk ∈ E be such that uk → 0 in L2 (Ω) and lim aμ,θ (un − uk , un − uk ) = 0. It is n,k→∞

clear that uk → 0 in H 1 (Ω). By [90, Theorem 2.1.3] applied to the relative capacity, the sequence (uk ) contains a subsequence (wk ) which converges to zero r.q.e. on Ω. Since μ + θˆ charges no set of zero relative capacity, it follows that uk|∂Ω → 0 μ a.e. and θˆ a.e. Since uk is a Cauchy sequence in L2 (∂Ω, μ), it follows that uk → 0 in L2 (∂Ω, μ). On the other hand we ˆ have θ(A × A) ≤ θ(A) for all Borel set A ⊂ ∂Ω. Since uk|∂Ω → 0 θˆ a.e, thus uk (x) − uk (y) → 0 θ a.e. Since uk (x) − uk (y) is a Cauchy sequence in L2 (∂Ω × ∂Ω \ d, θ), it follows that uk (x) − uk (y) → 0 in L2 (∂Ω × ∂Ω \ d, θ) and thus lim aμ,θ (uk , uk ) = 0, which means that the form (aμ.θ , E) is closable. k→∞

(⇒) Suppose that there exists a Borel set B such that CapΩ (B) = 0 and ˆ θ(B) > 0 or μ(B) > 0. We show that aμ,θ is not closable. We consider ˆ just the case where θ(B) > 0. The case where μ(B) > 0 can be proved simultaneously. By Lemma 5.2.1 we can choose disjoint compact K and C such that θ(C × K) > 0 and K ⊂ B(and therefore CapΩ (K) = 0). Since CapΩ (K) = 0, by [90, Theorem 2.2.4], there exists a sequence uk ∈ H 1 (Ω) ∩ Cc (Ω) such that 0 ≤ uk ≤ 1,

uk = 1 on K and uk H 1 (Ω) → 0

Let (Ai ) be a seqence of relatively open sets with compact closure satisfaying Ai = K K ⊂ Ai+1 ⊂ Ai ⊂ ∂Ω and i

77

There exists then a sequence vi ∈ D(RN ) such that supp[vi ] ⊂ Ai , vi = 1 on K and 0 ≤ vi ≤ 1. Clearly, vi|Ω ∈ H 1 (Ω) ∩ Cc (Ω) and uk vi H 1 (Ω) → 0 as k → ∞. For all i ≥ 1 we have uk vi ∈ H 1 (Ω) ∩ Cc (Ω). Moreover, for all i, k, 0 ≤ uk vi ≤ 1 and uk vi = 1 on K. For all i ≥ 1, we choose ki ∈ N such that uki vi H 1 (Ω) ≤ 21i . Let wi = uki vi . Then wi → 0 in H 1 (Ω), 0 ≤ wi ≤ 1 and wi = 1 on K. Moreover wi → χK pointwise, since supp[wi ] ⊂ Ai . One has supi wi 2μ,θ < ∞. In fact   ˆ sup (wi (x) − wi (y))2 dθ ≤ 2 sup wi2 (x)dθˆ ≤ 2θ(K) 0 lim sup 16 n→∞ and therefore lim sup aμ,θ (hn , hn ) > 0. the existence of such sequence contran→∞

dicts the closability of aμ,θ . We denote by −Δμ,θ the operator associated with the closure of aμ,θ . It follows that Δμ,θ is a realization of the Laplacian in L2 (Ω), and it generates a symmetric sub-Markovian semigroup on L2 (Ω). One can characterize the closure of the closable part of aμ,θ by using the same method as in [90] or in [15]. We have then the following theorem.

78

Theorem 5.2.3. Let μ (resp. θ) be a Radon measure on ∂Ω (resp. ∂Ω × ∂Ω\d ). Suppose that the pair (μ, θ) is admissible. Then the closure of (aμ,θ , D(aμ,θ )) is given by   1  1 (Ω) : | u|2 dμ+ ( u(x)− u(y))2 dθ < ∞} (5.8) D(aμ,θ ) = {u ∈ H 2 ∂Ω×∂Ω\d ∂Ω    1 aμ,θ (u, v) = ∇u∇vdx + u vdμ + ( u(x) − u (y))( v (x) − v(y))dθ 2 ∂Ω×∂Ω\d Ω ∂Ω (5.9) where u  is a relative quasi-continuous representation of u. Remark 5.2.4. 1- The proof of this theorem is based on the technics in the proofs of [90, Theorem 3.1.1] and [31, Theorem 4.1.2]. 2- One can consider a general θ (Not as Radon measure), but in this case the definition of admissible measures should change as follow: The pair (μ, θ) is said admissible if for any separated compact sets K1 and K2 on ∂Ω such that CapΩ (K1 )CapΩ (K2 ) = 0, we have θ(K1 × K2 ) = 0.(see [5] for more details)

5.2.2

Domination

Now, we want to verify if the semigroup generated by Δμ,θ is or is not sandwiched between the semigroups of Dirichlet Laplacian and the semigroup of Neumann Laplacian. The left hand side of the inequality is assured by the following theorem Theorem 5.2.5. We have etΔD ≤ etΔμ,θ Proof. By Ouhabaz’s domination criterion, it suffices to prove that H01 (Ω) is  an ideal of D(aμ,θ ) and aμ,θ (u, v) ≤ Ω ∇u∇vdx for all u, v ∈ H01 (Ω)+ . We  1 (Ω) are r.q.c. may assume that functions in H a) Let u ∈ H01 (Ω) and v ∈ D(aμ,θ ) such that 0 ≤ v ≤ u. Since Ω is relatively open, it follows form Proposition 2.1.5 that 0 ≤ v ≤ u r.q.c. on Ω. We have u = 0 r.q.e. on ∂Ω, then v = 0 r.q.e. on ∂Ω. Therefore v ∈ H01 (Ω). b) Let u, v ∈ H01 (Ω)+ . Then u = v = 0 r.q.e. on ∂Ω. Since the pair (μ, θ) ˆ is admissible, it follows that u = v = 0 (μ + θ)−a.e. on Γμ,θ . We finally

79

obtain that aμ,θ (u, v)





= Ω

+

∇u∇vdx +

1 2



(u(x) − u(y))(v(x) − v(y))dθ ∂Ω×∂Ω\d

 = 

Ω

=  ≤

Ω

Ω

uvdμ ∂Ω



 ˆ − uvd(μ + θ)

∇u∇vdx + ∂Ω

 ∇u∇vdx −

u(x)v(y)dθ ∂Ω×∂Ω\d

u(x)v(y)dθ ∂Ω×∂Ω\d

∇u∇vdx

and the proof is complete. The right hand side of the sandwiched inequality is not valid. We have then the following theorem, Theorem 5.2.6. We have etΔμ,θ ≤ etΔN

θ≡0

if and only if

Proof. We use the same technics of calculus on aμ,θ • Suppose that we have etΔμ,θ ≤ etΔN . By Ouhabaz’s domination criterion, we know that Ω ∇u∇vdx ≤ aμ,θ (u, v) for all u, v ∈ D(aμ,θ )+ . Which means that for all u, v ∈ D(aμ,θ )+ we have     ˆ − ∇u∇vdx ≤ ∇u∇vdx + uvd(μ + θ) u(x)v(y)dθ Ω

Ω

∂Ω

∂Ω×∂Ω\d

Thus for all u, v ∈ D(aμ,θ )+ ,   u(x)v(y)dθ ≤ ∂Ω×∂Ω\d

ˆ uvd(μ + θ) ∂Ω

We choose u, v ∈ D(aμ,θ )+ such that supp[u] ∩ supp[v] = ∅, then we get  u(x)v(y)dθ ≤ 0 

∂Ω×∂Ω\d

Which means that ∂Ω×∂Ω\d u(x)v(y)dθ = 0 for all u, v ∈ D(aμ,θ )+ such that supp[u] ∩ supp[v] = ∅. Thus θ ≡ 0 because supp(θ) ⊂ ∂Ω × ∂Ω \ d. 80

• Now if θ ≡ 0 the assertion follow from the sandwitched property proved in the last section.

The last Proposition is natural in view of Section 1 of this Chapter.

5.3

Nonlocal Robin Laplacian involving Bounded Operators

In this section we study another notion of nonlocal Robin Laplacian. The  nonlocality is traduced by the term Γ Buvdσ, where B is a bounded operator on the boundary Γ of Ω. A first result concerns the characterization of the positivity of the semigroup generated by the nonlocal Robin Laplacian, and then we will focus on domination results.

5.3.1

The Nonlocal Robin Laplacian

Let V be a normed space and a : V × V :→ C a sesquilinear form. Then a is continuous if and only if there exists a c > 0 such that |a(u, v) ≤ cuV vV ,

for all u, v ∈ V

Let H be a Hilbert space and V densly defined in H. The form a : V × V :→ C is called elliptic if there exist ω ∈ R and μ > 0 such that Rea(u) + ωu2H ≥ μu2V ,

for all u ∈ V

Let H be a Hilbert space and V a reflexive Banach space densely defined in H and let a : V × V :→ C a sesquilinear form. Let K ∈ K(V, V  ) to be a compact operator and define the following sesquilinear form b with domain V on H by b(u, v) = a(u, v)+V < u, Kv >V  We want to characterize the continuity and the ellipticity of (b, V ) in function of the one of (a, V ). We start with the following Lemma. Lemma 5.3.1. Let U be a reflexive Banach space, W a Banach space, assume that K ∈ K(U, U  ) and that T ∈ B(U, W ) is one to one. Then, for every > 0 there exist c > 0 such that |U < u, Ku >U  | ≤ u2U + c T u2W , 81

u∈V

Proof. Seeking a contradiction, assume that there exists > 0 along with a family of vectors un ∈ U , un  = 1, n ∈ N for which U < un , Kun >U   ≤ + nT un 2W ,

n∈N

Furthermore, since U is reflexive, there is no loss of generality assuming that there exists u ∈ U such that un  u in U . In addition, since T ( and hence T ) is bounded, one concludes that T un  T u(n → ∞) in W as is clear from W

< T un , v >W  =V < un , T v >V  →V < u, T v >V  =W < T u, v >W 

for all v ∈ W  . Moreover, since K is compact, we may choose a subsequence of (un ) (still noted the same) such that Kun → Ku strongly in U  . This, in turn, yealds that U < un , Kun >U  →U > u.Ku >U  , together with 1 |U < un , Kun >U  |, n ∈ N n This also shows that T un → 0in W . Hence T u = 0 in W which forces u = 0, since T is one-to-one. Given this facts, we note that on the one hand we have U < un , Kun >U  → 0 while on the other hand |U < un , Kun >U  | ≥ for every n ∈ N, which is a contradiction. T un 2W ≤

Theorem 5.3.2. Let a and b as described above, then we have the following: (i) a is continuous if and only if b is continuous, (ii) a is elliptic if and only if b is elliptic. Proof. (i) The continuity is clear, (ii) Applying Lemma 5.3.1 take U = V, W = H and T the canonical injection of V in H, then for all > 0, there exists c > 0 such that |V < u, Ku >V  | ≤ u2V + c u2H ,

u∈V

Suppose a is elliptic, then b(u) + ωu2H = a(u)+V < u, Ku >V  +ωu2H (Inequality (5.10)) ≥ a(u) − u2V − c u2H + ωu2H ≥ a(u) + (ω − c )u2H − u2V (Ellipticity of a) ≥ αu2V − u2V ≥ (α − )u2V

82

(5.10)

Let Ω ⊂ Rd be a bounded open set with Lipschitz boundary Γ := ∂Ω, σ is the (d − 1)− dimensional Hausdorff measure on Γ and B a bounded operator on L2 (Γ). Define the bilinear form aB with domain H 1 (Ω) on L2 (Ω) by   aB (u, v) = ∇u∇vdx + Bu|Γ v|Γ dσ Ω

Γ

The choice of the domain D(aB ) = H 1 (Ω) comes from the fact that B is bounded in L2 (Γ) and the continuity of the embedding H 1 (Ω) → L2 (Γ). Corollary 5.3.3. The billinear form (aB , H 1 (Ω)) is continuous and elliptic. Proof. This follows immediately from Theorem 5.3.2 by taking H = L2 (Ω), V = H 1 (Ω) and K = T r BT r which is a compact operator from H 1 (Ω) to H 1 (Ω) because the operator trace T r : H 1 (Ω) → L2 (Γ) is compact. Let −ΔB be the associated operator of the form aB , and we note etΔB the associated semigroup. It is natural to expect that −ΔB is a realization of the Laplacian in L2 (Ω). Theorem 5.3.4. Let u, f ∈ L2 (Ω). Then u ∈ D(ΔB ) and −ΔB u = f if and only if u ∈ H 1 (Ω), −Δu = f and ∂ν u + Bu = 0 on Γ. Proof. Let u ∈ D(ΔB ) such that −ΔB u = f . Then u ∈ H 1 (Ω) and    f vdx, v ∈ H 1 (Ω) ∇u∇vdx + Buvdσ = Ω

Γ

(5.11)

Ω

We take in particular v ∈ D(Ω) and we obtain −Δu = f . It follows from (5.11) that    ∇u∇vdx + Δuv = − Buvdσ, v ∈ H 1 (Ω) Ω

Ω

Γ

It follows from the integration by parts that ∂ν u + Bu|Γ = 0. Conversely, let u ∈ H 1 (Ω) such that −Δu = f and ∂ν u + Bu|Γ = 0, then    ∇u∇vdx − f v = − Buvdσ, v ∈ H 1 (Ω) Ω

Ω

Γ

which complete the proof. We call ΔB the Laplacian with nonlocal Robin boundary condition. In what remain in this section, we will see that one can write the form aB in a way where the nonlocality appears explicitly. It suffices to use the following Lemma, 83

Lemma 5.3.5. If S is a positive linear operator on L2 (Γ), then there exists a unique positive Radon measure σS on Γ × Γ satisfying the following property: for any Borel functions u, v ∈ L2 (Γ)  (Su|v)L2 (Γ) = u(x)vy)σS (dx, dy) Γ×Γ

If in addition S is Markovian σS (Γ × E) ≤ σ(E) and σS (E × Γ) ≤ σ(E),

∀E ∈ B(Γ)

The proof of the Lemma 5.3.5 can be found in [46, Lemma 1.4.1], which is done for positive symmetric bounded operators, but one can see that the proof work also for not symmetric operators, and in this case the measure σS need not to be symmetric. The following Proposition gives another way to write the form aB for a positive bounded operator B on L2 (Γ), but the same can be done for a negative bounded operator, Proposition 5.3.6. Let B a positive bounded linear operator on L2 (Γ), then there exists a unique positive Radon measure σB on Γ × Γ such that the form aB can be written as follow   ∇u∇vdx + u(x)v(y)σB (dx, dy), u, v ∈ H 1 (Ω) aB (u, v) = Ω

5.3.2

Γ×Γ

Positivity

Let H = L2 (X) where (X, Σ, μ) is a σ−finite measure space. Let (a, V ) be a densly defined, continuous, elliptic form on H with associated semigroup T . The first Beurling-Deny criterion asserts that T is positive (i.e., T (t)L2 (X)+ ⊂ L2 (X)+ for all t ≥ 0) if and only if u ∈ V implies u+ ∈ V and a(u+ , u− ) ≤ 0. Assume that T is positive. Then the second Beurling-Deny criterion asserts that T is L∞ −contractive (i.e., if f ∈ L2 (X) satisfies 0 ≤ f ≤ 1 then 0 ≤ T (t)f ≤ 1 for all t ≥ 0) if and only if 0 ≤ u ∈ V implies u ∧ 1 ∈ V and a(u ∧ 1) ≤ a(u) Now let b be a second densely defined continuous, elliptic form on L2 (X) such that the associated semigroup S is positive. We say that D(a) is an ideal of D(b) if 84

a) u ∈ D(a) implies |u| ∈ D(a) and, b) 0 ≤ u ≤ v, v ∈ D(a), u ∈ D(b) implies u ∈ D(a). Ouhabaz’s domination criterion [69] says that 0 ≤ T (t) ≤ S(t) (t ≥ 0) if and only if D(a) is an ideal of D(b) and b(u, v) ≤ a(u, v) for all u, v ∈ D(a)+ In what follows, we aim to find a characterization of the positivity of the semigroup etΔB . To do this we start with the following Lemma. Lemma 5.3.7. Let C ∈ L(L2 (Γ)). Then (Cu+ |u− )L2 (Γ) ≤ 0 for all u ∈ L2 (Γ) if and only if there exists a positive constant c such that C − cI ≤ 0. Proof. [67, C-II, Theorem 1.11. p:255] Theorem 5.3.8. The following assertions are equivalent to each other: (i) etΔB ≥ 0, (t ≥ 0) (ii) ∃c such that B − cI ≤ 0 on L2 (Γ) Proof. (i) ⇒ (ii) Suppose etΔB ≥ 0 then a(u+ , u− ) ≤ 0 for u ∈ H 1 (Ω). Moreover Ω ∇u+ ∇u− = 0 for u ∈ H 1 (Ω), hence  Bu+ u− dσ ≤ 0, Γ

1

for all u ∈H (Ω) and then for all u ∈ C 1 (Ω). It follows that, for all h ∈ L2 (Γ) one have Γ Bh+ h− dσ ≤ 0. Following Lemma 5.3.7, there exists b ≥ 0 such that B − bI ≤ 0. (ii) ⇒ (i) Suppose that there exists b ≥ 0 such that B − bI ≤ 0, then for u ∈ H 1 (Ω) we have a(u+ , u− ) =



Bu+ u− dσ

Γ = ≤ 0

Γ

(B − bI)u+ u− dσ

85

Remark 5.3.9. The trace operator T r : H 1 (Ω) → L2 (Γ) is continuous and T ru = u|Γ for u ∈ C(Ω) ∩ H 1 (Ω). In particular (T ru)+ = T ru+ for u ∈ C(Ω) ∩ H 1 (Ω). On the other hand, the mapping u → u+ is continuous from H 1 (Ω) into H 1 (Ω). Thus (T ru)+ = T ru+ for all u ∈ H 1 (Ω) Now, we consider the case where d = 1. Let Ω = (0, 1) then Γ = {0, 1} and B = (bij )i,j=1,2 ∈ R2×2 . In this simple situation, one can see easily when the semigroup etΔB ≥ 0 is positive or not. In fact, using Theorem 5.3.8, one can conclude that etΔB ≥ 0 if and only if bij ≤ 0 for i = j. The Example 4.5 of [15] corresponds to the choice   1 1 B= 1 1 and gives, 

1

a(u, v) =

u v  dx + u(0)v(0) + u(1)v(0) + u(0)v(1) + u(1)v(1)

0

fot all u, v ∈ H 1 (0, 1). This example is used in [15] to show that the locality condition in Theorem 4.1 and Corollary 4.2 can not be omitted. The authors used Ouhabaz’s domination criterion to prove that the semigroup etΔB is sandwiched between Dirichlet semigroup and Neumann semigroup, but etΔB is obviously not positive, the situation where Ouhabaz’s domination criterion can not be used, which means that the Example was not convenient. In fact one can prove that any positive semigroup dominated by Neumann semigroup is automatically local. Consider then the situation of Theorem 4.1 in [15]. Let T be a symmetric C0 −semigroup on L2 (Ω) associated with a positive closed form (a, D(a)) such that N 0 ≤ T (t) ≤ etΔ , (t ≥ 0) Thus, we have, simultaneously, for u ∈ D(a) that a(u+ , u− ) ≤ 0 and a(u+ , u− ) ≥ Ω ∇u+ ∇u− dx = 0, which means that a(u+ , u− ) = 0 for all u ∈ D(a) and then that a is local. One can take also, instead of Neumann semigroup, any symmetric C0 − semigroup associated with a positive closed local form (b, D(b).

86

5.3.3

Domination

Proposition 5.3.10. Suppose that etΔB is positive, then 0 ≤ e−tΔD ≤ etΔB ,

(t ≥ 0)

Proof. We use Ouhabaz’s domination criterion. It is clear that H01 (Ω) is an ideal of H 1 (Ω). To prove that e−tΔD ≤ etΔB (t ≥ 0), it suffice to prove that aB (u, v) ≤ aD (u, v) for all u, v ∈ H01 (Ω)+ . One can see easily that,   ∇u∇v + Buvdσ aB (u, v) = Γ Ω = ∇u∇v Ω

= aD (u, v) for all u, v ∈ H01 (Ω)+ . The positivity of e−tΔD is trivial. Proposition 5.3.11. Suppose that etΔB is positive, which means that B − cI ≤ 0 for some c ≥ 0. We have e−tΔD ≤ etΔcI ≤ etΔB ,

(t ≥ 0)

Proof. It is clear by Ouhabaz domination criterion that etΔcI ≤ etΔB (t ≥ 0). In fact, we have B ≤ cI and hence for all u, v ∈ H 1 (Ω)+ ,   ∇u∇v + Buvdσ aB (u, v) = Γ Ω  ≤ ∇u∇v + c uvdσ Ω

≤ acI (u, v)

Γ

It is clear that ΔcI is the Laplacian with Robin boundary condition associated with the measure μ = cσ. From [15, Therem 3.1.] or [1, Theorem 4.6.], we know that e−tΔD ≤ etΔμ (t ≥ 0). Let (Y, Σ, μ) be a σ−finite measure space, we recall the Zaanen’s characterization of bounded multiplication operators on (Lp (Y, μ) and we include a short proof for completeness [14, Proposition 1.7] Proposition 5.3.12. (Zaanen) Let S ∈ L(Lp (Y, μ)), 1 ≤ p ≤ ∞. The following assertions are equivalent. (i) There exists m ∈ L∞ (Y, μ) such that Sf = mf (f ∈ Lp (Y, μ)); (ii) S is local in the sens that (Sf )(x) = 0 a.e. on {x ∈ Y : f (x) = 0} for all f ∈ Lp (Y, μ). 87

Proof. (ii) ⇒ (i) 1. Assume that μ(Y ) < ∞. One can see from (ii) that T (1A f ) = 1A T f for all f ∈ Lp (Y, μ) and for each measurable set A ⊂ Y . Let m = T 1Y . We claim that m ∈ L∞ (Y, μ). Let c > 0. assume that A := {y ∈ Y : |m(y)| > c} has positive measure. Then cμ(A)1/p = c1A p ≤ 1A mp = 1A T 1Y p = T 1A p ≤ T μ(A)1/p Hence c ≤ T . We have  from T (1A f ) = 1A T f that T f = mf for each simple function f = ni=1 ci 1Ai , Ai ∈ Σ. Since simple functions are dense in Lp (Y, μ), the claim (i) follows. 2. Since (Y, μ) is σ−finite, there exists h ∈ L1 (Y, m) such that h(y) > f defines an isomorphism from Lp (Y, μ) onto 0 a.e. Then f → h1/p p L (Y, hμ). Now the claim follows from 1. This completes the proof and the converse is obvious.

Proposition 5.3.13. If there exists c ≥ 0 such that 0 ≤ B ≤ cI, then there exists 0 ≤ m ∈ L∞ (Γ) such that B = mI. Proof. A direct application of Zaanen’s characterization. One can see then that when 0 ≤ B ≤ cI, aB = aμ where μ = mσ which is the form associated with Robin Laplacian. The form aμ is evidently a local form. It is well known also that in this case we have that e−tΔD ≤ etΔμ ≤ e−tΔN ,

(t ≥ 0)

Now, we deal with the case where B ≤ 0, Proposition 5.3.14. If B ≤ 0, then (i) we have 0 ≤ etΔB  e−tΔN ,

(t ≥ 0)

(ii) In addition, 0 ≤ etΔB ≤ e−tΔN if and only if B = 0 Proof. Let u, v ∈ H 1 (Ω)+ then



aB (u, v) − aN (u, v) =

Γ

Buvdσ ≥ 0

which means that etΔB  e−tΔN except when B = 0. 88

Let B1 and B2 two bounded operators in L2 (Γ). We write B1 ≥ B2 if and only if B1 − B2 ≥ 0 Proposition 5.3.15. Let B1 and B2 two bounded operators in L2 (Γ) such that B1 ≥ B2 , suppose, in addition, that etΔB1 is positive, then e−tΔD ≤ etΔB1 ≤ etΔB2 ,

(t ≥ 0)

Proof. Again using Ouhabaz’s domination criterion, we have that aB1 (u, v) ≥ aB2 (u, v) for all u, v ∈ H 1 (Ω)+ .

89

90

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