Parabolic problems with general Wentzell boundary conditions and

arXiv:1504.05363v1 [math.AP] 21 Apr 2015

Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary Davide Guidetti Dipartimento di matematica, Piazza di Porta S. Donato 5, 40126 Bologna (Italy) e-mail: [email protected] Abstract We show a result of maximal regularity in spaces of H¨ older continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.

AMS Subject classification: 35K15, 35K20 Keywords:Maximal regularity, Dynamic boundary conditions; Wentzell boundary conditions; Diffusion term on the boundary.

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Introduction

In this paper we want to study second order parabolic systems in the forms  Dt u(t, x) = A(t, x, Dx )u(t, x) + f (t, x), (t, x) ∈ [0, T ] × Ω,      Dt u(t, x′ ) + B(t, x′ , Dx )u(t, x′ ) − L(t)(u(t, ·)|Γ )(x′ ) = h(t, x′ ), (t, x′ ) ∈ [0, T ] × ∂Ω,      u(0, x) = u0 (x), x ∈ Ω.

and  Dt u(t, x) = A(t, x, Dx )u(t, x) + f (t, x), (t, x) ∈ [0, T ] × Ω,      A(t, x′ , Dx )u(t, x′ ) + B(t, x′ , Dx )u(t, x′ ) − L(t)(u(t, ·)|Γ )(x′ ) = h(t, x′ ), (t, x′ ) ∈ [0, T ] × ∂Ω,      u(0, x) = u0 (x), x ∈ Ω.

(1.1)

(1.2)

Here for every t ∈ [0, T ], A(t, x, Dx ) is a second order linear strongly elliptic operator in the open, bounded subset Ω of Rn , L(t) is a second order linear strongly elliptic tangential operator in ∂Ω, B(t, x′ , Dx ) is a first order (not necessarily tangential) operator in ∂Ω. It is clear that, at least formally, (1.1) and (1.2) are strictly related. A large amount of papers has been devoted to parabolic problems with dynamic and Wentzell boundary conditions in the form (1.1)-(1.2) in the case that the summand L(t)(u(t, ·)|Γ ) does not appear. We refer to the bibliographies in [6] and [8]. In our knowledge, a problem in the form (1.1) was introduced for the first time in [10], in the particular case that A(t, x, Dx ) = α(x)∆x , with α positively valued, B(t, x′ , Dx ) = b(x′ )Dν , with ν unit normal vector to ∂Ω, pointing outside Ω, L(t) = a(x)∆LB u, where

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we indicate with ∆LB the Laplace-Beltrami operator. [10] contains a physical interpretation of the problem: briefly, a heat equation with a heat source on the boundary, that depends on the heat flow along the boundary, the heat flux across the boundary and the temperature at the boundary. The first paper where a problem in the form (1.2) is really studied seems to be [3]. In it it was considered the system  Dt u(t, x) = Au(t, x) = ∇ · (a(x)∇u)(t, x), (t, x) ∈ (0, T ) × Ω,      Au(t, x′ ) + β(x′ )DνA u(t, x′ ) + γ(x′ ) − q∆LB u(t, x′ ) = 0, (t, x′ ) ∈ (0, T ) × ∂Ω, (1.3)      u(0, x) = u0 (x),

with , A strongly elliptic, β(x′ ) > 0 in ∂Ω, DνA conormal derivative, q ∈ [0, ∞). It is proved that, if 1 ≤ p ≤ ∞ the closure a suitable realisation of the problem in the space Lp (Ω × ∂Ω) (1 ≤ p ≤ ∞), gives rise to an analytic semigroup (not strongly continuous if p = ∞). The continuous dependence on the coefficients had already been considered in [2]. In [1] the authors generalised some of the results in [3], considered also the case that the first equation in (1.3) is the telegraph equation (with two initial conditions) and studied the asymptotic behaviour of solutions. In [12] the author considered the case of a domain Ω with merely Lipschitz boundary, with a strongly elliptic operator A (independent of t). It was shown that a realisation of A with the general boundary condition (Au)|∂Ω − γ∆LB u + DνA u + βu = g in ∂Ω generates a strongly continuous compact semigroup in C(Ω). Semilinear problems were studied in [13] and [14]. Finally, in the paper [11] the authors treated (1.1) in the particular case A(t, x, Dx ) = ∆x , f ≡ 0, h ≡ 0, L(t) = l∆LB with l > 0 and B(t, x′ , Dx ) = kDν , with k which may be negative (in contrast with the previously quoted literature). They showed that, if the initial datum u0 is in H 1 (Ω) and u0|∂Ω ∈ H 1 (Γ), (1.1) has a unique solution u in C([0, ∞); H 1 (Ω)) ∩ C 1 ((0, ∞); H 1 (Ω)) ∩ C((0, ∞); H 3 (Ω)), with u|Γ in C([0, ∞); H 1 (Γ)) ∩ C 1 ((0, ∞); H 1 (Γ)) ∩ C((0, ∞); H 3 (Γ)). The main aim of this paper is to show that, in a suitable functional setting, the role of the operator B(t, x′ , Dx ) in (1.1) and (1.2) is minor, in the sense that these equations can be treated as perturbations of the corresponding problems with B(t, x′ , Dx ) ≡ 0. In fact, we shall see that B(t, x′ , Dx ) may be, apart some limitations on the regularity of its coefficients, an arbitrary first order linear differential operator. Moreover we shall consider problems with coefficients depending on t and we shall obtain results of maximal regularity, that is, results establishing the existence of linear and topological isomorphisms between classes of data and classes of solutions. Following the lines of [7] and [8], we shall work in spaces of H¨ older continuous functions. Now we are going to state our main results. We begin by introducing the following assumptions: (A1) Ω is an open, bounded subset of Rn (n ∈ N, n ≥ 3), lying on one side of its topological boundary Γ, which is a compact submanifold of Rn of class C 2+β , for some β ∈ (0, 1). P (A2) A(t, x, Dx ) = |α|≤2 aα (t, x)Dxα , with aα ∈ C β/2,β ((0, T ) × Ω); in case |α| = 2, aα is real valued P and there exists ν > 0 such that, ∀(t, x) ∈ [0, T ] × Ω, ∀ξ ∈ Rn , |α|=2 aα (t, x)ξ α ≥ ν|ξ|2 (we shall briefly say that A(x, Dx ) is strongly elliptic). P (A3) B(t, x′ , Dx ) = |α|≤1 bα (t, x′ )Dxα , with bα ∈ C β/2,β ((0, T ) × Γ).

(A4) ∀t ∈ [0, T ] L(t) is a second order, partial differential operator in Γ. More precisely: for every local chart (U, Φ), with U open in Γ and Φ C 2+β − diffeomorphism between U and Φ(U ), with Φ(U ) open in Rn−1 , ∀v ∈ C02+β (Γ) with compact support in U , X lα,Φ (t, x′ )Dyα (v ◦ Φ−1 )(Φ(x′ )); L(t)v(x′ ) = |α|≤2

we suppose, moreover, that, if |α| = 2, lα,Φ is real valued, for every open subset V of U , with V ⊂⊂ β/2,β U ), and there exists ν(V ) positive such that, ∀(t, x′ ) ∈ V , ∀η ∈ Rn−1 , P, lα,Φ|V ∈ C ′ α ((0, T ) × V 2 |α|=2 lα,Φ (t, x )η ≥ ν(V )|η| . We want to prove the following

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Theorem 1.1. Suppose that (A1)-(A4) are fulfilled. Then the following conditions are necessary and sufficient in order that (1.1) have a unique solution u belonging to C 1+β/2,2+β ((0, T ) × Ω): (a) f ∈ C β/2,β ((0, T ) × Ω); (b) h ∈ C β/2,β ((0, T ) × Γ); (c) u0 ∈ C 2+β (Ω); (d) A(0, x′ , Dx )u0 (x′ ) + f (0, x′ ) = −B(0, x′ , Dx )u0 (x′ ) + L(0)(u0|Γ )(x′ ) + h(0, x′ ), ∀x′ ∈ Γ. Theorem 1.2. Suppose that (A1)-(A4) are fulfilled. Then the following conditions are necessary and sufficient in order that (1.2) have a unique solution in C 1+β/2,2+β ((0, T ) × Ω): (a) f ∈ C β/2,β ((0, T ) × Ω); (b) h ∈ C β/2,β ((0, T ) × Γ); (c) u0 ∈ C 2+β (Ω); (d) A(0, x′ , Dx )u0 (x′ ) + B(0, x′ , Dx ) − L(0)(u0|Γ )(x′ ) = h(0, x′ ), ∀x′ ∈ Γ. Now we are going to describe the organisation of the paper. We begin by considering in Section 2 the parabolic problem   Dt g(t, x′ ) = Lg(t, x′ ) + h(t, x′ ), (t, x′ ) ∈ (0, T ) × Γ, (1.4)  g(0, x′ ) = g0 (x′ ), x′ ∈ Γ, with L strongly elliptic in Γ. We do not impose the variational form of L. We show that the operator L, defined as   D(L) = {g ∈ ∩1≤p 0), α which can be again defined by local charts. In Section 2 we shall employ Besov spaces B∞,∞ (Γ), with α ∈ [0, 2]. We shall need the following facts, which can be easily deduced from analogous statements in Rn−1 (see [5]):

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Lemma 1.4. Let Γ be a compact submanifold of Rn , of class C 2+β , for some β > 0. Then θ (I) if θ ∈ [0, 2], C θ (Γ) ⊆ B∞,∞ (Γ); θ (II) in case θ 6∈ Z, C θ (Γ) = B∞,∞ (Γ); 1 (III) ∀θ ∈ (0, 1) \ { 2 }, 0 2 2θ (C(Γ), C 2 (Γ))θ,∞ = (B∞,∞ (Γ), B∞,∞ (Γ))θ,∞ = B∞,∞ (Γ) = C 2θ (Γ).

(a) (b) in any case, We shall employ the following version of the continuation principle: Proposition 1.5. Let X, Y be Banach spaces and L ∈ C([0, 1]; L(X, Y )). Assume the following: (a) there exists M ∈ R+ , such that, ∀x ∈ X, ∀ǫ ∈ [0, 1], kxkX ≤ M kL(ǫ)xkY ; (b) L(0) is onto Y . Then, ∀ǫ ∈ [0, 1] L(ǫ) is a linear and topological isomorphism between X and Y .

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Parabolic problems in Γ

In this section, we study the parabolic system (1.4). We introduce the following as (A4’). L is a second order, partial differential operator in Γ. More precisely: for every local chart (U, Φ), with U open in Γ, ∀v ∈ C02+β (Γ) with compact support in U , X lα,Φ (x′ )Dyα (v ◦ Φ−1 )(Φ(x′ )); Lv(x′ ) = |α|≤2

we suppose, moreover, if |α| = 2, lα,Φ is real valued, for every open subset V of U , with V ⊂⊂ U , lα,Φ|V ∈ P C β/2,β ((0, T ) × V ), and, there exists ν(V ) positive such that, ∀x′ ∈ V , ∀η ∈ Rn−1 , |α|=2 lα,Φ (x′ )η α ≥ ν(V )|η 2 . We begin by considering the elliptic system depending on the parameter λ λg(x′ ) − Lg(x′ ) = h(x′ ),

x′ ∈ Γ.

(2.1)

We shall prove the following Theorem 2.1. Suppose that (A1) and (A4) hold. Then: (I) for every φ0 ∈ [0, Tπ) there exists R(φ0 ) > 0 such that, if |Arg(λ)| ≤ φ0 and |λ| ≥ R(φ0 ), (1.4) has a unique solution g in 1≤p 0, depending on φ0 and γ, such that kgkC γ (Γ) ≤ C|λ|−1+γ/2 khkC(Γ); (III) if h ∈ C β (Γ), g ∈ C 2+β (Γ). Proof. We take an arbitrary x0 ∈ Γ and consider a local chart (U, Φ) around x0 , with U open subset of Γ and Φ diffeomorphism between U and Φ(U ), open subset in Rn−1 . We introduce in Φ(U ) the strongly elliptic operator B ♯ , L♯ v(y) := L(v ◦ Φ)(Φ−1 (y)), y ∈ Φ(U ). By shrinking U (if necessary), we may assume that the coefficients of L♯ are in C β (Φ(U )) and are extensible to elements lβ in C β (Rn ), in such a way that the operator which we continue to call L♯ = P β n |α|≤2 lβ (y)Dy is strongly elliptic in R . Now we consider the problem λv(y) − L♯ v(y) = k(y), 5

y ∈ Rn−1 .

(2.2)

Then, (see Chapter 3 in [9]), for every φ0 ∈ [0, π) there 1 (φ0 ) > 0 such that, if |Arg(λ)| ≤ φ0 T exist R2,p and |λ| ≥ R1 (φ0 ), (2.2) has a unique solution v in 1≤p 0, depending on φ0 and γ, such that kvkC γ (Rn−1 ) ≤ C1 (φ0 , γ)|λ|−1+γ/2 kkkC(Rn−1) ; finally, if k ∈ C β (Rn−1 ), v ∈ C 2+β (Rn−1 ). Now we fix U1 open subset of U , with U1 contained in U , x0 ∈ U1 and φ ∈ C 2+β (Γ), with compact support in U , φ(x) = 1 ∀x ∈ U1 . For every h ∈ C(Γ), we indicate with k the trivial extension of (φh) ◦ Φ−1 to Rn−1 . If λ is such that (2.2) is uniquely solvable for every k in C(Rn−1 ), we set [S(x0 , λ)h](x) := φ(x)v(Φ(x)),

x ∈ Γ,

(2.3)

with v solving (2.2). We observe that 2 (α1 ) : S(x0 , λ)h ∈ ∩1≤p