Geometric regularity estimates for fully nonlinear elliptic equations with ...

Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries by J.V. da Silva∗,

R.A. Leitão,†

G.C. Ricarte‡

Abstract In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators (which can be either degenerate or singular when “the gradient is small”) under strong absorption condition of the general form: (0.1)

G(x, Du, D2 u) = f (u)χ{u>0}

in

Ω,

where the mapping u 7→ f (u) fails to decrease fast enough at the origin, so allowing that non-negative solutions may create plateau regions, that is, a priori unknown subsets where a given solution vanishes κ regularity along the set F = ∂ {u > 0} ∩ Ω (the free identically. We establish improved geometric Cloc 0 boundary of the model). The sharp value of κ 1 is obtained explicitly, and depends only on structural parameters. Non-degeneracy, finiteness of the (N − 1)-Hausdorff measure of the free boundary and among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings. The results proven in this article are new even for simple equation |Du|γ ∆u = λ0 uµ χ{u>0} , for certain constants λ0 > 0, µ ∈ [0, γ + 1) and 0 6= γ > −1. Finally, this work extends previous ones by allowing deal with more general classes of operators, making use of different methods. Keywords: Fully nonlinear elliptic operators of degenerate/singular type, sharp and improved regularity estimates, Hausdorff measure estimates. AMS Subject Classifications: 35B65, 35J60, 35J70.

1 1.1

Introduction Statement of main results

In this manuscript we study regularity issues for reaction-diffusion problems governed by second order nonlinear elliptic equations (possibly of degenerate or singular type) for which a Minimum Principle is not available (see [8] for an essay on this topic): ( F(x, Du, D2 u) + |Du|γ hb(x), Dui = f (u)χ{u>0} in Ω (1.1) u(x) = g(x) on ∂ Ω, ∗ J OÃO V ÍTOR DA S ILVA . Universidad de Buenos Aires. FCEyN, Department of Mathematics. Ciudad Universitaria-Pabellón I-(C1428EGA) - Buenos Aires, Argentina. E-mail address: [email protected] † R AIMUNDO A LVES L EITÃO J ÚNIOR . Universidade Federal Ceará - UFC. Department of Mathematics. Fortaleza - CE, Brazil 60455-760. E-mail address: [email protected] ‡ G LEYDSON C HAVES R ICARTE . Universidade Federal Ceará - UFC. Department of Mathematics. Fortaleza - CE, Brazil - 60455760. E-mail address: [email protected]

1

R EGULARITY ESTIMATES FOR DEGENERATE / SINGULAR TYPE EQUATIONS

2

where Ω ⊂ RN is a smooth, open and bounded domain, 0 ≤ g ∈ C0 (∂ Ω), b ∈ C0 (Ω, RN ), f is a continuous and increasing function with f (0) = 0 and F : Ω × (RN \{0}) × Sym(N) → R is a second order fully nonlinear elliptic operator satisfying: −p ) ∈ Ω × (F1) [(λ , Λ, γ)-Ellipticity condition] There exist constants Λ ≥ λ > 0 such that for any (x, → N (R \ {0}) and M, P ∈ Sym(N), with P ≥ 0 and γ > −1 there holds −p , M + P) − F(x, → −p , M) ≤ |→ −p |γ P + (P), −p |γ P − (P) ≤ F(x, → |→ λ ,Λ λ ,Λ where Pλ±,Λ denote the Pucci’s extremal operators: Pλ+,Λ (M) := λ ·

∑ ei + Λ · ∑ ei

ei 0

Pλ−,Λ (M) := λ ·

∑ ei + Λ · ∑ ei

ei >0

ei 0} (x) in

Ω,

where 0 ≤ µ < γ + 1 is the absorption factor and λ0 ∈ C0 (Ω) (Thiele modulus - bounded away from zero and infinity). Such models are mathematically interesting because they enable the formation of dead-core sets, i.e. a priori unknown regions where non-negative solutions vanish identically. By way of contrast, the study of (1.2) is relevant, not only for its several applications, but specially for its innate relation with a number of relevant free boundary problems in the literature (cf. [1], [6], [7], [27, Chapter 1], [29], [32] and [34] for some variational examples and [15], [16], [37] and [40] for a non-variational counterpart). For this reason, understanding the “geometry” of the former model is an important step in comprising the behaviour of dead-core solutions near their free boundary points. We will provide a brief discussion of this soon. In our first result we establish the precise asymptotic behavior at which non-negative viscosity solutions leave their dead-core sets. This is an important piece of information in several free boundary problems (cf. [3], [4], [32], [35] and [38]), and it plays a pivotal role in establishing many weak geometric properties. Theorem 1.1 (Non-degeneracy). Let u be a nonnegative, bounded viscosity solution to (1.2) in B1 and let x0 ∈ {u > 0} ∩ B 1 be a point in the closure of the non-coincidence set. Then for any 0 < r < 21 , there holds 2

γ+2 sup u(x) ≥ C] N, inf λ0 , γ, µ .r γ+1−µ . Br (x0 )

Ω

In our next result we establish a sharp and improved regularity estimate at free boundary points.


3

Theorem 1.2 (Improved regularity along free boundary). Let u be a nonnegative and bounded viscosity 0 0 solution and n to (1.2) o consider Y0 ∈ ∂ {u > 0} ∩ Ω a free boundary point with Ω b Ω. Then for r0 dist(Ω0 ,∂ Ω) min 1, and any X ∈ Br0 (Y0 ) ∩ {u > 0} there holds 2 γ+2

u(X) ≤ C] .kukL∞ (Ω) .|X −Y0 | γ+1−µ , where C] > 0 depends only on N, λ , Λ, γ, µ, inf λ0 , kbkL∞ (Ω) , kλ0 kL∞ (Ω) and dist(Ω0 , ∂ Ω). Ω

To the best of our knowledge and as far as degenerate/singular elliptic models in non-divergence form are concerned, Theorem 1.2 is a novelty even for dead-core problems as follows |Du|γ Tr(A(x)D2 u) + hb(x), Dui = λ0 (x)uµ χ{u>0} for 0 6= γ > −1. The insights behinds the proof of Theorem 1.2 one resume to three important steps: a (finer) asymptotic blow-up analysis, a precise control of the growth rate of solutions close to their free boundaries and the use of a sharp a priori bounds device (Harnack type inequality). Heuristically, for an appropriate family of normalized and scaled solutions, if the “magnitude” of the Thiele modulus is under control (with small enough bounds), then we must expect the following iterative geometric decay sup u(x) ≤ C0 N, λ , Λ, µ, γ, λ0 , kbkL∞ (Ω) B 1 (x0 )

1 2k

γ+2 γ+1−µ

∀ x0 ∈ ∂ {u > 0} ∩ Ω0 and k ∈ N,

2k

which will yield the desired geometric regularity estimate along free boundary points (see Lemma 4.3). As one of consequences of our findings, we are able to establish a finer control for any viscosity solution to (1.2) close its free boundary. Such a kind of information is crucial in a number of quantitative features for many free boundary problems (cf. [5], [22], [35] and [40]). Precisely, we prove that (near their free boundaries) solutions are “trapped” between the graph of two suitable multiples of dist(·, ∂ {u > 0}). Theorem 1.3. Let u be a non-negative, bounded viscosity solution to (1.2) in Ω. Given x0 ∈ {u > 0} ∩ Ω0 with Ω0 b Ω, then γ+2

γ+2

c] dist(x0 , ∂ {u > 0}) γ+1−µ ≤ u(x0 ) ≤ c] dist(x0 , ∂ {u > 0}) γ+1−µ , where c] , c] > 0 are universal constants1 . We also able to provide a crystal clear answer as to what will happen with solutions of (1.2) when µ = γ + 1. It is worthwhile to mention that in such a scenery, the former regularity estimates “collapse”. For this reason, analysing such a borderline setting is a delicate and challenging task. Theorem 1.4. Let u be a non-negative, bounded viscosity solution to (1.2) with µ = γ + 1. Then, the following dichotomy holds: either u > 0 or u ≡ 0 in Ω. We also find the sharp (and improved) rate which gradient decays at interior free boundary points. Theorem 1.5 (Sharp gradient decay). Let u be a bounded non-negative viscosity solution to (1.2). Then, for any point z ∈ ∂ {u > 0} ∩ Ω0 for Ω0 b Ω, there exists a universal constant C > 0 such that 1+µ dist(Ω0 , ∂ Ω) . sup |Du(x)| ≤ Cr γ+1−µ for all 0 < r min 1, 2 Br (z) 1 Throughout

this manuscript, we will refer to universal constants when they depend only on dimension and structural properties of the problem, i.e. on N, λ , Λ, γ, µ and the bounds of λ0 and kbkL∞ (Ω)


4

In the following, we present a sharp Liouville type result, which holds provided that entire solutions satisfy a controlled growth condition at infinity (compare with [9, Theorem 3.1]). Theorem 1.6 (Liouville). Let u be a non-negative viscosity solution to (1.2) in RN . Then, u ≡ 0 provided that " # 1 γ+1−µ u(x) (γ + 1 − µ)γ+2 (1.3) lim sup < inf λ0 (x) . γ+2 RN N (µ + 1) (γ + 2)γ+1 |x|→∞ |x| γ+1−µ Finally, our last main result concerns the finiteness of (N − 1)-Hausdorff measure of the free boundary. Theorem 1.7 (Hausdorff measure estimates). Let u be a viscosity solution to (1.2) in B1 with F a “concave” operator. There exists a universal constant C > 0 such that for all x0 ∈ ∂ {u > 0} ∩ B 1 , there holds 2

H N−1

∂ {u > 0} ∩ Bρ (x0 ) ≤ Cρ N−1 ,

for all ρ 1, where H N−1 is the (N − 1)-dimensional Hausdorff measure. Up to our knowledge, there has been no prior mathematical results regarding such Hausdorff measure estimates to degenerate/singular (in non-divergence form) models like (1.2) (compare with [20, Theorem 1.3], [24, Theorem 7.2], [32, Theorem 3.3] and [34] and therein references).

1.2

Motivation and first insights into the theory

Throughout the last decades a number of reaction-diffusion elliptic equations have emerged as models of many phenomena coming from pure and applied sciences, where some remarkable examples appear in chemical reactions, physical-mathematical phenomena, biological processes and population dynamics just to name a few. Notwithstanding, reaction-diffusion processes with one-phase transition, i.e. with sign constraint, are often more interesting from the applied point of view, since they constitute the only significant (realistic) situation. An illustrative example coming from certain (stationary) isothermal, and irreversible catalytical reaction-diffusion process (cf. [6], [7] and [27, Chapter 1] for remarkable surveys) is ( −∆u(x) + λ0 . f (u)χ{u>0} = 0 in Ω (1.4) u(x) = 1 on ∂ Ω, where Ω ⊂ RN is a regular and bounded domain, f : R+ → R+ is a continuous and increasing reaction term, with f (0) ≥ 0. For such a model, u stands the concentration of a certain chemical reagent (or gas) under a prescribed isothermal flow on the boundary. Furthermore, f (u) represents the ratio of reaction rate at concentration u to reaction rate at concentration unity, and λ0 > 0 (the Thiele modulus) controls the ratio of reaction rate to diffusion-convection rate. Recall that when f ∈ C0,1 (R+ ), it follows from the Maximum Principle that nonnegative solutions of (1.4) must be strictly positive. However, if f fails to be Lipschitz (or even not decays fast enough at the origin), e.g. as f (t) = t q with q ∈ [0, 1), then nonnegative solutions may exhibit plateaus zones, also known as dead core sets, i.e. regions of positive measure Ω0 b Ω where solutions vanish identically. From a physical-mathematical point of view, such a phenomenon revels that where such a solution is vanishing, it delineates a region (a priori unknown) where no diffusion process is present (i.e. the chemical substance is “wasted”). Such a feature enable us to treat (1.4) as a problem with free boundaries. In a heuristic analysis, the set ∂ {u > 0} ∩ Ω (the free boundary) represents the “phase-transition for diffusivity of model”, and writing (1.4) as f −1 (u).∆u = λ0 .χ{u>0}

in

Ω


5

hints that the blowing-up rate of f −1 (u) and the order in which ∆u vanishes along the free boundary points are strongly related. For this reason, the proper understanding of such a (mathematical) balance should yield decisive geometric information about solutions and their free boundaries. Now, let us move towards the theory of non-divergence form elliptic equations of degenerate/singular type. Recall that solutions to such PDEs in general have a lack of good regularity estimates (e.g. pLaplacian operator). Indeed, several mathematical models involving such operators have their “ellipticity factor” collapsing along a unknown set, the free boundary. Particularly, this phenomenon implies less diffusivity for the model near such a set, and consequently regularity properties of solutions one become, not quite so simple to be established. A typical case of such a phenomenon occurs as the governing operator is anisotropic and the “modulus of ellipticity” degenerates along the set of critical points of solution, i.e. n → −o CΩ (u) := x ∈ Ω : Du(x) = 0 . An interesting and simple example is given by model G(x, Du, D2 u) = g(x, |Du|)Tr(A(x)D2 u) = f(u) in Ω, where f : R → [0, ∞) and g : Ω × [0, ∞) → [0, ∞] are continuous functions with g(x, 0) = 0 (or g(x, 0) = +∞), A ∈ C0 (Ω, R2N ) (uniformly elliptic matrix) and u 7→ f(u) is non-decreasing with f(s) = 0 for every s ≤ 0. Notice that solutions for such a model might only be “irregular” if |Du| is small or large enough. So the delicate question here consists of understanding the precise (local) behavior of solutions close theirs two free boundaries, namely the physical free boundary ∂ {u > 0} and the “non-physical” one ∂ {|Du| > µ 0} when such a model presents “absorption factor” ( f (t) ≈ t+ ) lesser than the homogeneity degree of governing operator (e.g. as g(x, s) ≈ sγ ), so enabling the absence of strong maximum principle (compare with [8] for the validity of the strong maximum principle). Therefore, the understanding of such an interplay is fundamental in order to study (finer) regularity issues to (1.2). Despite of the fact that there is a huge amount of literature on dead-core problems in divergence form and theirs qualitative features, quantitative counterparts for non-variational elliptic models like (1.2) are far less studied due to the rigidity of the structure of such operators (cf. [3], [20] and [38] as enlightening examples). Therefore, the treatment of such free boundary problems requires the development of new ideas and modern techniques. This lack of investigation has been our main impetus in studying fully nonlinear models with non-uniformly elliptic (anisotropic) structure under strong absorption conditions, which focus on a modern, systematic and non-variational approach for a general class of problems with free boundaries.

1.3

An overview on our results and their connections with other theories

Our analysis relies strongly on methods from the so-termed Geometric Regularity Theory: an approach to regularity theory born in the fully nonlinear setting, with the Caffarelli’s seminal work (cf. [13], see also [14]). Currently, Teixeira’s outstanding survey [36] summarizes the state of the art for such techniques and methods coming from Geometric Measure Theory, Harmonic Analysis, Nonlinear Analysis and PDEs, and Free boundary problems (see also [2], [4], [19], [25], [37], [39] for further current examples this subject). It is worth highlighting that our results generalizes and/or recovers and/or complements previous ones (see [38] (dead-core problems with uniformly elliptic structure), [36, Section 4] (p−dead core problems) and [40] (quenching type problems), see also [3]) by allowing deal with more general classes of operators. Different from the former results, we suggest completely different approaches and techniques to address such regularity issues, which enable us to establish a number of significant applications for our findings. Now, let us connect our results with certain equations from the theory of superconductivity. For that end, consider the simple prototype G(x, Du, D2 u) = |Du|γ F(x, D2 u), where γ > 0 and F : Ω × Sym(N) → R


6

is a (λ , Λ)−elliptic operator. In face of (1.2) we observe that {|Du| = 0} ∩ Ω ⊂ {u = 0} ∩ Ω. For this reason, instead of (1.1) we may consider the most realistic accuracy (physical-mathematical) model |Du|γ F(x, D2 u) = f (u)χ{|Du|>0} (x),

(1.5)

which can be naturally thought, to some extent, as a more general prototype (with the adding of a degeneracy factor) in the theory of superconductivity (i.e. as γ = 0), where fully nonlinear equations with patches of zero gradient drives the mathematical model. Recall (cf. [15], [16] and [17]) that (1.5) (for γ = 0) stands the stationary situation for the mean field theory of superconductivity (vortices), provided the scalar stream function admits a functional dependence on the scalar magnetic potential. In this scenery, it was proved (see [15, Corollary 7]) that viscosity solutions to (1.5) are merely C0,α for some α ∈ (0, 1). Moreover, under the concavity assumption on F was proved (cf. [15, Corollary 8]) that solutions are in W 2,p . Nevertheless, even 2

for γ = 0, our results (see Theorem 1.2) are striking, because in this setting we obtain C 1−µ local regularity along free boundary points, which represents an unexpected gain of smoothness along gradient degenerate points (compare also with [4, Theorem 4.1] and [37, Theorem 1 and Theorem 3]). Thanks to Theorem 1.2 (resp. Theorem 1.5) we are able to access better regularity estimates (at free boundary points) than those currently available. As a matter of fact, in our approach we impose just continuous coefficients for the governing operator F. Nevertheless, the modulus of continuity improves upon the expected Hölder regularity coming from Krylov-Safonov type estimates (cf. [26] and [30]). Furthermore, even for constant coefficient problems, F(Du, D2 u) = f (x, u) ∈ L∞ , our result is remarkable, because in this n o − 1 1,min αH , 1+γ

scenery Cloc -estimates are the best expected regularity (cf. [4], [10], [11], [12], [13], [14] and [31]). By way of illustration, taking into account the model G(x, Du, D2 u) = |Du|γ F(x, D2 u), with F a con1, 1

cave/convex operator, it was proved in [4, Corollary 3.2] sharp Clocγ+1 regularity estimates for degenerate elliptic equations. Nevertheless, Theorem 1.2 (resp. Theorem 1.5) presents a sharp and improved modulus of continuity (along free boundary points), i.e. γ +2 1 1+µ 1 κ(γ, µ) = ≥ 1+ resp. ≥ (sharp and improved exponents). γ +1−µ γ +1 γ +1−µ γ +1 We must also compare our estimates with ones coming from classical Schauder theory. By way of simpleness, let us suppose that F(x, Du, D2 u) = Tr(A(x)D2 u), for A ∈ C0,1 (Ω, Sym(N)) and λ0 > 0 a 2,µ constant. Notice that λ0 uµ ∈ C0,µ (Ω). Therefore, the Schauder estimates imply that u ∈ Cloc (Ω) (see [14], and [19] and therein references). Notwithstanding, the estimates from Theorem 1.2 says us that u has 2

ω ∗ (s) = s 1−µ as modulus of continuity at points on the free boundary. Finally, a surprising fact is: κ(0, µ) =

2 > 2+µ 1−µ

for any

µ ∈ (0, 1),

i.e. we get improved estimates (at free boundary points) when compared with classical Schauder ones. Our paper is organized as follows: in Section 2 we present a appropriate notion of viscosity solutions to our context. Yet in Section 2, we deliver few results about a priori estimates, comparison and existence of dead core solutions. Section 3 is devoted to prove some sharp non-degeneracy results, in particular Theorem 1.1. In Section 4, we prove a central result, namely Lemma 4.4, which allows us to place solutions in a flatness improvement regime. Yet is Section 4 we deliver a proof of Theorem 1.2. At the end of Section 4, we analyse the borderline case, i.e., Theorem 1.4. In Section 5 is dedicated to applications of the our main results, e.g. Theorem 1.3. Moreover, some weak geometric properties such as uniform positive density and porosity of the free boundary are established. In Section 6 we prove our Liouville type result, Theorem 1.6. In Section 7, we study the (N − 1)-Hausdorff measure estimates for a certain class of operators.


2

7

Background results

Let us review the definition of viscosity solution for our operators. For G : Ω × (RN \ {0}) × Sym(N) → R and f : Ω × R → R continuous functions we have the following definition. Definition 2.1 (Viscosity solutions). u ∈ C0 (Ω) is a viscosity super-solution (resp. sub-solution) to G(x, Du, D2 u) = f (x, u) in Ω if for every x0 ∈ Ω we have the following 1. Either ∀ φ ∈ C2 (Ω) such that u − φ has a local minimum at x0 and |Dφ (x0 )| 6= 0 holds G(x0 , Dφ (x0 ), D2 φ (x0 )) ≤ f (x0 , φ (x0 )) (resp. ≥ f (x0 , φ (x0 ))) 2. Or there exists an open ball B(x0 , ε) ⊂ Ω, ε > 0 where u is constant, u = K and holds f (x, K) ≥ 0 ∀ x ∈ B(x0 , ε) (resp. f (x, K) ≤ 0) Finally, u is said to be a viscosity solution if it is simultaneously a viscosity super-solution and a viscosity sub-solution. Hereafter, we shall adopt the following notation S(r,x0 ) [u] := sup u(x)

I(r,x0 ) [u] := inf u(x).

and

Br (x0 )

Br (x0 )

Moreover, we will omit the center of the ball as x0 = 0. The following Harnack inequality will be an important tool for our arguments. Theorem 2.2 (Harnack inequality, [30, Theorem 7]). Let u be a non-negative viscosity solution to G(x, Du, D2 u) = f ∈ C0 (B1 ) ∩ Lq (B1 ), q > N. Then, S 1 [u] ≤ C N, γ, q, λ , Λ, kbkL∞ (Ω)

2

1 I 1 [u] + k f kLγ+1 q (B ) . 2

1

The next result plays a fundamental role in obtaining sharp and improved estimates along free boundary points of solutions. Such a result can be found in the references [4], [10], [11], [12], [13], [14] and [31]. Theorem 2.3 (Gradient estimates). Let u be a bounded viscosity solution to G(x, Du, D2 u) = f ∈ L∞ (B1 ). Then, S 1 [|Du|] ≤ C N, γ, λ , Λ, kbkL∞ (Ω) 2

i h 1 S1 [u] + S1 [| f |] γ+1 .

The next result is pivotal in order to prove existence of viscosity solutions for our problem, as well as in proving some weak geometric properties soon. The proof holds the same lines as [3, Lemma 3.2] (see also [9, Theorem 1.1]). For this reason, we will omit the proof here.


8

Lemma 2.4 (Comparison Principle). Let u1 and u2 be continuous functions in Ω and f ∈ C0 ([0, ∞)) increasing with f (0) = 0 fulfilling G(x, Du1 , D2 u1 ) − λ0 (x) f (u1 ) ≤ 0 ≤ G(x, Du2 , D2 u2 ) − λ0 (x) f (u2 )

in

Ω

in the viscosity sense. If u1 ≥ u2 on ∂ Ω, then u1 ≥ u2 in Ω. Let us now comment on the existence of a viscosity solution to the Dirichlet problem (1.1). It follows by an application of Perron’s method since a version of the Comparison Principle is available. In fact, let us consider functions u] and u[ that are solutions to the following boundary value problems: ( ( µ G(x, Du[ , D2 u[ ) = kgkL∞ (∂ Ω) in Ω, G(x, Du] , D2 u] ) = 0 in Ω, and u] (x) = g(x) on ∂ Ω. u[ (x) = g(x) on ∂ Ω. Existence of such solutions follows of standard arguments. Moreover, notice that u] and u[ are respectively, super-solution and sub-solution to (1.1). Consequently, by Comparison Principle, Lemma 2.4, it is possible, under a direct application of Perron’s method, to obtain the existence of a viscosity solution in C(Ω) to (1.1), more precisely we have the following theorem. Theorem 2.5 (Existence and uniqueness). Let f ∈ C([0, ∞)) be a bounded, increasing real function with f (0) = 0. Suppose that there exist a viscosity sub-solution u[ ∈ C(Ω) ∩ C0,1 (Ω) and a viscosity supersolution u] ∈ C(Ω) ∩C0,1 (Ω) to G(x, Du, D2 u) = f (u) satisfying u[ = u] = g ∈ C(∂ Ω). Define the class of functions   v is a viscosity super-solution to   . Sg (Ω) := v ∈ C(Ω) G(x, Du, D2 u) = f (u) in Ω such that u[ ≤ v ≤ u]   and v = g on ∂ Ω Then, u(x) := inf v(x), for x ∈ Ω Sg (Ω)

is (the unique) continuous viscosity solution to G(x, Du, D2 u) = f (u) in Ω with u = g continuously on ∂ Ω.

3

Non-degeneracy results

This Section is devoted to prove some geometric non-degeneracy properties that play an essential role in the description of solutions to free boundary problems of dead core type. Proof of Theorem 1.1. Notice that, due to the continuity of solutions, it is sufficient to prove that such a estimate is satisfied just at point within {u > 0} ∩ Ω0 for Ω0 b Ω. +rx) First of all, for x0 ∈ {u > 0} ∩ Ω0 let us define the scaled function ur (x) := u(x0γ+2 for x ∈ B1 . r γ+1−µ

" Now, let us introduce the comparison function: Ξ(x) := inf λ0 Ω

(γ + 1 − µ)γ+2 γ+1

N (µ + 1) (γ + 2)

Straightforward calculus shows that G (x, DΞ, D2 Ξ) + |DΞ|γ hbr (x), DΞi − λˆ 0 (x) .Ξµ (x) ≤ 0

in

B1

and µ G (x, Dur , D2 ur ) + |Dur |γ hbr (x), Dur i − λˆ 0 (x) .(ur )+ (x) ≥ 0

in B1

#

1 γ+1−µ

γ+2

|x| γ+1−µ .


9

in the viscosity sense, where γ−2µ γ−2µ −p , r− γ+1−µ −p , M) := r γ+1−µ F x0 + rx, → M , br (x) = rb(x0 + rx) and G (x, →

λˆ 0 (x) := λ0 (x0 + rx)

Moreover, G satisfies the same structural assumptions than F, namely (F1)-(F3). Finally, if ur ≤ Ξ on the whole boundary of B1 , then the Comparison Principle (Lemma 2.4), would imply that ur ≤ Ξ in B1 , which clearly contradicts the assumption that ur (0) > 0. Therefore, there exists a point Y ∈ ∂ B1 such that " ur (Y ) > Ξ(Y ) = inf λ0 Ω

(γ + 1 − µ)γ+2

#

1 γ+1−µ

N (µ + 1) (γ + 2)γ+1

and scaling back we finish the proof of the Theorem. Theorem 1.1 provides the following finer (lower) control for point near the free boundary in terms of the distance up to the free boundary. Proof of the Lower bound in Theorem 1.3. The proof is based on reductio ad absurdum argument. Thus, suppose for sake of contradiction that c] does not exist. Then, there exists a sequence xk ∈ {u > 0} ∩ Ω0 with dk := dist(xk , ∂ {u > 0} ∩ Ω0 ) → 0

γ+2

as k → ∞

and

Now, let us define the auxiliary function vk : B1 → R by: vk (y) :=

u(xk ) ≤ k−1 dkγ+1−µ u(xk +dk y) γ+2 γ+1−µ

∀ k ∈ N.

.

dk

Hence, we have that 1. vk ≥ 0 in B1 .

2. G (y, Dvk , D2 vk ) + |Dvk |γ hbk (y), Dvk i = λ0 (xk + dk y)vk (y) in B1 in the viscosity sense, where µ

γ−2µ γ−2µ −p , M) := d γ+1−µ F x + d y, → −p , d − γ+1−µ M G (y, → k k k k

and

bk (y) = dk b(xk + dk y),

with G fulfilling the structural assumptions (F1)-(F3). 3. According to local Hölder regularity of solutions (cf. [30, Corollary 2]) we have (3.1)

1 vk (y) ≤ C N, λ , Λ, γ, kbkL∞ (B1 ) .dkα + ∀ y ∈ B 1 . 2 k

From the Non-degeneracy (Theorem 1.1) and sentence (3.1) we obtain that 0 < C] .

γ+2 1 1 γ+1−µ ≤ S 1 [vk ] ≤ max{1,C(N, λ , Λ, γ)}. dkα + →0 2 2 k

as

k → ∞,

which yields a contradiction. This completes the proof. In our last result this section we establish an average control for the µ−power of dead-core solutions. Such an estimate will be useful in order to prove our Hausdorff measure estimates (see Section 7).


10

0 b Ω. For each x ∈ ∂ {u > Theorem 3.1. Let u be a nonnegative, bounded viscosity solution to Ωo 0 n (1.2) and 0 0} ∩ Ω0 , there exist universal constants C∗ > 0 and 0 < r0 min 1, dist(Ω2 ,∂ Ω) such that (γ+2)µ

Z Br (x0 )

uµ (x)dx ≥ C∗ r γ+1−µ ,

for any

r ≤ r0 .

Proof. One more time we will proceed via a contradiction argument. If such a C∗ > 0 does not exist, then there would exist a sequence xk ∈ ∂ {u > 0} ∩ Ω0 such that for any sequence rk → 0+ as k → ∞ we have (γ+2)µ

Z

uµ (x)dx < k−1 rkγ+1−µ .

Brk (xk )

Now, define the function vk : B1 → R by: vk (y) :=

u(xk +rk y)

. It is readily to verify that

γ+2 γ+1−µ

rk

Gk (x, Dvk , D2 vk ) + |Dvk |γ hbk (y), Dvk i = λk (x).(vk )+ µ

in B1

in the viscosity sense, where γ−2µ γ−2µ −p , X ) := r γ+1−µ F x + r y, r− γ+1−µ X , b (y) = r b(x + r y) and λ (y, s) := λ (x + r y), Gk (y, → 0 k k k k k k k k k k k with Gk fulfilling (F1)-(F3). On the one hand, using the contradiction assumption Z

vk (y)dy = 2N µ

(3.2) B 1 (0) 2

uµ (x)

Z Brk (xk )

dx
0}) rk

(γ+2)µ

γ+1−µ

dy.

2

Now, let us denote dk (y) := dist(xk + rk y, ∂ {u > 0} ∩ Ω0 ). Under such a notation we define n o Dk := y ∈ B 1 (0) | dk (y) < ak rk , 2

where ak :=

γ+1−µ γ+1−µ γ+1−µ γ+1−µ 1 (γ+2)µ N (γ+2)µ −µ (γ+2)µ L B 1 (0) C] (2αN ) (γ+2)µ 2 k

and α > 0 to be chosen so that 2αN N L B 1 (0) > 2N . 2 10 Notice that for k 1 large enough Dk ∩ Brk (xk ) ∩ {u > / Moreover, since ak → 0 as k → ∞, we have 0} 6= 0.

for k 1 large enough that LN (Dck ) ≥ µ

Z B 1 (0) 2

µ vk (y)dy

Z C ] ≥ c LN B 1 (0) Dk

1 N 10 L

B 1 (0) . Therefore, we can estimate for k 1

(γ+2)µ

dk (y) rk

2

γ+1−µ

dy ≥

2N 2αN N c 2αN N L (Dk ) ≥ L B 1 (0) > , 2 k 10k k

2

which is a contradiction with (3.2). This finishes the proof of the Theorem.


4

11

Improved regularity estimates In this Section we prove an improved regularity result to viscosity solutions along their free boundaries.

4.1

A geometric iterative approach

Before proving the main result this section, let us introduce some important definition for our approach. Definition 4.1 (The class J(F, λ0 , µ)(B1 )). For a fully nonlinear operator F fulfilling (F1)-(F3) we say that u ∈ J(F, λ0 , µ)(B1 ) if

µ µ X F(x, Du, D2 u) + |Du|γ hb(x), Dui = λ0 (x)u+ (x) in B1 for 0 ≤ µ < γ + 1 and λ0 u+ L∞ (Ω) 1. X 0 ≤ u ≤ 1, 0 < m ≤ λ0 ≤ M

in B1

and

b ∈ C0 (B1 , RN ).

X u(0) = 0. Definition 4.2. We define for u ∈ J(F, λ0 , µ)(B1 ) the following set γ+2 1 Vγ,µ [u] := j ∈ N ∪ {0}; S 1 [u] ≤ 2 γ+1−µ max 1, .S 1 [u] , C] 2j 2 j+1 where the constant C] > 0 is the one coming from Non-degeneracy property, Theorem 1.1. Notice that Vγ,µ [u] is not empty. Indeed, j = 0 ∈ Vγ,µ [u] since, in view of Theorem 1.1 γ+2 γ+2 1 γ+1−µ 1 γ+1−µ ≥ C] S1 [u] S 1 [u] ≥ C] 2 2 2

S1 [u] ≤ 2

=⇒

γ+2 γ+1−µ

1 max 1, C]

S 1 [u]. 2

Next result regards a machinery of sharp geometric decay, which is a powerful device in nonlinear (geometric) regularity theory and plays a pivotal role in our approach. The core idea was inspired in [32], as well as in the flatness reasoning from [38]. Notwithstanding, the general class of operators which we are dealing with imposes some significant adjusts in such strategies. As a matter of fact, different from [32] and [38] we are not allowed to use a stability argument for viscosity solutions in order to conclude the proof via a strong maximum principle. We overcome such an obstacle by invoking a Harnack type inequality for general fully nonlinear elliptic equations, and by showing that under a suitable control of Thiele modulus (with small enough bounds) solutions fall into a flatness improvement regime (near their free boundaries) (compare with [21, Lemma 3.1], [23, Lemma 4.3] and [40, Lemma 4]). Lemma 4.3. There exists a universal C0 > 0 such that

1 S 1 [u] ≤ C0 . 2j 2 j+1

γ+2 γ+1−µ

for all u ∈ J(F, λ0 , µ)(B1 ) and j ∈ Vγ,µ [u]. Proof. Let us suppose for sake of contradiction that the thesis of Lemma fails to hold. Then, for each k ∈ N we might to find uk ∈ J(Fk , λ0k , µ)(B1 ) and jk ∈ Vγ,µ [uk ] such that   γ+2 −1  1 γ+1−µ  γ+2 1 γ+1−µ max 1, S (4.1) S 1 [uk ] ≥ max k. , 2 . 1 [u]   2 jk C] 2 jk 2 jk +1 Now, let us to define the auxiliary function: vk (x) :=

1 2 jk S 1 2 jk +1

uk

x

[uk ]

in B1 . Hence, vk fulfils

R EGULARITY ESTIMATES FOR DEGENERATE / SINGULAR TYPE EQUATIONS S 1 [uk ]

X 0 ≤ vk (x) ≤

S

2 jk 1 2 jk +1

[uk ]

n o γ+2 ≤ A := 2 γ+1−µ max 1, C1]

12

in B1 and vk (0) = 0.

X S 1 [vk ] = 1 2

µ X Gk x, Dvk , D2 vk + |Dvk |γ hbk (x), Dvk i = λˆ0k (x)(vk )+ (x) in B1 in the viscosity sense, where −p , M) := Gk (x, →

1 22 jk S

1 2 jk +1

[uk ]

Fk

1 1 x bk (x) = j b 2k 2 jk

1 → x, −p , 22 jk S 1 [uk ]M 2 jk 2 jk +1 λˆ0k (x) :=

and

1

(Gk fulfilling (F1) − (F3)), 1

2(γ+2) jk S γ+1−µ [uk ] 1

λ0k

1 x 2 jk

2 jk +1

Therefore, by using (4.1) γ+1−µ 1 . ≤ A .M. k L∞ (B1 )

ˆk µ

λ0 (vk )+

µ

According to Harnack inequality (Theorem 2.2 we have that γ+1−µ 1 1 1 γ+1 µ γ+1 µ ˆ k γ+1 1 = S 1 [vk ] ≤ C N, γ, λ , Λ, kbkL∞ (B1 ) I1 [vk ] + S1 [λ0 (vk )+ ] , ≤ C. (A .M) . 2 k which clearly yields a contradiction as k → ∞. Lemma 4.4. There exists a universal constant C > 0 such that for all u ∈ J(F, λ0 , µ)(B1 ) γ+2

u(x) ≤ C.|x| γ+1−µ

∀ B1 2

Proof. The proof will be by induction process. First of all, we claim that S 1 [u] ≤ C0 .

(4.2)

2j

1

γ+2 γ+1−µ

∀ j ∈ N,

2 j−1

where C0 > 0 is the constant coming from Lemma 4.3. Note that if C0 ≥ 1, which we can suppose without loss of generality, then (4.2) holds for j = 0. Suppose now that (4.2) holds for some j ∈ N. We will verify the ( j + 1)th step of induction. In fact, if j ∈ Vγ,µ [u] then the result holds directly by Lemma 4.3. On the other hand, if (4.2) fails, then we obtain by using the induction hypothesis γ+2 γ+2 γ+2 γ+2 γ+1−µ 1 γ+1−µ 1 1 γ+1−µ 1 γ+1−µ S 1 [u] ≤ .S 1 [u] ≤ C0 . = C . 0 2 2 2 j−1 2j 2j 2 j+1 Therefore, (4.2) holds for all j ∈ N. Finally, for r ∈ (0, 1) let j ∈ N the greatest integer such that Sr [u] ≤ S 1 [u] ≤ C0 . 2j

1 2 j−1

γ+2 γ+1−µ

1 2 j+1

≤r
0} ∩ Ω0 with Ω0 b Ω, let us define v(x) :=

u (x0 + Rx) τ

in B1 ,

for τ, R > 0 constants to be determined universally a posteriori. From the equation satisfied by u, we easily verify that v fulfils in the viscosity sense µ ˆ G (x, Dv, D2 v) + |Dv|γ hb(x), Dvi = λˆ 0 (x).v+ (x),

where −p , M) := G (x, →

R2 −p , τ M , b(x) ˆ F x0 + Rx, → = Rb(x0 + Rx) and τ R2

Rγ+2 λˆ 0 (x) := γ+1−µ λ0 (x0 + Rx) τ

with G fulfilling the assumptions (F1)-(F3). Now, let κλ0 ,µ > 0 be the greatest universal constant, granted by the Definition 4.1 such the Lemma 4.4

2

holds provided G (x, Dv, D v) L∞ (B ) ≤ κλ0 ,µ . Then, we make the following choices in the definition of v: 1

1

τ := max 1, kukL∞ (Ω) , kλ0 kLγ+1−µ ∞ (Ω)

and

1 dist(Ω0 , ∂ Ω) γ+2 , κλ ,µ R := min 1, 0 2

Finally, with such selections v fits into the framework of Lemma 4.4. Now, we are in position to supply the proof of Theorem 1.2. Proof of Theorem 1.2. Let u be a non-negative and bounded viscosity solution to (1.2), Y0 ∈ ∂ {u > 0}∩Ω0 n o 0 any free boundary point and r0 := min 1, dist(Ω2 ,∂ Ω) . According to Remark 4.5, we may assume, without loss of generality that Y0 = 0 and r0 = 1. Therefore, Lemma 4.4 can be applied. The final fashion follows by re-scaling back the normalized and scaled solution, i.e. γ+2

u(X) ≤ C(N, λ , Λ, γ, µ, m, M, kbkL∞ (Ω) , dist(Ω0 , ∂ Ω)).kukL∞ (Ω) .|X −Y0 | γ+1−µ , for points X ∈ {u > 0} ∩ Ω0 close enough to Y0 . Finally, we are in position to complete the proof of Theorem 1.3. Proof of the Upper bound in Theorem 1.3. Fix x0 ∈ {u > 0} ∩ Ω0 and denote d := dist(x0 , ∂ {u > 0}). Now, select z0 ∈ ∂ {u > 0} a free boundary point which achieves the distance, i.e., d = |x0 − z0 |. From Theorem 1.2 we have that γ+2 u(x0 ) ≤ sup u(x) ≤ sup u(x) ≤ C] N, λ , Λ, γ, µ, m, M, kbkL∞ (Ω) d γ+1−µ , Bd (x0 )

B2d (z0 )

which finishes the proof.

4.2

The case µ = γ + 1: Proof of Theorem 1.4

In the next, we will prove the following regularity estimate which will play a significant role in obtaining Theorem 1.4. The next result is inspired to one in [18, Proposition 3.2]. For this reason, we will write the modifications for the reader’s convenience (see also [24, Theorem 1.2] for similar results for quasi-linear elliptic equations with p−Laplacian structure).


14

Theorem 4.6. Let u be a bounded non-negative viscosity solution to (1.2). Then, given 0 < τ0 1 there exists a constant C > 02 depending only on N, λ , Λ, γ, µ, kbkL∞ (Ω) , kλ0 kL∞ (Ω) and τ0 such that for any x0 ∈ Ω fulfilling Bτ0 (x0 ) b Ω and any r ≤ τ20 , the following estimate holds: γ+2 γ+1−µ Sr,x0 [u] ≤ C max Ir,x0 [u], r . Particularly, if x0 ∈ ∂ {u > 0} ∩ Ω0 with Ω0 b Ω (an interior free boundary point), then γ+2 dist(Ω0 , ∂ Ω) γ+1−µ Sr,x0 [u] ≤ Cr for all 0 < r < min 1, . 2 Proof. Firstly, notice that the scaled and normalized function vr (x) :=

u(x0 +rx) γ+2

, satisfies the equation

r γ+1−µ µ G (x, Dvr , D2 vr ) + |Dvr |γ hbr (x), Dvr i = λ˜ 0 (x)(vr )+ = 0

in B1 ,

in the viscosity sense, where γ−2µ − γ−2µ G (x, s, M) := r γ+1−µ F x0 + rx, s, r γ+1−µ M , br (x) = rb(x0 + rx) and λ˜ 0 (x) := λ0 (x0 + rx). Observe that G fulfils the same structural conditions as F. By applying Harnack inequality (Theorem 2.2) we obtain µ 1 (4.3) S 1 [vr ] ≤ C(N, λ , Λ, γ, kbkL∞ (B1 ) ) I 1 [vr ] + M γ+1 S1 [vr ] γ+1 . 2

2

For µ = 0, by scaling back (4.3) in terms of u, the proof is immediate. Thus, consider 0 < µ < γ + 1 and τ0 τ0 := 2ck for 0 ≤ k ≤ n0 such that rn0 = r for some n0 ∈ N, 4 ≤ c ≤ 2 . In order to iterate (4.3) in relation to rk we must consider two possibilities. First, if it holds that µ γ+1 γ+2 γ+2 1 − γ+1−µ − γ+1−µ Irk [u](x0 ) ≤ M γ+1 rk−1 Srk−1 [u](x0 ) for 1 ≤ k ≤ n0 , rk we arrive at − γ+2 rn0 γ+1−µ Srn0 [u](x0 ) ≤ C(N, λ , Λ, γ, µ, M)

µ γ+1 γ+2 − γ+1−µ r0 ≤C Sr0 [u](x0 )

γ+2

from where we deduce that Sr [u](x0 ) ≤ Cr γ+1−µ . Conversely, if for some k0 ≤ n0 it holds that 1 − γ rk0 γ+1−µ Irk [u](x0 ) ≥ M γ+1 0

and 1 − γ+2 rk γ+1−µ Irk [u](x0 ) ≤ M γ+1

µ γ+1 γ+2 − γ+1−µ rk0 −1 Sr0 −1 [u](x0 )

µ γ+1 γ+2 − γ+1−µ rk−1 Srk−1 [u](x0 )

for

k0 < k ≤ n0 ,

then we arrive at − γ+2

− γ+2

− γ+2

rn0 γ+1−µ Srn0 [u](x0 ) ≤ crk0 γ+1−µ Irk [u](x0 ) ≤ crn0 γ+1−µ Irn0 [u](x0 ), 0

where c = c(N, λ , Λ, γ, µ, M), from where we obtain the desired result. 2 It

is worth noting that, different from Theorem 1.2, such a constant C > 0 does not depend on inf λ0 (x). Such a subtle detail is Ω

one of key points, which will enable us to prove Theorem 1.4.


15

In the next we shall analyse the “extremal case” obtained as µ → γ + 1, in other words, F(x, Du, D2 u) + |Du|γ hb(x), Dui = λ0 (x)u+ (x) in Ω. γ+1

(4.4)

By means a barrier argument for the critical equation (4.4) and by using the Theorem 4.6 we shall prove that a non-negative solution to (4.4) can not vanish in an interior point, unless it is identically zero. This is a sort of strong maximum principle result. Proof of Theorem 1.4. The prove will follow by reductio ad absurdum. For that purpose suppose that {u = 0} $ Ω, and let x0 ∈ Ω such that u(x0 ) > 0. We can suppose without loss of generality that d0 := dist(x0 , ∂ {u > 0})
0 and s > 0 (large enough) we define the following barrier function: −s

ΘA ,s (x) := A

e

|x−x0 |2 d2 0

− 4s

e

− e−s , − e−s

for which, a straightforward calculation shows (in the viscosity sense) that   G [ΘA ,s ](x) ≥ 0 in Bd0 (x0 ) \ B d0 (x0 )   2 ΘA ,s = A in ∂ B d0 (x0 )  2   Θ = 0 in ∂ B (x ), A ,s

0

d0

where G [u](x) = F(x, Du, D2 u) + |Du|γ hb(x), Dui − λ0 (x)u+ (x). Notice that γ+1

(4.5)

inf

Bd (x ) \B d0 (x0 ) 0 0

|D ΘA ,s (x)| ≥

−s A sd−1 0 e s e− 4

− e−s

≥

10A se−s := ι > 0. s dist(x0 , ∂ Ω)(e− 4 − e−s )

2

On the other hand, for any constant ζ > 0, the barrier ζ ΘA ,s still being a sub-solution in Bd0 (x0 ) \ B d0 (x0 ). In consequence, we have in the viscosity sense 2

G [ζ ΘA ,s ](x) ≤ 0 ≤ G [u](x) in Bd0 (x0 ) \ B d0 (x0 ). 2

Moreover, by taking ζ0 ∈ (0, 1) small enough such that ζ0 A ≤ I d0 [u](x0 ) we obtain 2

ζ0 ΘA ,s ≤ u

in

∂ Bd0 (x0 ) ∪ ∂ B d0 (x0 ). 2

Thus, by using Comparison Principle (Lemma 2.4) we obtain that ζ0 ΘA ,s ≤ u

(4.6)

in

Bd0 (x0 ) \ B d0 (x0 ). 2

On the other hand, for any max{0, γ} < µ < γ + 1 fixed we can rewrite equation (4.4) as F(x, Du, D2 u) + |Du|γ hb(x), Dui = h(x)u+ (x) in µ

γ+1−µ

where h(x) := λ0 (x)u+

B1 ,

(x). Thus, for z ∈ ∂ Bd0 ∩ ∂ {u > 0} we obtain invoking Theorem 4.6 that

γ+2 Sr,z [u] ≤ C N, λ , Λ, γ, µ, kbkL∞ (B1 ) , khkL∞ (B1 ) .r γ+1−µ ≤ C.rγ+2 .


16

for r 1, since 1 > γ + 1 − µ. Now, since γ + 2 > 1 we can select 0 < r0 1 small enough such that γ+2

C.r0

1 ≤ ζ .ι.r0 . 7

Finally, according to sentences (4.5) and (4.6) we obtain γ+2

ζ .ι.r0 ≤ sup ζ · |Θ(|x|) − Θ(|z|)| ≤ sup ζ .Θ(|x|) ≤ sup u(x) ≤ C.r0 Br0 (z)

Br0 (z)

Br0 (z)

1 ≤ ζ .ι.r0 , 7

which yield a contradiction. Therefore, either u > 0 or u ≡ 0 in Ω. Example 4.7. Theorem 1.4 assures that non-trivial viscosity solutions to (1.2) cannot dead-core sets, i.e, they must be strictly positive, provided µ = γ + 1. Indeed, fixed a direction i = 1, · · · , N and λ0 > 0 we have √ √ γ+2 γ+2 λ0 .xi for b(x) = λ0 that u(x) = e is a strictly positive viscosity solution to |Du(x)|γ (∆u(x) + hb(x), Du(x)i) = λ0 .uγ+1 (x) in

5

B1 .

Consequences and further results

Throughout this section we will present further consequences arising from our main results. An important application of the Theorem 1.2 (see also Theorem 4.6), we obtain a finer gradient control to solutions of (1.2) near their free boundary points (compare with [20, Corollary 4.1], [21, Lemma 3.8] and [23, Lemma 3.8]). Proof of Theorem 1.5. Firstly, let x0 ∈ ∂ {u > 0} ∩ Ω0 be an interior free boundary point. Now, we define +rx) the scaled auxiliary function Φ : B1 → R+ by: Φ(x) := u(x0γ+2 . Notice that Φ fulfils in the viscosity sense r γ+1−µ

ˆ G (x, DΦ, D2 Φ) + |Du(x)|γ hb(x), Dui = λˆ 0 (x)Φµ (x) in B1 , where γ−2µ γ−2µ − γ+1−µ → − → − γ+1−µ ˆ G (x, p , M) := r F z + rx, p , r M , b(x) = rb(z + rx) and

λˆ 0 (x) := λ0 (z + rx).

From Theorem 1.2 (resp. Theorem 4.6) we get that S1 [Φ] ≤ C. Finally, by invoking the gradient estimates (Theorem 2.3) we obtain that 1 1+µ

r γ+1−µ

µ 1 sup |Du(x)| = sup |DΦ(y)| ≤ C N, γ, λ , Λ, kbkL∞ (B1 ) . S1 [Φ] + M γ+1 S1 [Φ] γ+1 ≤ C0 ,

B r (x0 )

B 1 (x0 )

2

2

which finishes the proof. As a result, we obtain the following result in terms of distance up to the free boundary (see also [24, Corollary 5.1]). The proof is similar to the one employed in Theorem 1.3 (Upper bound). Corollary 5.1. Let u be a bounded non-negative viscosity solution to (1.2) in B1 . Then, for any point z ∈ {u > 0} ∩ B 1 , there exists a universal constant C > 0 such that 2

1+µ

|Du(z)| ≤ Cdist(z, ∂ {u > 0}) γ+1−µ ,


17

Now, we will obtain some weak geometric (or measure theoretical) properties of the phase-transition. Next result establishes that the positiveness region enjoys uniform positive density property along the free boundary. In particular, the development of cusps along free boundary points is inhibited. Corollary 5.2 (Uniform positive density). Let u be a nonnegative, bounded viscosity solution to (1.2) in B1 and x0 ∈ ∂ {u > 0} ∩ B 1 a free boundary point. Then for any 0 < ρ < 12 , 2

LN (Bρ (x0 ) ∩ {u > 0}) ≥ θ .ρ N , for a constant θ > 0 that depends only upon universal parameters and kukL∞ . Proof. Because of Theorem 1.1, for some 0 < r < that,

1 2

fixed, it is possible to select a point Y0 ∈ Br (x0 ) such γ+2

u(Y0 ) = sup u(x) ≥ C] .r γ+1−µ .

(5.1)

Br (x0 )

In order to complete the proof the following inclusion Bς .r (Y0 ) ⊂ {u > 0} ∩ B1

(5.2)

holds, for some ς > 0 (universal) small enough. In fact, from Theorem 1.2, for Z0 ∈ ∂ {u > 0}, we get γ+2

u(Y0 ) ≤ C] .|Y0 − Z0 | γ+1−µ .

(5.3) Thus, by (5.1) and (5.3) we obtain

γ+2

γ+2

C] .r γ+1−µ ≤ C] .|Y0 − Z0 | γ+1−µ and consequently,

C] C]

γ+1−µ γ+2

.r ≤ |Y0 − Z0 |.

For this reason, by taking 0 < ς 1 small enough, the inclusion in (5.2) is fulfilled. Therefore, LN (Bρ (x0 ) ∩ {u > 0}) ≥ LN (Bρ (X0 ) ∩ Bς r (Z0 )) ≥ θ .rN ,

Next, we shall prove that the free boundary is a porous set. For the reader’s convenience, we shall recall the definition of this notion. Definition 5.3 (Porous set). A set S ⊂ RN is said to be porous with porosity constant ε ∈ (0, 1) if there is an R0 > 0 such that ∀ x ∈ S, ∀ r ∈ (0, R0 ) , ∃ y ∈ RN

such that Bεr (y) ⊂ Br (x) \ S.

Corollary 5.4 (Porosity of the free boundary). Let u be a bounded, non-negative solution of (1.2). Then, there exists a universal constant ξ > 0 such that H N−ξ ∂ {u > 0} ∩ Ω0 < ∞.


18

Proof. Let R > 0 and x0 ∈ Ω be such that B4R (x0 ) ⊂ Ω. We will prove that ∂ {u > 0} ∩ BR (x0 ) is a ε2 porous set for some ε ∈ (0, 1]. For this purpose, let x ∈ ∂ {u > 0} ∩ BR (x0 ). For each r ∈ (0, R) we have Br (x) ⊂ B2R (x0 ) ⊂ Ω. Now, let y ∈ ∂ Br (x) such that u(y) = sup u. By Non-degeneracy (Theorem 1.1) ∂ Br (x) γ+2

u(y) ≥ C] .r γ+1−µ .

(5.4)

On the other hand, near the free boundary (Theorem 1.3 upper bounds) γ+2

u(y) ≤ c] .d(y) γ+1−µ ,

(5.5)

where d(y) := dist(y, ∂ {u > 0} ∩ B2R (x0 )). Now, from (5.4) and (5.5) we get d(y) ≥ ε.r

(5.6) for a constant 0 < ε :=

c] C]

γ+1−µ γ+2

≤ 1.

Let now yˆ ∈ [x, y] (the segment connection x to y) be such that |y − y| ˆ =

εr 2,

then there holds

ˆ ⊂ Bεr (y) ∩ Br (x). B ε r (y)

(5.7)

2

In effect, for each z ∈ B ε r (y) ˆ 2

|z − y| ≤ |z − y| ˆ + |y − y| ˆ
0}, then Bεr (y) ∩ Br (x) ⊂ {u > 0}, which together with (5.7) implies ˆ ⊂ Bεr (y) ∩ Br (x) ⊂ Br (x) \ ∂ {u > 0} ⊂ Br (x) \ ∂ {u > 0} ∩ BR (x0 ). B ε r (y) 2

Therefore, ∂ {u > 0} ∩ BR (x0 ) is a ε2 -porous set. Finally, the desired (N − ξ )-Hausdorff measure estimate follows from [33]. Remark 5.5. Notice that, particularly the Hausdorff estimate from Corollary 5.4 assures that LN (∂ {u > 0} ∩ Ω0 ) = 0 (cf. [33]).

6

A Liouville type result: Proof of Theorem 1.6 The main purpose of this section is to prove that a global solution to

(6.1)

F(x, Du, D2 u) + |Du|γ hb(x), Dui = λ0 (x)u+ (x) µ

γ+2

in

RN .

must grow faster than C|x| γ+1−µ as |x| → ∞ for a suitable constant C > 0, unless it is identically zero.


19

For this end, fix x0 ∈ RN , ς > 0 and 0 < r0 < r, we consider for ρ < r (to be considered) the quantity r0 = r − ρ. Then, as in the proof of Theorem 1.1, it can be seen that the radially symmetric function v : Br (x0 ) → R+ given by γ+2

v(x) = Θ(N, λ0 , γ, µ)(|x − x0 | − r0 )+γ+1−µ is a viscosity super-solution to  µ 2 γ   F(x, Du, D u) + |Du| hb(x), Dui = λ0 (x)u+ (x) u(x) = ς   u(x) = 0

Br (x0 ) ∂ Br (x0 ) , Br0 (x0 )

in on in

where " Θ(N, λ0 , γ, µ) = λ

#

(γ + 1 − µ)γ+2

1 γ+1−µ

and

γ+1

N (µ + 1) (γ + 2)

ρ=

ς Θ(N, λ0 , γ, µ)

γ+1−µ γ+2

.

Such a explicit expression of v allow us to prove our sharp (quantitative) Liouville type result. Proof of Theorem 1.6. Fixed s0 > 0 (large enough), let us consider w : Bs0 → R the unique (see Theorem 2.5) viscosity solution to  µ  F(x, Dw, D2 w) + |Dω|γ hb(x), Dωi = λ0 (x)w+ (x) in Bs0 (6.2) w(x) = sup u(x) on ∂ Bs0 .  ∂ Bs0

According to the Comparison Principle (Lemma 2.4) u ≤ w in (6.3)

sup ∂ Bs0

u(x) γ+2 γ+1−µ

≤ sup

s0

Bs0

u(x) γ+2 γ+1−µ

Bs0 . Moreover, due to hypothesis (1.3)

≤ cΘ(N, λ0 , γ, µ)

s0

for some c 1 (small enough) and s0 1 (large enough). As above, the function 

(6.4)

 γ+1−µ γ+2

sup u(x)     ∂ Bs0 v(x) = Θ(N, λ0 , γ, µ) |x| − s0 +   Θ(N, λ0 , γ, µ)  

γ+2  γ+1−µ

    +

is a viscosity super-solution to (6.2). Thus, w ≤ v in Bs0 . Therefore, by (6.3) and (6.4) we conclude that γ+2 γ+1−µ γ+1−µ →0 u(x) ≤ Θ(N, λ0 , γ, µ) |x| − (1 − c γ+2 )s0

as

s0 → ∞.

+

Example 6.1. It is worth highlighting that the constant in Theorem 1.6 is optimal in the sense that we can not remove the strict inequality in (1.3). In fact, the function given by γ+2

u(x) = Θ(N, λ0 , γ, µ)(|x| − r0 )+γ+1−µ solves (6.1) (for λ0 (x) = λ a constant) and it clearly attains the equality in (1.3) for the explicit value " Θ(N, λ0 , γ, µ) = λ

(γ + 1 − µ)γ+2 N (µ + 1) (γ + 2)γ+1

#

1 γ+1−µ

.


20

7 (N − 1)-Hausdorff measure estimates: Proof of Theorem 1.7 In this section, we will proceed to estimate the Hausdorff measure of the free boundary to solutions of (1.2). To this end, we need first some preliminary results, which are based on the set of assumptions in Section 2 on the operator F, together with the following additional hypothesis (C) [(A, b)-Concavity] There exist a bounded symmetric positive definite Lipschitz matrix ( ) n A = [ai j (x)]1≤i, j≤N ∈ Aλ ,Λ := A ∈ Sym(N) λ kξ k2 ≤ ∑ ai j ξi ξ j ≤ Λkξ k2 , ∀ ξ ∈ RN i, j=1

and b ∈ C0 (Ω, RN ) so that −p , X ) = G(x, →

−p |γ (F(x, X ) + hb(x), → −p i) if |→ → − γ | p | F(x, X ) if

(

γ ≥0 γ < 0,

where F(x, X ) ≤ Tr(A(x)X ) in the viscosity sense (see [5, Section 6], [20, Theorem 1.3], [22, Section 5] and [35, Section 6] for similar insights). Moreover, by technical considerations we must suppose the following upper bound control: |b| ≤ mC−(γ+1) , where C > 0 is the universal constant from sharp gradient estimate (Theorem 1.4), and max 0, 2γ ≤ µ < γ + 1 as γ ≥ 0. Before proving the main result of this section, let us present some useful notions and preliminaries. Definition 7.1. Let A b Ω be a subset. We say that A has the (δ , ζ )-density property if there is δ ∈ (0, 1) so that there corresponds ζ > 0 with the property (7.1)

LN (Bδ (x) ∩ A ) ≥ ζ LN (Bδ (x)),

for all x ∈ ∂ A ∩ Ω. If (7.1) holds for all δ in (0, 1), then we say that A has uniform density in Ω along ∂ A . As a consequence of the above definition, we derive the following facts which will be used in our proof of Hausdorff estimates for the free boundary . Proposition 7.2. Let A b Ω be open. Then (a) If A has the (δ , ζ )-density property, then there is a constant M = M(N) so that the following holds LN (Nδ (∂ A ) ∩ Bρ (x0 )) ≤

M(N) N L (Nδ (∂ A ) ∩ Bρ (x0 ) ∩ A ) +C(N, ζ )δ ρ N−1 ζ

for x0 ∈ ∂ A ∩ Ω and δ ρ, where Nδ (O) = {x ∈ RN : dist(x, O) < δ }. (b) If A has uniform density in Ω along ∂ A , then LN (∂ A ∩ Ω) = 0. Proof. Property (a) follows by a standard covering argument and (b) is a consequence of the Lebesgue Differentiation Theorem. We start with the series of preliminary results needed in the proof of Theorem 1.7. The first lemma contains an L2 estimate on the gradient near free boundary points.


21

Lemma 7.3. There exists a constant C > 0 such that for all x0 ∈ ∂ {u > 0} ∩ B 1 and ρ < 14 , the following 2 holds: Z |Du|2 dx ≤ Cερ N . (

γ+2

)

Bρ (x0 )∩ 0 0 to be determined a posteriori. Moreover, we impose that [ B j ⊂ N 1 B 1 ∩ Bρ (x0 ) j

4

2


22

Note that, from Heine-Borel Lemma there exists a constant l > 0 (with dimensional dependence) such that ∑ χB j (x) ≤ l. We first prove the following estimate j

Z

|Du|2 dx ≥ CLN (B j ),

(7.3) (

γ+2

) ∩B j

0 0 large enough so that K :=

|Du| ≥

sup

N1 B1 4

1 C∗

γ+2

γ+2

C] .(C∗ ) γ+1−µ > 4 γ+1−µ .

and

2

Next, we choose ε > 0 small enough so that if r1j = Φ≥

(7.4)

3ε 4

ε K

and r2j = Kε , then

in B 1j := Br1 (x1j ) j

and Φ
0 and for at

least one of the balls B 1j and B 2j . Indeed, if this is not the case, then we can find a sequences (xk )k∈N ⊂ B 1j and (yk )k∈N ⊂ B 2j such that |Φ(xk ) − m j | 1 < ε k

and

|Φ(yk ) − m j | 1 < . ε k

Letting k → ∞, we obtain that |Φ(xk ) − Φ(yk )| → 0. ε This contradicts (7.4) and (7.5). Thus, by Poincaré inequality, we get σ 2ε 2 ≤

Z Bj

Z

|Φ(x) − m j |dx ≤ (C∗ ε)2

Bj

|DΦ|2 dx

and thus, for a universal constant C2 > 0, we conclude that Z (

γ+2

|Du|2 ≥ C2 LN (B j ). )

0 0}) ≥ ζ LN (Bδ (x0 )). Hence, the set {u > 0} has the (δ , ζ )-density property. From Proposition 7.2 there is a constant M > 0 so that LN Nδ (∂ {u > 0}) ∩ Bρ (x0 ) ≤ C2 LN Nδ (∂ {u > 0}) ∩ Bρ (x0 ) ∩ {u > 0} + Mδ ρ N−1 . Thus, using (7.6) we derive that L

N

Nδ (∂ {u > 0}) ∩ Bρ (x0 ) ≤ C2 LN

γ+2 ] γ+1−µ 0 0).


24

Finally, we will supply the proof of Hausdorff estimate for the free boundary. Proof of Theorem 1.7. . Let 0 < δ < ρ < 14 , and consider a covering B j by balls of the set ∂ {u > 0} ∩ Bρ (x0 ) centered at points in ∂ {u > 0} ∩ Bρ (x0 ) and with radius δ . Hence, [

B j ⊂ Nδ (∂ {u > 0}) ∩ Bρ+δ (x0 ).

j

Thus, we derive that HδN−1 ∂ {u > 0} ∩ Bρ (x0 ) ≤ C ∑ δ N−1 j

1 = C ∑ LN (B j ) δ j C N ≤ L Nδ (∂ {u > 0}) ∩ Bρ+δ (x0 ) δ ≤ C(ρ + δ )N−1 where we have used Theorem (7.5) to obtain the last inequality. By letting δ → 0 we finish the proof. Remark 7.6. As a consequence of Theorem 1.7 we conclude that F0 (u, B1 ) := ∂ {u > 0} ∩ B1 has locally finite perimeter. Moreover, the reduced free boundary F∗0 (u, B1 ) := ∂red {u > 0} ∩ B1 has a total H N−1 measure in the sense that H N−1 (F0 (u.B1 ) \ F∗0 (u, B1 )) = 0. Particularly, the free boundary has an outward vector for H N−1 almost everywhere point in F∗0 (u, B1 ) (cf. [28, Section 5] for such properties).

Acknowledgments This work has been partially supported by CNPq (Brazilian government program Ciência sem Fronteiras), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina) and by ANPCyT under grant PICT 2012-0153. The authors would like to thank Eduardo V. Teixeira by his comments and suggestions that benefited a lot the final outcome of this manuscript. J.V. da Silva and G.C. Ricarte would like to thank respectively Research Group on Partial Differential Equations from Universidad de Buenos Aires and Analysis Research Group of Centro de Matemática da Universidade de Coimbra for fostering a pleasant and productive scientific atmosphere during their Postdoctoral programs. J.V da Silva is a researcher of CONICET.

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