Boundary regularity for a degenerate elliptic equation with mixed boundary conditions

We consider a function U satisfying a degenerate elliptic equation on (0,+\infty)\times R^N with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain \Omega\subset R^N of class C^{1;1}, whereas the Dirichlet data is on the exterior of \Omega. We prove Holder regularity estimates of U/d^s, where d is a distance function defined as d(z) := dist(z;R^N\setminus\Omega), for z\in (0,+\infty)\times R^N. The degenerate elliptic equation arises from the Caffarelli-Silvestre extension of the Dirichlet problem for the fractional Laplacian. Our proof relies on compactness and blow-up analysis arguments.


Introduction
This paper is concerned with regularity estimates of solutions to degenerate mixed elliptic problems. More precisely, for s ∈ (0, 1), we consider the differential operators M s and N s given by M s U (t, x) := div t,x (t 1−2s ∇ t,x U )(t, x) and N s U (t, x) := −t 1−2s ∂U ∂t (t, x), for (t, x) ∈ (0, +∞) × R N . Now let f ∈ L ∞ (R N ) and U ∈Ḣ 1 (t 1−2s ; R N +1 Equation (1.1) is understood in the weak sense, see (1.3). Here and in the following, R N +1 and Ω is a bounded domain of class C 1,1 in R N . Problem (1.1) is a weighted (singular or degenerate, depending on the value of s ∈ (0, 1)) elliptic equation on R N +1 + with mixed boundary conditions. The weight t 1−2s belongs to the Muckenhoupt class A 2 , i.e. for any ball B ⊂ R N +1 , there exists a constant C such that see [21] for more details. Regularity estimates and Harnack inequalities for solutions to degenerate elliptic equations with mixed boundary conditions have been studied by many authors, we refer to [1,3,5,8,9,16,17,19,25,29]. Important applications to these equations can be found in [2,10,20].
In the present paper, we are interested in the regularity of U d s Ω up to the interface {0} × ∂Ω of Dirichlet and Neumann data. Equation (1.1) can be seen as a local version of the following fractional elliptic equation (−∆) s u = f in Ω, u = 0 in R N \ Ω, (1.2) where u is the trace of U . Here (−∆) s is the fractional Laplacian defined as |x−y| N+2s dy, with C a positive normalization constant. Indeed, in 2007, Caffarelli and Silvestre [5] obtained an extension theorem that renders (1.2) somewhat local. They proved that for every u ∈Ḣ s (R N ), there exists a unique U ∈Ḣ 1 (t 1−2s ; R N +1 + ) satisfying where k s is a constant depending only on s, see e.g. [3], anḋ The Caffarelli-Silvestre extension, because of its local nature, is very often used to prove qualitative properties of solutions to problems involving the fractional Laplacian, see for instance [3,4,12,16,18,27]. Equation (1.2) is a special case of integro-differential equations called nonlocal equations. The study of nonlocal equations have attracted several researchers in the last years since they appear in different physical models; from water waves, signal processing, materials sciences, financial mathematics etc. We refer to [7] and the references therein for further motivations. Let us now recall some of the main boundary regularity results in the case of problem (1.2) itself. In [22], Ros-Oton and Serra first proved that for f ∈ L ∞ (R N ) and Ω of class C 1,1 , u/δ s Ω belongs to C 0,α (Ω), for some α ∈ (0, 1) and u satisfying (1.2). Here and in the following δ Ω (x) = dist(x, R N \ Ω). Exploiting Hörmander's theory for pseudo-differential operators, Grubb [13,14] proved that u/δ s Ω ∈ C ∞ (Ω) if f is C ∞ −regular and Ω of class C ∞ , for the fractional Laplacian. More recently, Ros-Oton and Serra [23,24] extended and generalised their result to fully nonlinear nonlocal operators. They showed that if f ∈ C 0,α (R N ) (f ∈ L ∞ (R N )) and Ω is of class C 2,α (Ω of class C 1,1 ) then u/δ s Ω ∈ C s+α (Ω) (u/δ s Ω ∈ C s−ε for any ε > 0) for α > 0. Recently in [11], the author proves Hölder estimates up to the boundary of Ω, for u and the ratio where Ω is of class C 1,γ , for γ > 0 and u is a weak solution of a nonlocal Schrödinger equation, with f in some Morrey spaces.
The main goal of this paper is to study the same type of regularity for problem (1.1). Our main result is stated in the following Then, for any 0 < ε < s, there exists a function Ψ ∈ C s−ε ([0, 1] × Ω) such that Here C is a positive constant depending only on Ω, N , s and ε.
The result in Theorem 1.1 was known in the case s = 1/2, Ω of class C ∞ and f ∈ C ∞ (Ω), see e.g. [6,15]. It does not seem to be an immediate task to derive Theorem 1.1 from the nonlocal result in [23,24], by e.g. the Poisson kernel representation. We therefore have to study in details (1.4), although our argument is inspired by [23].
The proof of Theorem 1.1 is inspired by [23], which we explain in the following. First, we let h + : (1.5) In particular h + (0, r) = max(r, 0) s , see [23]. Let ν(x 0 ) be the unit interior normal to ∂Ω at x 0 . Given x 0 ∈ ∂Ω, the function H x 0 ,ν + (t, x) = h + (t, (x − x 0 ) · ν(x 0 )) satisfies (1.1), for f = 0. For an explicit expression of h + , see Section 3. We note that H x 0 ,ν + belongs to the space for an open set B ⊂⊂ R N +1 The main goal is then to derive the estimate and C is a positive constant depending only on N , s, ε and Ω. Moreover |Q(x 0 )H x 0 ,ν + (z)| ≤ C for every x 0 ∈ ∂Ω ∩ B 1/2 and z ∈ B + 1 (x 0 ). We note that (1.6) can be seen as a Taylor expansion of W near the interface {0} × ∂Ω. To reach (1.6), we use blow up analysis combined with a regularity estimate on R N +1 + and the Liouville-type result on the half-space contained in Lemma 5.4. This argument was developped by Serra [26] to prove interior regularity results for fully nonlinear nonlocal parabolic equations and by Ros-Oton and Serra [23] to prove boundary regularity estimates for integro-differential equations. Once we get (1.6), we now deduce the result in the main theorem.
The paper is organized as follows. In Section 2, we give some notations and definitions of functional spaces and their associated norms for the need of this work. We state some preliminaries in Section 3. In Section 4, we prove an intermediate boundary regularity result for solution to equation (1.1) on R N +1 + with the Neumann boundary condition only. We use blow up analysis and compactness arguments to prove (1.6) in Section 5. In Section 6, we prove some regularity estimates in the neighbourhood of the interface set ∂Ω. In Section 7, we give the complete proof of Theorem 1.1. Acknowledgement: The author would like to thank Diaraf Seck and Mouhamed Moustapha Fall for helpfull discussions and encouragements. This work is supported by the NLAGA Project of the Simons foundation and the Post-AIMS bursary of AIMS-SENEGAL.

Definitions and Notations
We start by introducing some spaces and their norms. Let s ∈ (0, 1), we define This space is endowed with the norm We let We recall the fractional Laplacian of u ∈ L s (R N ) ∩ C 2 loc (R N ), where C N,s = π 2s+N/2 Γ(s + N/2) Γ(−s) , Γ is the usual Gamma function and P.V. is the Cauchy Principal Value. For s ∈ (0, 1),Ḣ s (R N ) coincides with the trace ofḢ 1 (t 1−2s ; R N +1 In particular, every function U ∈Ḣ 1 (t 1−2s ; R N +1 + ) has a unique trace function u = U |R N ∈Ḣ s (R N ), see [3].
Let f be a function and α ∈ (0, 1), the Hölder seminorm of f is given by For k ∈ N, f ∈ C k,α (Ω) means that the quantity is finite. In this work, instead of writing C k,α , we will put C k+α sometimes for the same definition.
Let us now introduce some notations used throughout the paper, and is the ball of center z 0 = (t 0 , x 0 ) ∈ R N +1 and radius R. We will use the variables x and z for the spaces R N and R N +1 respectively. For simplicity, when x 0 = 0 and z 0 = 0, we simply write B R (or B + R ) and B R respectively. We also define the distance functions δ and d by Finally, for x 0 ∈ ∂Ω, we let ν(x 0 ) be the interior normal to ∂Ω at x 0 . We then definē In this paper, all constants C or C(N, s) that we do not specify are positive universal constants.

Preliminaries
and ⋆ denotes the convolution product. For every δ ∈ R and t > 0, we let See for instance [23], using polar coordinates, letting t = r sin θ and δ = r cos θ, with θ ∈ (0, π) and r > 0, we have . Here 2 F 1 is the Hypergeometric function which can be expressed by the power series, for 0 < x < 1, with a n > 0. Next, we consider a bounded domain Ω of class C 1,1 . We denote by ν the interior normal to ∂Ω. For x 0 ∈ ∂Ω, we will consider the function

Regularity estimate up to the boundary for the degenerate equation with the Neumann boundary condition
The following result is stronger than needed since the C s−ε estimate for the solution V in B + 1 will be enough for our purpose.
Then V ∈ C 2s−ε (B + 1 ) for all 0 < ε < 2s. Moreover, where C is a positive constant depending only on N , s and ε.
Proof. Consider the cut-off function η ∈ C ∞ c (B 3 ) such that η ≡ 1 in B 2 and 0 ≤ η ≤ 1 in R N . Let v be the (unique) solution to the equation where C > 0 is a constant that depends only on N , s and ε. Now consider the Caffarelli- By a change of variable, we have By (4.1), it is clear that where the constant C > 0 depends only on N , s and ε.

Toward regularity by blow up analysis
For local boundary regularity results in C 1,1 domains, we fix the geometry of the domain as follows: Definition 5.1. We define G the set of all interfaces Γ with the following properties: there are two disjoint domains Ω + and Ω − satisfying For Γ ∈ G, we let x 0 ∈ Γ ∩ B 1/2 and W, H x 0 ,ν + ∈ L 2 (t 1−2s ; B + r (x 0 )), for all r > 0. Consider the 1-dimensional subspace of L 2 (t 1−2s ; B + r (x 0 )) spanned by H x 0 ,ν + and given by Moreover P x 0 r W has the property that We now state the following lemma which will be useful later.
Proof. The proof is similar to the one of [23, Lemma 6.2]. We skip the details.
The main result of this section is contained in the following on Ω − .

(5.3)
Then, for all x 0 ∈ Γ ∩ B 1/2 , where P x 0 W is given by (5.2) and the positive constant C depends only on N, s, ε and Γ.
Remark that we can replace R N +1 ) and (f k ) ⊂ L ∞ (R N ) be sequences such that W k satisfies (5.3) on R N +1 + , the Neumann data on Ω + k is f k and the Dirichlet condition is on Ω Clearly, Θ is a monotone nonincreasing function, it verifies Θ(r) ր +∞ as r ց 0 and Θ(r) < +∞ for r > 0, because W k L ∞ (R N+1 + ) ≤ 1. Thus, by definition of the supremum, there exist sequences r m ց 0, k m and x m → x 0 ∈ Γ ∩ B 1/2 such that Let us consider the sequence .
Then by (5.5), we get Also by (5.1), we obtain the orthogonality condition Now, let R ≥ 1 be fixed, m large enough so that r m R < 1 2 and z ∈ B + 2R (x m ), we have that where we used (3.2) and (5.3). We have also that where Ω ⋆ m := {x ∈ R N : . By Lemma 5.4, see below, up to a subsequence, Passing to the limit in (5.7) and (5.8), we get a contradiction.
The following result was used in the proof of Proposition 5.3. Proof. For m fixed, consider the function v m , the trace of the function V m such that is a unit normal vector to Γ km at x m pointing towards Ω + km and f m → 0 as m → +∞. In particular, from the first and the last equation above, we have that We notice that from [23, Proof of Proposition 8.3] (for α = 0), the sequence v m satisfies the following estimates, v m L ∞ (B R ) ≤ R β , for every R > 1 and 0 < β < 2s (5.9) and for every m such that r m R ≤ 1, v m C 0,α (B R ) ≤ C(R) for some α ∈ (0, 1). (5.10) First, let us prove that V m L ∞ (B + R ) ≤ CR β , for every R > 1 and that for every m ∈ N such that r m R ≤ 1, . We set For every R > 1 and (t, x) ∈ B + R , by using (5.9). Now, for every m ∈ N such that r m R ≤ 1 and z 1 , z 2 ∈ B + R , we get where we have used (5.10) in the last inequality. Thus, for every m ∈ N such that r m R ≤ 1, we have Notice that v m ∈ L s (R N ) (see (2.1)) and ( in the sense of distribution. Applying the result in [4, Proposition 2.1], we find that there exists a positive constant C = C(N, s) and α ∈ (0, β) such that where we have used (5.9), the change of variable y = 2 i z and the fact that β < 2s in (5.14) so that the summation is finite. It follows that (5.15) Using (5.15) in (5.13), we get ≤ CR β for every R > 1, with r m R ≤ 1, by (5.11) and (5.15). Using (5.12) and (5.16), we also have . We then conclude that, up to a subsequence, the sequence (V m ) converges uniformly to some function V on compact subsets of R N +1 + by Arzelã-Ascoli theorem. Recall that N+2s 2

dy.
Since V m L ∞ (B + 1 ) is bounded then, by the dominated convergence theorem, as m → +∞, recall that v m → Kδ s x 0 ,ν uniformly on compact subsets of R N . Now put We now prove that V m → P(t, ·) ⋆δ s x 0 ,ν uniformly on compact subsets of R N +1 Since v m → Kδ x 0 ,ν uniformly on compact subsets of R N then, by the dominated convergence theorem, we have that Let r > 0 and z = (t, x) ∈ B + r fixed. With similar arguments as in (5.14), we have that since β − 2s < 0. Consequently, by the dominated convergence theorem, Finally, for every z = (t, x) ∈ B + r , we conclude that = KH x 0 ,ν + (t, x). Since r is arbitrary, we get the desired result.

Regularity up to the Dirichlet-Neumann interface
Note that the estimates in the following lemmas hold in a tubular neighbourhood of the interface set ∂Ω, the boundary of Ω. We define H + Ω (t, x) := h + (t, δ(x)), ∀(t, x) ∈ (0, +∞) × Ω. (6.1) In this section, we assume that for anyx 0 ∈ Ω ∩ B 1/2 , there exists a unique x 0 ∈ ∂Ω such that |x 0 − x 0 | = δ Ω (x 0 ). We have the following result comparing H + Ω and H x 0 ,ν and where the positive relabelled constant C depends only on N, s, ε and Ω.
Proof. Define d + (t, δ(x)) := d(t, x) = t 2 + δ 2 (x) 1/2 . Hence, we have that To prove estimates (6.6) and (6.7) for the quantities I and II, we use the same argument as in Lemma 6.1 by remarking that for I ∇d s + (t, δ) ≤ r s−1 . For the quantity II, we note that there are two positive constants C 1 and C 2 such that , similarly as in the proof of Lemma 6.1, see also the definition of h + in Section 3. For (6.8), recall first that for z 1 , z 2 ∈ B + r (x 0 ), We have 1 up to relabeling the positive constant C that depends only on N , s, ε and Ω. Lemma 6.3. Letx 0 ∈ Ω ∩ B 1/2 and x 0 ∈ ∂Ω be the unique point such that 2r : r (x 0 )) ≤ CC 0 r s , for some constant C > 0 depending only on N , s and ε.

Proof. Set
Then, by Theorem 4.1, We now prove the following result.
where the positive constant C depends only on N , s, ε and Ω.
Proof. By (6.2) and Proposition 5.3, we have that for r as in Lemma 6.3. Now, for z 1 , On one hand, using (6.11), we obtain by noting that H + Ω ∼ r s in B + 2r (x 0 ), up to relabeling the positive constant C. On the other hand, by (6.10) and (6.4), we infer that Therefore, by (6.12) and (6.13),

Proof of Theorem 1.1
Regularity of Set : Let k ∈ N and α ∈ (0, 1]. A set Ω ⊂ R N is of class C k,α if there exists M > 0 such that for any x 0 ∈ ∂Ω, there exist a ball B = B r (x 0 ), r > 0 and an isomorphism ϕ : Q −→ B such that : [7,Section 1]. In order to complete the proof of Theorem 1.1, we will need the following result.
where C is a positive constant depending only on N , s, ε and Ω.