Second order regularity for degenerate nonlinear elliptic equations

We investigate the second order regularity of solutions to degenerate nonlinear elliptic equations.

1. Introduction and results. We are interested in establishing Hessian summability for weak solutions of for nonlinear degenerate elliptic equations in divergence form. In particular we consider weak solutions to − div(A(|∇u|)∇u) + b(x)|∇u| q = f (x) in Ω, (1.1) in an open set Ω ⊆ R N . The real valued function A : R + → R + is of class C 1 , with lim sup We also assume that When b(x) is not identically zero, we assume that q >θ +1 2 . We shall consider solutions of class C 1,α . This natural in general according to [3,5,4,12,13]. Therefore we give the following Definition 1.1. We say that u ∈ C 1,α (Ω) is a weak distributional solution to (1.1), if Ω A(|∇u|)(∇u, ∇φ) dx + Ω b(x)|∇u| q φ dx = Ω f φ dx (1.5) for every φ ∈ C ∞ c (Ω).

ANNAMARIA CANINO, ELISA DE GIORGIO AND BERARDINO SCIUNZI
We will frequently exploit the fact that the equation is no longer degenerate outside the critical set Z u , Z u := {∇ u = 0} . Consequently it is also natural to assume that the solution is of class C 2 outside the critical set.
Remark 1.2. We will use the notation u i := u xi , i = 1 . . . , N , to indicate the partial derivative of u with respect to x i . This are the classic derivatives since u is of class C 1 . The second derivatives will be indicated with u ij , i, j = 1 . . . , N . In this case, since u is of class C 2 only far from the singular set Z u , we agree that u ij coincides with the second derivatives far from the singular set Z u , while u ij = 0 on the singular set Z u . It is important to note that, at this stage and without the needed regularity information, this is just a notation inspired by the Stampacchia's theorem. The rigorous proof of the fact that u ij actually represent the second distributional derivatives will be a consequence of our Theorem 1.5.
Our aim is to study the summability of the second derivatives of the solutions. When A(t) = t p−2 the operator reduces to standard p-Laplacian. In this case, from [2,10,11] (see also [1,6]), it follows that u ∈ W 2,2 loc (Ω) if 1 < p < 3, and that if p ≥ 3 and the source term f is strictly positive then u ∈ W 2,q loc (Ω) for q < p−1 p−2 . We may look at this type of regularity as an issue in the context of the Calderón-Zygmund theory for nonlinear degenerate problems. We refer the reader to [7,8] and the references therein.
Here we shall extend the results in [2,10,11]. The setting described above is really more general than the case of the p-Laplacian. Then the proofs in [2,10,11] and the results as well needs appropriate modifications. We start with the following Theorem 1.3. Let u ∈ C 1,α (Ω) ∩ C 2 (Ω \ Z u ) be a solution to (1.1) with f, b ∈ W 1,∞ (Ω). Assume that B 2ρ (x 0 ) ⊂ Ω and y ∈ Ω. Then, for 0 ≤ β < 1 and The local regularity result of Theorem 1.3 holds without sign assumption on the source term f . If else a sign assumption on f is imposed, than we can prove a summability result regarding the inverse of the weight A(|∇u|). We have In particular L({A(|∇u|) = 0}) = 0 .
Theorem 1.4 is actually an estimate on the way the operator degenerate near the critical set. It might have future applications in the study of the qualitative properties of the solutions. Here, as a consequence, we shall point out a further correlated regularity result, see Theorem 1.5 below. Before we start observing that the estimates in Theorem 1.4 and in Theorem 1.3 (namely (1.6) and (1.7)), hold in a general compact set of Ω. The same regularity holds all over the domain once we assume that there are no critical points of the solutions up to the boundary, namely Z u ∩ ∂Ω = ∅. This an abstract assumption always verified each time we may exploit the Hopf boundary lemma, see [9]. The global regularity results follow via a covering argument and the proofs and the statements are postponed in Section 4, see Theorem 4.1 and Theorem 4.2.
As mentioned here above, the estimates on the second derivatives and the estimates of the summability of the weight can be exploited jointly in order to obtain the following . Assume that f is positive in Ω (possibly vanishing on the boundary). Then The paper is organized as follows: We prove the local regularity of the second derivatives in Section 2, while the summability of the weight is studied in Section 3. The covering argument needed to obtain the global results is developed in Section 4, where e also prove Theorem 1.5.
2. Local regularity. We start noticing that, if u ∈ C 1,α (Ω) ∩ C 2 (Ω \ Z u ) is a solution of (1.1), then the derivatives of the solution are solutions to the linearized equation, i. e.
. This follows just putting φ i as test function in (1.1) and integrating by parts.
To exploit such equation we will use a regularization argument. For denoting the characteristic function of a set). We will assume that the ball B 2ρ (x 0 ) is contained in Ω and we will consider a cut-off function Also, for β ∈ [0, 1) and γ < N − 2 if N ≥ 3 (γ = 0 for N = 2) fixed, we set Proof of Theorem 1.3. Let us consider the test function According to (2.3), it follows that such a test function can be plugged in the linearized equation (2.1), since it vanishes in a neighbourhood of the critical set Z u . Consequently by (2.1) we get It is convenient to set: and Regarding the terms I 1 and I 2 , exploiting (1.3), we note that when A (|∇u|) is nonnegative, while when A (|∇u|) is negative. Therefore Thence, from (2.5), we get Now we estimate the right hand side of (2.9) letting δ → 0. Indeed, using the fact that |T ε (t)| ≤ t 1−β and the Young inequality ab ≤ ϑa 2 + b 2 4ϑ , we deduce that where we also used the fact that A(t)t is locally bounded and we have set |∇u| .

(2.11)
Exploiting the fact that |∇ϕ| ≤ 2 ρ , |T ε (t)| ≤ t 1−β and the Young inequality, we also get that lim sup (2.12) Now we set B = sup x∈B2ρ(x0) |b(x)| and we get lim sup (2.13) (2.14) Taking into account (2.9), letting δ → 0, exploiting the above estimates and evaluating T ε , we get Now we fix ϑ sufficiently small such that 3. Local summability of the weight. We exploit here the summability properties of the second derivatives of the solution proved in Theorem 1.3 to obtain information on the summability of (A(|∇u|)) −1 .

4.
Global results. In this section we deduce global regularity information, starting from the local regularity results already proved. Let us define the neighborhood I δ (∂Ω) = {x ∈ Ω|d(x, ∂Ω) ≤ δ}. Without loss of generality, in all the section, we will assume that and x i ∈ Ω\I 3δ (∂Ω) and ρ < δ. We will state our results under the general assumption This assumption is actually verified in all the situations when the Hopf boundary Lemma holds, we refer therefore to [9] .
for any compact set K ⊂ Ω. If we also assume that Z u ∩ ∂Ω = ∅, then Proof. The proof follows via a covering argument. We prove directly the estimate in (4.2), since the estimate in (4.1) follows with the same proof more easily. In all the proof the reader should take into account that we are integrating with respect the x-variable, and the center of the kernel y is varying all over the domain. Under our assumptions, we can take δ > 0 such that there are no critical points of the solution in the neighborhood I 3δ (∂Ω). It follows therefore in this case that A(|∇u|) > 0 in I 3δ (∂Ω) and u ∈ C 2 (3δ(∂Ω)). We set  Therefore Ω\I 3δ (∂Ω) we also have that Finally, by (4.5) and (4.7), we deduce that (4.8) We prove here a global summability result for (A|∇u|) −1 using Theorem 4.1.
Let as above I δ be the neighborhood of ∂Ω of radius δ and consider the same covering with x i ∈ Ω\I 3δ (∂Ω) and ρ < δ. We set (4.9) Theorem 4.2. Let Ω ⊂ R N a bounded smooth domain and let u ∈ C 1,α (Ω) ∩ C 2 (Ω \ Z u ) be a solution to (1.1) with f, b ∈ W 1,∞ (Ω). Assume that f is positive in Ω (possibly vanishing on the boundary). Then, for every compact set K ⊂ Ω, we have K 1 (A(|∇u|)) α 1 |x − y| γ ≤ C * (K) (4.10) with 1 < α < 1 + 1 ϑ , γ < N − 2, if N ≥ 3 and γ = 0 if N = 2 and C * = C * (K, γ, µ,λ, m A , M A , α, f, ||∇u|| ∞ ). If we further assume that Z u ∩ ∂Ω = ∅, then Proof. We deal directly with the more difficult case, namely we prove (4.11). In all the proof the reader should take into account that we are integrating with respect the x-variable, and the center of the kernel y is varying all over the domain. Under our assumptions, we can take δ > 0 such that there are no critical points of the solution in the neighborhood I 3δ (∂Ω). It follows therefore thatλ > 0 in I 3δ and u ∈ C 2 (3δ(∂Ω)). Furthermore, since f is positive in the interior of Ω, we can also assume that µ > 0. By Theorem 4.2, since µ > 0, we get that