Regularity and weak comparison principles for double phase quasilinear elliptic equations

We consider the Euler equation of functionals involving a term of the form \begin{document}$ \int_{\Omega}(| \nabla u|^p+a(x)| \nabla u|^q) \,{ {\rm{d}}} x, $\end{document} with \begin{document}$ 1 and \begin{document}$ a(x)\geq 0 $\end{document} . We prove weak comparison principle and summability results for the second derivatives of solutions.


1.
Introduction. For n ≥ 2 we consider a bounded and smooth domain Ω ⊂ R n and we are interested in the study of properties of positive solutions of the equation: where: • p, q ∈ R are such that 0 < p < q < p + 1 • a(x) ∈ C 1 (Ω) satisfies: a(x) ≥ 0 ∀x ∈ Ω • f is a strictly positive and locally Lipschitz continuous function. ∆ p u = div(p|∇u| p−2 ∇u) is the usual p-Laplace operator. Equation (1) is related to the Euler equation of functionals involving a term of the form F (u) = Ω (|∇u| p + a(x)|∇u| q ) dx, (2) which were introduced by Zhikov [29,30] in order to study the behavior of anisotropic materials. Properties of minimizers for these variational integrals were firstly studied by Marcellini ([16,17]) and recently by several authors (see for instance [1,3,4,5,6,7] and the references quoted there). In particular in [4] C 1,α regularity is established. As pointed out in [4,5,12,29,30], condition p < q < p + 1 excludes the Lavrentiev phenomenon. In [9,10] were introduced several useful tools to handle equations involving the p-Lapace operator. Then these strategies were widely developed (see for instance [2,13,15,18,20,23,24,25,26] and the reference therein) in order to achieve properties of equations (and systems of equations) involving the p-Laplace operator in several frameworks (in bounded and unbounded domains, with lower order terms 4864 GIUSEPPE RIEY or singular data). In this paper, by means of the linearized equation of (1), we give some bounds on the hessian of the solutions, which will allow us to achieve an estimate of the inverse of the gradient of the solutions (see Section 3). Using these tools, we are able to give some regularity results and to recover a weighted Poincaré inequality, useful to obtain a weak comparison principle (see Section 4). We recall that a weak comparison principle for the classical p-Laplace equation (−∆ p u = f (u)) was proved for p ∈ (1, 2) in [8] and for p ∈ [2, +∞) in [9].
2. Notation and preliminary results. For x ∈ R n , |x| stands for the euclidean norm. ·, · is the euclidean scalar product. If A = (a ij ) ∈ R n×n is a matrix, we ij . For r > 0 and x 0 ∈ R n , B r (x 0 ) denotes the open ball of radius r and centered at x 0 . Given a function u : R n → R, we denote by u i the partial derivative of u with respect to the i-th variable. c and C will be positive constants, which can vary from line to line. For m > 1, we will use the standard notation W 1,m (Ω) for the usual Sobolev spaces. In order to well define the so called linearized equation of (1), we need some properties about the weighted Sobolev spaces (for more details about them see for instance [19,21,28]). We recall here their definition and some basic tools. For m ≥ 1 and µ ∈ L 1 (Ω) the weighted Sobolev space W 1,m µ (Ω) (with respect to the weight µ) is defined as the set of functions v ∈ L m (Ω) which are bounded with respect to the norm: where ∇v is the distributional derivative of v, ||v|| L m (Ω) = Ω |v| m m . According to [19], equivalently it is possible to define such a space as the completion of C ∞ (Ω) with respect to the norm defined in (3). As for the usual Sobolev spaces, the space W 1,m 0,µ is defined as the closure of C ∞ c (Ω) in W 1,m µ (Ω). We set H 1 µ (Ω) = W 1,2 µ (Ω) and H 1 0,µ (Ω) = W 1,2 0,µ (Ω), which are the Hilbert spaces where the linearized operator associated to equation (1) is defined. Notice that, for p > 2, the space W 1,p (Ω) is continuously embedded in H 1 µ (Ω) if µ = |∇u| p−2 .
3. Main estimates. In this section we state and prove our main results, starting with some local estimates of the second derivatives of solutions of equation (1).
We remark that the results in this section hold also if in the righthand side of equation (1) the function f depends on x (instead on u). Actually, the same proofs work with minor modifications. For u ∈ W 1,p (Ω), the weak form of (1) is: Our main step is to compute the so called linearized equation of (1). We set Z = {x ∈ Ω : ∇u(x) = 0} and we considerψ ∈ C ∞ c (Ω \ Z). For any i = 1, ..., n, we take ψ =ψ i in (4) and, integrating by parts, we get: In the sequel we denote by ∇u i and u ij the second derivatives of u outside Z and thought extended equal to 0 inside Z. Actually, at the end of our accounts, we achieve the suitable regularity to ensure that these derivatives coincide with the distributional second derivatives of u in the whole Ω. We remark that, in order to achieve the estimate for the Hessian, it is required the local Lipschitz regularity of solutions, ensured if we take for example minima of functional in (2), which are C 1,α for some α ∈ (0, 1), as proved in [4]. Moreover, we remark that the strict positivity of f does not need to prove the following proposition, while it is required in order to give an estimate of the inverse of the weight.
Thanks to the above estimate on the Hessian of u we are able to prove the following proposition, for which the positivity of f will be needed. We first prove an estimate of 1 |∇u| on balls and then we conclude by means of a covering argument.
Theorem 4.2 is a crucial tool to prove the following weak comparison principle in small domain. Theorem 4.3 (Weak comparison principle). Let u, v ∈ C 1 (Ω) be positive solutions of (4) and let D ⊂⊂ Ω be such that u ≤ v on ∂D. Then there exists δ > 0 such that, if |D| < δ, there holds u ≤ v in D.
Proof. We recall that (see [8, Lemma 2.1]) for every p > 1 there exists c > 0 such that for every ξ, η ∈ R n . By (4) written for u and v with ψ = (u − v) + χ D (being χ D the characteristic function of D), we get: By (33) and (34) and recalling that a(x) is positive and that f is locally Lipshitz, we infer: If p ∈ (1, 2), using classical Poincaré inequality, by (35) and recalling that ∇u and ∇v are bounded, we get where C(|D|) → 0 as |D| → 0 and hence the thesis follows choosing |D| small enough.