Harnack’s inequality and Hölder continuity for weak solutions of degenerate quasilinear equations with rough coefficients

Abstract

Communicated by Enzo Mitidieri MSC: 35J70 35J60 35B65 Keywords: Quasilinear equations Degenerate elliptic partial differential equations Degenerate quadratic forms Weak solutions Regularity Harnack's inequality Hölder continuity Moser method a b s t r a c tWe continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form div (A(x, u, ∇u)) = B(x, u, ∇u) for x ∈ Ω as considered in our paper Monticelli et al. (2012). There we proved only local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin (1964) andN. Trudinger (1967) for quasilinear equations, as well as ones for subelliptic linear equations obtained in Wheeden (2006, 2010).

Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients

“…For clarity we will always write QH 1 ( ) and QH 1 0 ( ) in place of W 1,2 Q ( ) and W 1,2 Q,0 ( ) respectively, taking the isomorphism in context. We adopted this notation in lieu of W 1,2 Q ( ) and W 1,2 Q,0 ( ), as used in [1,11,13,14], in order to agree with classical literature, see for example [12], where it is convention that "W " spaces refer to Sobolev spaces defined with respect to distributional derivatives. We also mention that it is possible to introduce definitions and make similar considerations for the spaces QH 1,p ( ), QH…”

“…We invite the interested reader to see [1,11,14] for the construction of these spaces and related objects.…”

“…Before continuing we briefly describe how the paper is organized. Our results are developed in a general axiomatic framework similar to those used in [1,Section 3], [11,13,14], that we outline in Sections 1.1 through 1.6. This axiomatic setting includes the definition of geometric homogeneous space, of degenerate Sobolev spaces associated with a nonnegative definite symmetric measurable matrix function comparable to P and also gives Poincaré-Sobolev type inequalities.…”

“…Section 4 contains a maximum principle for weak solutions of Xu ≤ 0 and in Section 5 we demonstrate a relationship between compact embeddings of Sobolev spaces and global Poincaré inequalities with gain; we refer the reader to Theorems 4.3 and 5.1 for these results. All of our results are developed in the spirit of [2] and [4] using ideas presented in [1,[9][10][11][13][14][15][16][17], and other related works.…”

Global Sobolev inequalities and degenerate p-Laplacian equations

“…We prove the existence of solutions using Minty's theorem in Section 4 and we prove boundedness using ideas from [CRW,MRW1] in Section 5. Finally, in Section 6 we prove Theorem 2.7.…”