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).
We study degenerate Sobolev spaces where the degeneracy is controlled by a matrix A p weight. This class of weights was introduced by Nazarov, Treil and Volberg, and degenerate Sobolev spaces with matrix weights have been considered by several authors for their applications to PDEs. We prove that the classical Meyers-Serrin theorem, H = W , holds in this setting. As applications we prove partial regularity results for weak solutions of degenerate p-Laplacian equations, and in particular for mappings of finite distortion.
Abstract:We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative quadratic forms on R n . Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegenerate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains in R n , the latter for weights equal to powers of the distance to the boundary. We also derive a compactness result for Lebesgue spaces on quasimetric spaces unrelated to R n and possibly without any notion of gradient.
The General TheoremThe main goal of this paper is to generalize the classical Rellich-Kondrachov theorem concerning compact embedding of Sobolev spaces into Lebesgue spaces. Our principal result applies not only to the classical Sobolev spaces on open sets Ω ⊂ R n but also allows us to treat the degenerate Sobolev spaces defined in [SW2], and to obtain compact embedding of them into various L q (Ω) spaces. These degenerate Sobolev spaces are associated with quadratic forms Q(x, ξ) = ξ ′ Q(x)ξ, x ∈ Ω, ξ ∈ R n , which are nonnegative but may vanish identically in ξ for some values of x. Such quadratic forms and Sobolev spaces arise naturally in the study of existence and regularity of weak solutions of some second order subelliptic linear/quasilinear partial differential equations; see, e.g., [SW1, 2]The Rellich-Kondrachov theorem is frequently used to study the existence of solutions to elliptic equations, a famous example being subcritical and critical Yamabe equations, resulting in the solution of Yamabe's problem; see [Y], [T], [A], [S]. Further applications lie in proving the existence of weak solutions to Dirichlet problems for elliptic equations with rough boundary data and coefficients; see [GT]. In a sequel to this paper, we will apply our compact embedding results to study the existence of solutions for some classes of degenerate equations.
Abstract. We prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p-Laplacian.e Poincaré inequalities are formulated in the context of degenerate Sobolev spaces de ned in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.
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