Matrix $$\mathcal {A}_p$$ Weights, Degenerate Sobolev Spaces, and Mappings of Finite Distortion

Abstract

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.

Two Weight Bump Conditions for Matrix Weights

“…Remark 1.20. As a corollary to Theorem 1.18 we prove two weight estimates for convolution operators and approximations of the identity, generalizing one weight results from [8]. See Corollary 6.1 below.…”

“…This was proved in the one weight case in [8,Theorem 4.9]. The proof is essentially the same, bounding the convolution operator by averaging operators and then applying Theorem 1.18.…”

“…In the one weight, scalar case when p = q Theorem 1.18 was implicit in Jawerth [21]; for the general result in the scalar case, see Berezhnoȋ [1]. In the one weight matrix case, again when p = q, Theorem 1.18 was proved in [8].…”

Poincaré Inequalities and Neumann Problems for thep-Laplacian

“…We refer the interested reader to [CMN,CMR,CRW,MR,MRW1,MRW2,SW1,SW2] for definitions and discussions of these and similar spaces, including examples where the gradient is uniquely defined.…”

“…In this paper we show that this is equivalent to the existence of a regular solution of a Neumann boundary value problem for a degenerate p-Laplacian. We formulate our result in the very general setting of degenerate Sobolev spaces: elliptic operators have been considered in this setting by a number of authors: see, for instance, [CMN,CMR,MR,MRW1,MRW2,SW1,SW2] and the references they contain.…”