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Zaremba problem with degenerate weights

Anna Kh. Balci, Ho-Sik Lee

Abstract

We establish Zaremba problem for Laplacian and $p$-Laplacian with degenerate weights when the Dirichlet condition is only imposed in a set of positive weighted capacity. We prove weighted Sobolev-Poincaré inequality with sharp scaling-invariant constants involving weighted capacity. Then we show higher integrability of the gradient of the solution (Meyers estimate) with minimal conditions on the part of the boundary where the Dirichlet condition is assumed. Our results are new both for the linear $p=2$ and nonlinear case and include problems with the weight not only as a measure but also as a multiplier of the gradient of the solution.

Zaremba problem with degenerate weights

Abstract

We establish Zaremba problem for Laplacian and -Laplacian with degenerate weights when the Dirichlet condition is only imposed in a set of positive weighted capacity. We prove weighted Sobolev-Poincaré inequality with sharp scaling-invariant constants involving weighted capacity. Then we show higher integrability of the gradient of the solution (Meyers estimate) with minimal conditions on the part of the boundary where the Dirichlet condition is assumed. Our results are new both for the linear and nonlinear case and include problems with the weight not only as a measure but also as a multiplier of the gradient of the solution.
Paper Structure (7 sections, 10 theorems, 136 equations)

This paper contains 7 sections, 10 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Zaremba problem with degenerate weights as measures
  4. Weighted Sobolev-Poincaré inequality with optimal constants
  5. Main settings, existence and uniqueness
  6. Higher integrability of the gradient of the solution
  7. Zaremba problem with degenerate weights as multipliers

Key Result

lemma 1

For $1<p<\infty$ and $\mu(\cdot)\in\mathcal{A}_p$, there exist $c=c(n,p,[\mu]_{\mathcal{A}_p})\geq 1$ and $q_0=q_0(n,p,[\mu]_{\mathcal{A}_p})<p$ such that for any $q\in[q_0,p]$, $Q_r\subset\setR^n$ and $u\in W^{1,q}_{0,\mu}(Q_r)$. Moreover, we have where $A=\mean{u}_{Q_r}$ or $A=\frac{1}{\mu(Q_r)}\int_{Q_r}u(x)\mu\,dx$.

Theorems & Definitions (23)

  • definition 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • definition 2
  • proposition 1
  • proof
  • remark 1
  • corollary 1
  • ...and 13 more