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Regularity and Separation for $p$-Laplace operators

Daniel Hauer, Adam Sikora

Abstract

We analyze $p$-Laplace operators with degenerate elliptic coefficients. This investigation includes Grušin type $p$-Laplace operators. We describe a \emph{separation phenomenon} in elliptic and parabolic $p$-Laplace type equations, which provides an illuminating illustration of simple jump discontinuities of the corresponding weak solutions. Interestingly validity of an isoperimetric inequality for considered setting does not imply continuity of elliptic equations. On the other hand, we are able to establish global $L^1$-$L^\infty$-regularization and decay estimates of every mild solution of the parabolic Grušin type $p$-Laplace equation.

Regularity and Separation for $p$-Laplace operators

Abstract

We analyze -Laplace operators with degenerate elliptic coefficients. This investigation includes Grušin type -Laplace operators. We describe a \emph{separation phenomenon} in elliptic and parabolic -Laplace type equations, which provides an illuminating illustration of simple jump discontinuities of the corresponding weak solutions. Interestingly validity of an isoperimetric inequality for considered setting does not imply continuity of elliptic equations. On the other hand, we are able to establish global --regularization and decay estimates of every mild solution of the parabolic Grušin type -Laplace equation.
Paper Structure (21 sections, 17 theorems, 142 equations)

This paper contains 21 sections, 17 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries - our framework
  3. A Riemannian structure
  4. The elliptic setting.
  5. The gradient associated with a Riemannian structure
  6. The degenerate setting
  7. Some examples of degenerate Riemannian structures
  8. Lebesgue and mixed Sobolev spaces
  9. Degenerate p-Laplace operators
  10. Examples of p-Laplace operators
  11. The associated carré du champ
  12. The semigroup generated by the p-Laplace operator
  13. Nash and Sobolev inequalities in Grušin and spaces with monomial weights
  14. Sub-elliptic estimates in Grušin spaces
  15. Sobolev inequality in spaces of monomial weights
  16. ...and 6 more sections

Key Result

Proposition 2.1

For a given Riemannian structure $(\mathbb{R}^{d},\bm{A})$ and $f\in C^{1}(\mathbb{R}^d)$, the gradient$\nabla_{\!\bm{A}}f$ in local coordinates can be rewritten by

Theorems & Definitions (48)

  • Remark
  • Definition 2.1
  • Remark
  • Definition 2.2
  • Proposition 2.1: The Gradient in local coordinates
  • Proposition 2.2
  • Example 2.1
  • Definition 3.1
  • Proposition 3.1
  • proof
  • ...and 38 more