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Regularity and symmetry results for the vectorial p-Laplacian

Luigi Montoro, Luigi Muglia, Berardino Sciunzi, Domenico Vuono

Abstract

We obtain some regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $Ω$}\,.$$ More precisely we address the issue of second order estimates for the stress field. As a consequence of our regularity results we deduce a weighted Sobolev inequality that leads to weak comparison principles. As a corollary we run over the moving plane technique to deduce symmetry and monotonicity results for the solutions, under suitable assumptions.

Regularity and symmetry results for the vectorial p-Laplacian

Abstract

We obtain some regularity results for solutions to vectorial -Laplace equations More precisely we address the issue of second order estimates for the stress field. As a consequence of our regularity results we deduce a weighted Sobolev inequality that leads to weak comparison principles. As a corollary we run over the moving plane technique to deduce symmetry and monotonicity results for the solutions, under suitable assumptions.
Paper Structure (3 sections, 7 theorems, 114 equations)

This paper contains 3 sections, 7 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Regularity results for second derivative
  3. Comparison Principles and Symmetry

Key Result

Theorem 1.1

Let $\Omega$ be a bounded smooth domain and $p>1$. Let $\boldsymbol{u}\in C^1({\Omega})$ be a weak solution of system1, with where $0<s<n-\gamma$ and $\gamma<n-2$ ($\gamma=0$ if $n=2$). If $1<p\leq 2$ let us assume $0\leq \alpha <p-1$ and if $p>2$ let $\alpha\in [0,1)$. Then, for any $\tilde{\Omega} \subset\subset \Omega$, it follows that uniformly for any $y\in \tilde{\Omega}$, with $C=C(\bolds

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Weak Comparison Principle in small domains
  • Theorem 1.6
  • Remark 1.7
  • proof : Proof of Theorem \ref{['teosecond']}
  • proof : Proof of Theorem \ref{['Reg2']}
  • Theorem 2.1
  • ...and 7 more