Global Sobolev inequalities and Degenerate P-Laplacian equations

TL;DR

This paper establishes that local weak Sobolev inequalities imply global Sobolev inequalities for degenerate P-Laplacian equations, using existence and regularity results for solutions of associated boundary value problems.

Contribution

It provides sufficient conditions under which local Sobolev inequalities extend to global ones in the context of degenerate P-Laplacian equations with zero order terms.

Findings

Global Sobolev inequalities follow from local inequalities under certain conditions.

Existence and boundedness of solutions to degenerate P-Laplacian Dirichlet problems are proven.

The results apply to equations involving a quasi-metric and semi-definite matrix functions.

Abstract

We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$, and let $Q$ be an $n\times n$ semi-definite matrix function defined on $\Omega$. For an open set $\Theta\Subset\Omega$, we give sufficient conditions to show that if the local weak Sobolev inequality % \[ \Big(\fint_B |f|^{p\sigma}dx\Big)^\frac{1}{p\sigma} \leq C\Big[ r(B)\fint_B |\sqrt{Q}\nabla f|^pdx + \fint_B |f|^pdx\Big]^\frac{1}{p} \] holds for some $\sigma>1$, all balls $B\subset \Theta$, and functions $f\in Lip_0(\Theta)$, then the global Sobolev inequality \[ \Big(\int_\Theta |f|^{p\sigma}dx\Big)^\frac{1}{p\sigma} \leq C\Big(\int_\Theta |\sqrt{Q}\nabla f(x)|^pdx\Big)^\frac{1}{p} \] also holds. Central to our proof is showing the existence and boundedness of solutions of the Dirichlet problem \[ \begin{cases} \mx_{p,\tau} u & = \varphi \text{in} \Theta \\ u & = 0 \text{in} \partial \Theta, \end{cases} \] where $\mx_{p,\tau}$ is a degenerate $p$-Laplacian operator with a zero order term: \[ \mx_{p,\tau} u = \text{div}\Big(\big|\sqrt{Q} \nabla u\big|^{p-2}Q\nabla u\Big) - \tau |u|^{p-2}u. \]