Global Sobolev inequalities and Degenerate P-Laplacian equations

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: \[ \mx_{p,\tau} u = \text{div}\Big(\big|\sqrt{Q} \nabla u\big|^{p-2}Q\nabla u\Big) - \tau |u|^{p-2}u. \]