Global Sobolev inequalities and Degenerate P-Laplacian equations
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 $Ω\subset\mathbb{R}^n$, let $ρ$ be a quasi-metric on $Ω$, and let $Q$ be an $n\times n$ semi-definite matrix function defined on $Ω$. For an open set $Θ\SubsetΩ$, we give sufficient conditions to show that if the local weak Sobolev inequality % \[ \Big(\fint_B |f|^{pσ}dx\Big)^\frac{1}{pσ} \leq C\Big[ r(B)\fint_B |\sqrt{Q}\nabla f|^pdx + \fint_B |f|^pdx\Big]^\frac{1}{p} \] holds for some $σ>1$, all balls $B\subset Θ$, and functions $f\in Lip_0(Θ)$, then the global Sobolev inequality \[ \Big(\int_Θ |f|^{pσ}dx\Big)^\frac{1}{pσ} \leq C\Big(\int_Θ |\sqrt{Q}\nabla f(x)|^pdx\Big)^\frac{1}{p} \] also holds. Central to our proof is showing the existence and boundedness of solutions of the Dirichlet problem \[ \begin{cases} \mx_{p,τ} u & = φ\text{in} Θ\\ u & = 0 \text{in} \partial Θ, \end{cases} \] where $\mx_{p,τ}$ is a degenerate $p$-Laplacian operator with a zero order term: \[ \mx_{p,τ} u = \text{div}\Big(\big|\sqrt{Q} \nabla u\big|^{p-2}Q\nabla u\Big) - τ|u|^{p-2}u. \]