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 Ω⊂Rn, let ρ be a quasi-metric on Ω, and let Q be an n×n positive semi-definite matrix function defined on Ω. For an open set Θ⋐Ω, we give sufficient conditions to show that if the local weak Sobolev inequality(⨏B|f|pσdx)1pσ≤C[r(B)⨏B|Q∇f|pdx+⨏B|f|pdx]1p holds for some σ>1, all balls B⊂Θ, and functions f∈Lip0(Θ), then the global Sobolev inequality(∫Θ|f|pσdx)1pσ≤C(∫Θ|Q∇f(x)|pdx)1p also holds. Central to our proof is showing the existence and boundedness of solutions of the Dirichlet problem{Xp,τu=φ in Θu=0 in ∂Θ, where Xp,τ is a degenerate p-Laplacian operator with a zero order term:Xp,τu=div(|Q∇u|p−2Q∇u)−τ|u|p−2u.