Global Sobolev inequalities and Degenerate P-Laplacian equations

Original · EN

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