The Poisson equation involving surface measures

2021/02/27 by Müller, Marius
#35R06 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.00303

Abstract

We prove the (optimal) W^1,∞-regularity of weak solutions to the equation -Δu = Q H^n-1 \llcorner Γ in a domain Ω⊂ ℝⁿ with Dirichlet boundary conditions, where Γ⊂ ⊂ Ω is a compact (Lipschitz) manifold and Q ∈ L^∞(Γ). We also discuss optimality and necessity of the assumptions on Q and Γ. Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt-Caffarelli Problem.

Related

• On elliptic equations involving surface measures
• Lipschitz regularity for Poisson equations involving measures supported on C^1,Dini interfaces
• Lipschitz regularity for p-harmonic interface transmission problems
• Polyharmonic equations involving surface measures
• Lipschitz continuity of solutions of Poisson equations in metric measure spaces