On elliptic equations involving surface measures

2022/12/13 by Marius Müller, Müller, Marius · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2212.06494

openalex publication_date 2022/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE -div(A(x) ∇ u) = Q H^n-1 \llcorner Γ in a smooth domain Ω⊂ ℝⁿ. Here Γ is a C^1,α-regular hypersurface, Q∈ C^0,α is a density on Γ, and the coefficient matrix A is symmetric, uniformly elliptic and W^1,q-regular (q > n). We also discuss optimality of these assumptions on the data. The equation can be understood as a special coupling of two A-harmonic functions with an interface Γ. As such it plays an important role in several free boundary problems, as we shall discuss.

Cited by

• An anisotropic Alt-Caffarelli problem of higher order

Related

• The Poisson equation involving surface measures
• Lipschitz regularity for p-harmonic interface transmission problems
• Lipschitz regularity for Poisson equations involving measures supported on C^1,Dini interfaces
• Lipschitz regularity for solutions of a general class of elliptic equations
• Regularity of Weak Solutions for Singular Elliptic Problems Driven by\n m-Laplace Operator