Lipschitz regularity for Poisson equations involving measures supported on C^1,Dini interfaces

2024/05/09 by Iñigo U. Erneta, Erneta, Iñigo U., María Soria-Carro +1
Computer Science · Mathematics · #35B65 #35J25 (Primary) 35J05 (Secondary) #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.2405.06028

openalex publication_date 2024/05/09 · openalex created_date 2024/05/14 · openalex updated_date 2026/07/28

Abstract

We prove optimal Lipschitz regularity of solutions to Poisson's equation with measure data supported on a C^1,Dini interface and with C^0,Dini density. We achieve this by deriving pointwise gradient estimates on the interface, further showing the piecewise differentiability of solutions up to this surface. Our approach relies on perturbation arguments and estimates for the Green's function of the Laplacian. Additionally, we provide sharp counterexamples highlighting the minimality of our assumptions.

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