The prescribed mean curvature measure equation in non-parametric form

2023/02/21 by Gian Paolo Leonardi, Leonardi, Gian Paolo, Giovanni E. Comi +1 · 1 citation
Mathematics · #46N10 (Secondary) #49Q10 (Primary) 35J93 #49Q20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2302.10592

openalex publication_date 2023/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a weak formulation of the non-parametric prescribed mean curvature equation with measure data and show the existence and several properties of BV solutions under natural assumptions on the prescribed measure. Our approach does not rely on approximate or viscosity-type solutions. It requires combining various ingredients, including Anzellotti's pairing theory for divergence-measure fields and its recent developments, a refinement of Anzellotti-Giaquinta approximation, and convex duality theory. We also prove a Gamma-convergence result valid for suitable smooth approximations of the prescribed measure, and a maximum principle for continuous weak solutions. We finally construct some examples of non-uniqueness, showing at the same time the need for the continuity assumption in the maximum principle and an unexpected feature of weak solutions.

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• Existence theory for linear-growth variational integrals with signed measure data

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