The prescribed mean curvature measure equation in non-parametric form

Gian Paolo Leonardi; Giovanni Eugenio Comi

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arXiv · 2302.10592

The prescribed mean curvature measure equation in non-parametric form

Gian Paolo Leonardi·Giovanni Eugenio Comi

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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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Gian Paolo Leonardi, Giovanni Eugenio Comi. 2023-02-21. The prescribed mean curvature measure equation in non-parametric form. https://arxiv.org/abs/2302.10592

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