The prescribed mean curvature equation in weakly regular domains

TL;DR

This paper extends the existence and uniqueness results for solutions to the prescribed mean curvature equation to domains with very mild regularity, using advanced tools like a generalized Gauss-Green theorem and weak Young's law.

Contribution

It generalizes prior smooth-domain results to weakly regular domains, applying to capillary problems and introducing new analytical tools.

Findings

Existence and uniqueness of solutions in weakly regular domains

Application to capillary problems for wetting fluids

Development of a generalized Gauss-Green theorem and weak Young's law

Abstract

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda,r_{0})$-minimizers of the perimeter.