The prescribed mean curvature equation under low regularity assumptions

Abstract: given an open set Omega and a positive constant H, does it exist a cartesian hypersurface defined on Omega whose mean curvature is constantly H? Equivalently, can one find a function u on Omega, whose graph has mean curvature constantly H? This question leads to the nonlinear elliptic prescribed mean curvature PDE.

Foundational results by Concus, Finn, and Giusti establish that, assuming Omega is Lipschitz, there exists a geometric threshold h(Omega) such that existence of solutions is guaranteed if H>h(Omega), while non existence occurs for H<h(Omega). Interesting phenomena arise at the threshold. In the physically relevant case, that is, Omega is 2-dimensional, and assuming C^2 convexity, an elegant geometric criterion on the curvature of Omega characterizes the regimes of existence and non-existence..

In a series of works partly in collaboration with Gian Paolo Leonardi, we extend these results to low regularity settings by removing the Lipschitz assumption on Omega. This necessitates developing a refined functional framework, including the introduction of Gauss—Green formulas under weak regularity conditions. Moreover, we generalize the two-dimensional geometric criterion by relaxing convexity assumptions and relying solely on appropriate one-sided bounds on the reach of Omega.