An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow
Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll, Lisandra Sauer, Rodrigo Soares
TL;DR
The paper develops an explicit barrier-based solvability criterion for the Dirichlet problem of the minimal surface equation on domains with non-mean-convex boundaries, using an ODE-generated barrier $w=\varphi+\psi(d)$ where $d$ is a distance to the boundary. This approach produces practical conditions in the Euclidean setting, extends to unbounded domains, and clarifies the geometric dependence of solvability in Hadamard manifolds by introducing exterior sphere and Hadamard-type conditions. A key contribution is the separation of domain geometry from boundary data, enabling solvability in cases where Jenkins–Serrin fails. The same ODE-derived barrier also provides sub- and supersolutions for the mean curvature flow of graphs, yielding short-time existence for non-mean-convex boundaries and recovering classical results in the mean-convex regime, thereby linking elliptic solvability to parabolic evolution.
Abstract
We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. This condition is derived from a second-order ordinary differential equation whose solution produces a barrier that appears to be novel in the context of barrier constructions. It admits an explicit formulation and, in the setting of Hadamard manifolds, reveals a direct and transparent relationship between the geometry of the domain and the behavior of the boundary data required for solvability.The condition also extends naturally to unbounded domains. In the Euclidean case, it is not only more practical to verify but also less restrictive than the classical Jenkins-Serrin criterion, ensuring the existence of solutions in situations where that approach fails. Furthermore, unlike the Jenkins-Serrin condition, our approach separates the domain's geometric properties from its boundary data, providing a clearer and more manageable framework for solvability analysis. Beyond the stationary theory, the same ODE-generated barrier also yields natural sub- and supersolutions for the mean curvature flow of graphs, allowing one to initiate and control the short-time evolution from non-mean convex boundaries. As a consequence, our barrier construction extends a classical result of Ecker-Huisken on short-time existence of graphical mean curvature flow to domains with non-mean convex boundary; in the mean convex setting, their original result is recovered.