The Dirichlet problem for a prescribed mean curvature equation

TL;DR

This paper investigates a prescribed mean curvature problem, establishing existence of solutions near minimal surfaces when the prescribed vector field is small in a specific Sobolev norm.

Contribution

It provides a new existence result for prescribed mean curvature surfaces under small perturbations in a sharp Sobolev norm.

Findings

Existence of solutions near minimal surfaces under small prescribed vector fields

Solution existence depends on the smallness in a dimensionally sharp Sobolev norm

Theoretical framework for prescribed mean curvature problems

Abstract

We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.