Spacelike graphs with prescribed mean curvature on exterior domains in the Minkowski spacetime

TL;DR

This paper studies the existence of spacelike graphs with prescribed mean curvature in Minkowski spacetime, providing necessary and sufficient conditions for solutions over exterior domains with bounded mean curvature functions.

Contribution

It establishes a complete characterization for the existence of spacelike solutions with prescribed mean curvature on exterior domains in Minkowski spacetime.

Findings

Derived necessary and sufficient conditions for solution existence.

Extended results to mean curvature functions with bounded $L^p$-norm.

Addressed the Dirichlet problem in unbounded exterior domains.

Abstract

We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a function defined on a domain equal to the complement in $\mathbb R^n$ of the union of a finite number of bounded Lipschitz domains. The mean curvature $H=H(x,t)$ is assumed to have absolute value controlled from above by a locally bounded, $L^p$-function, $p\in [1,2n/(n+2)]$, $n\geq 3$.