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

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 R n \mathbb {R}^n of the union of a finite number of bounded Lipschitz domains. The mean curvature H = H ( x , t ) H=H(x,t) is assumed to have absolute value controlled from above by a locally bounded, L p L^p -function, p ∈ [ 1 , 2 n / ( n + 2 ) ] p\in [1,2n/(n+2)] , n ≥ 3 n\geq 3 .