A uniqueness result for maximal surfaces in Minkowski 3-space

In this paper, we study the Dirichlet problem associated to the maximal surface equation. We prove the uniqueness of bounded solutions to this problem in unbounded domain in R. Introduction We consider the Minkowski space-time L3 i.e. R3 with the following pseudoeuclidean metric 〈x, y〉 = x1y1 + x2y2 − x3y3. We define |x| 2 L = 〈x, x〉. A vector is said to be spacelike if |x|2 L > 0 and a surface S of class C1 is said to be spacelike if | · |2 L is positive definite on the tangent space to S. Such a surface is locally the graph of a function over a domain in R2. If v is a function in a domain Ω in R2 (in the paper, we always assume that Ω has smooth boundary), the graph of v is spacelike if and only if |∇v| < 1. The function v is then Lipschitz continuous and it extends to the closure Ω. We denote by φ the trace v|∂Ω of v on the boundary. The maximal area problem in the class of spacelike surfaces consists in solving the following variational problem: