The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

Rafael López

doi:10.3390/math7121211

Abstract

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

Highlights

We investigate the differences and similarities in the study of the solvability of the Dirichlet problem for the constant mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space
We review briefly the state of the art of the Dirichlet problem for the constant mean curvature equation in both spaces
In the Euclidean space and for the minimal case H = 0, the Dirichlet problem in Equation (1) was solved for n = 2 by Finn [6] and in arbitrary dimension by Jenkins and Serrin [7] proving that the mean convexity of the domain Ω yields a necessary and sufficient condition of the solvability of the Dirichlet problem for all boundary values φ: a domain Ω is said to be mean convex if the mean curvature κ∂Ω of

Summary

Introduction

H, there are solutions of Equation (1) defined in the plane R2. In addition, the cylinders u( x1 , x2 ) = 1/H 2 + x12 are other examples of entire solutions of Equations (1)–(3). If u is a solution of Equation (1) with H 6= 0 in E3 , Ω does not contain the closure of a disk of radius 1/| H |. S2 (r ) be a sphere of radius r whose center lies on the straight-line through x and perpendicular to the ( x1 , x2 )-plane. Let us descend S2 (r ) until the first point p of contact with M. Since D ⊂ Ω and M is a graph on Ω, the contact point p must be interior in both surfaces. M and S2 (r ) agree on an open set around p, M is included in a sphere of radius 1/H: this is a contradiction because the orthogonal projection onto R2 would give Ω ⊂ D

Preliminaries

The Constant Mean Curvature in Equation

The Solvability Techniques of the Dirichlet Problem

Height and Gradient Estimates

The Dirichlet Problem with Zero Boundary Values