A Free Boundary Problem for Minimal Surface Equation

Abstract

In this note we introduced a free boundary problem for minimal surface equation. The following variational problem had been treated by H.W.Alt, L.A.Caffarelli and A.Freedman $[1],[2]$: \{\begin{array}{l}\int_{\Omega}(F(|\nabla u|^{2})+Q^{2}\chi_{u>0})dL^{n}arrow\min u\in IC\equiv\{w\in L_{1oc}^{2}(\Omega)|\nabla w\in L^{2}(\Omega),w=u^{0}on S\}\end{array} (1.1) where $\Omega$ is a connected Lipshitz domain contained in n-dimensional Euclidean space $R$“. $F=F(t)$ is