L ^p Regularization of the Non-Parametric Minimal Surface Problem

Abstract

The non-parametric minimal surface problem is an ill-posed variational problem, even in its relaxed form. Indeed, the relaxed problem is an L ^p type problem, and it is not strictly convex so that it may have more than one solution. Following [2], we use the L ^p regularization technique with p → 1. Under fairly general assumptions, we show that the approximate solutions (u _p ) _p converge strongly in W^l,l to a particular solution of the relaxed problem. Indeed, the so-selected solution is characterized as the unique solution of an auxiliary variational problem involving the integrand t↦ |t|ln|t|.