On a singular variational integral with linear growth. I: Existence and regularity of minimizers (Q1106463)

The authors discuss the problem of minimizing the variational integral \[ (1)\quad F(u)\quad =\quad \int_{\Omega}u\quad \cdot \quad \sqrt{1+| Du| \quad 2}dx \] in a class of functions u: \({\mathbb{R}}\) \(n\supset \Omega \to {\mathbb{R}}\) with prescribed Dirichlet-data and the additional side condition \(u\geq 0\) a.e. The integral is of linear growth and also degenerates at points where u vanishes. In order to prove existence of a minimizer the authors replace (1) by the formally equivalent but less degenerate integral \[ (2)\quad G(v)\quad =\quad \int_{\Omega}\sqrt{v+(1/4)| Dv| \quad 2}dx \] with \(v=u\) 2 and show that a minimizer of (2) leads to a solution of the original problem, which (for \(n\leq 6)\) is continuous throughout \(\Omega\) and real analytic when it is strictly positive. These regularity results are based on a regularity theorem from geometric measure theory concerning minimal surfaces of codimension 1 in a Riemannian manifold.