The prescribed mean curvature equation for nonparametric surfaces
Consider the Dirichlet problem in a bounded \(C^{1,1}\) domain \(\Omega \subset \mathbb{R}^n\), for an embedded surface given by the graph of a function \(u:\overline {\Omega} \to R\), for the prescribed mean curvature equation: \({\text div} ({\nabla u \over \sqrt{1 +|\nabla u|^2}})=nH(x,u)\) in \(\Omega\) and \(u=g\) in \(\partial \Omega\). The authors obtain existence and uniqueness results in the Sobolev space \(W^{2,p}\). Moreover, under appropriate conditions, it is proved that the set of surfaces with mean curvature \(H\) is a connected subset of \(W^{2,p}\).
scientific article; zbMATH DE number 1879997