On a parameter dependence of solvability of the Dirichlet problem for non-parametric surfaces of prescribed mean curvature
Let \(\Omega\) be a bounded domain in \(R^ n\). Let \(c\geq 0\) be a given function on \(\Omega\), and \(\phi\) a function on \(\partial \Omega\). The author studies the problem of finding a function in BV(\(\Omega)\) which minimizes the functional \[ J_ T(u)=\int_{\Omega}(1+| \nabla u|^ 2)^{1/2} dx-T\int_{\Omega}cu dx+\int_{\partial \Omega}| u-\phi | dH_{n-1}, \] where \(T>0\) is a parameter. He proves that \(J_ T(u)\) is unbounded from below if \(T>T^*=\inf \{H_{n-1}(E)/\int_{E}c dx;\quad E\subset \Omega \}.\) He also discusses the existence, regularity and dependence on T of the solution of the variational problem when \(T<T^*\), and presents some results for the critical case \(T=T^*\).
non-parametric surfaces of prescribed mean curvature
scientific article; zbMATH DE number 3899647