Mean convexity of the zero set of symmetric minimal surfaces
Let \(\Omega\subset\mathbb R^n\) be a bounded open set, \(\alpha>0\) a given constant. The author considers the functional \[ \mathcal F(u):=\int_{\Omega}u^{\alpha}\sqrt{1+| Du|^2}\, dx \] defined on the class \[ BV_+^{1+\alpha}(\Omega):=\big\{u\in L^{1+\alpha}(\Omega):u\geq 0,\, u^{1+\alpha}\in BV(\Omega)\big\} \] as an \(n\)-dimensional generalization of the catenary problem, that is, the problem to find radially symmetric minimal surfaces bounded by two disks in \(3\)-dimensional space, and studies the geometric and analytic properties of the boundary of the coincidence set \(\{u = 0\}\) of a local minimizer \(u\). He shows that minimizers are elements of \(W^{1,1}_{\mathrm{loc}}(\Omega)\), and that the coincidence set is a set of locally finite perimeter in \(\Omega\), which, in case \(\alpha\geq 1\), has nonnegative inward mean curvature in the variational sense, i.e., is mean convex. In particular, if \(\alpha\geq 1\), the author proves the inequality \[ \int_{\Omega}| D\chi_{\{u=0\}\cap E}|\leq\int_{\Omega}| D\chi_E|-\int_{E\cap\{u>0\}}\frac{\alpha}{u\sqrt{1+| Du|^2}}\, dx \] for all sets \(E\subset\subset\Omega\) of finite perimeter.
scientific article; zbMATH DE number 6830040