Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set
The main result in this paper, stated as Theorem 1, is the following: Let \(\Omega\subset\mathbb{R}^n\) be a bounded open set (no boundary regularity is assumed), \(\psi\in C_c^2(\mathbb{R}^n)\) a positive function, and \(u\geq 0\) in \(C^0(\overline{\Omega})\cap C^2(\{u>0\})\) a solution of the equation \[ \text{div}\left({Du \over \sqrt{1+|Du|^2} }\right)={\alpha \over u\sqrt{1+|Du|^2}}\quad \text{in }\{u>0\}, \quad u=\psi\quad\text{on }\partial\Omega, \tag{*} \] where \(\alpha>0\) is a given constant. Then \(u\in C^{0,1/2}(\overline{\Omega})\) and the free boundary \(\partial\{u=0\}\) is strictly mean convex assuming it is of class \(C^2\) and \(\alpha\geq 1\). The first equation in (*) is equivalent to the Euler equation for \(\mathcal{F}(u)=\int_\Omega u^\alpha \sqrt{1+|Du|^2}\) when \(u>0\). If \(\alpha\) is an integer number, the functional \(\mathcal{F}\) is (up to a constant) the area of the rotated graph of \(u\) in \(\mathbb{R}^{n+\alpha+1}\). Theorem 1 is a consequence of Proposition 1 in the paper, whose proof uses a method in [\textit{N. Korevaar} and \textit{L. Simon}, Math. Z. 197, No. 4, 457--464 (1988; Zbl 0625.35034)].
scientific article; zbMATH DE number 6705788
scientific article