Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set
type: Item
instance of: scholarly article
title: Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set
zbMATH Open document ID: 1384.49034
author: Tobias Tennstädt
published in: Calculus of Variations and Partial Differential Equations
publication date: 2017-04-13T00:00:00Z
review text: 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)].
reviewed by: Manuel Ritoré
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35H30
zbMATH DE Number: 6705788
zbMATH Keywords: Hölder continuity
zbMATH Keywords: singular minimal surface equation
zbMATH Keywords: mean convexity
MaRDI profile type: Publication
full work available at URL: s00526-016-1106-4
OpenAlex ID: W2561478678
Wikidata QID: Q115387311
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DOI: 10.1007/S00526-016-1106-4
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description: scientific article; zbMATH DE number 6705788