Mean convexity of the zero set of symmetric minimal surfaces
type: Item
instance of: scholarly article
title: Mean convexity of the zero set of symmetric minimal surfaces
zbMATH Open document ID: 1386.35117
author: Tobias Tennstädt
published in: Manuscripta Mathematica
publication date: 2018-01-26T00:00:00Z
review text: 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.
reviewed by: Atsushi Fujioka
Mathematics Subject Classification ID: 35J75
Mathematics Subject Classification ID: 35J93
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 6830040
zbMATH Keywords: minimal surfaces
zbMATH Keywords: catenary problem
MaRDI profile type: Publication
full work available at URL: s00229-017-0940-9
OpenAlex ID: W2617975481
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DOI: 10.1007/S00229-017-0940-9
description: scientific article; zbMATH DE number 6830040