On a parameter dependence of solvability of the Dirichlet problem for non-parametric surfaces of prescribed mean curvature
type: Item
instance of: scholarly article
title: On a parameter dependence of solvability of the Dirichlet problem for non-parametric surfaces of prescribed mean curvature
zbMATH Open document ID: 0564.49023
author: Shōichirō Takakuwa
published in: Tokyo Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: 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 \(T=T^*\).
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 3899647
zbMATH Keywords: non-parametric surfaces of prescribed mean curvature
zbMATH Keywords: existence
zbMATH Keywords: regularity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270151736
OpenAlex ID: W2061833926
DOI: 10.3836/TJM/1270151736
description: scientific article; zbMATH DE number 3899647