On two free boundary problems in potential theory
type: Item
instance of: scholarly article
title: On two free boundary problems in potential theory
zbMATH Open document ID: 0760.31003
DOI: 10.1016/0022-247X(91)90333-U
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: The authors consider two free boundary value problems for the equation \[ \sum^ N_{k=1}{\partial\over\partial x_ k}\left(g\left(|\nabla u|^ 2\right){\partial u\over\partial x_ k}\right)=0 \tag{1} \] in the cylinder \(\Omega\subset R^ N\) with the free bases \(\Gamma_ 0\), \(\Gamma_ 1\) and the lateral surface \(\Gamma_ c\). It is assumed that \(g\in C^ \infty\) and \(g(\xi)+2\xi g'(\xi)>0\) \(\forall\xi\geq 0\). On the free boundary simultaneously the Dirichlet and Neumann conditions with constant data are prescribed. It is proved that the boundary value problem is solvable and that the free boundaries are contained in two different horizontal hyperplanes and that the solution depends only on \(x_ N\). The second problem is considered for the equation (1) with the right-hand side equal to \(-2\) in \(\Omega=\Omega_ 0\backslash\bigcup^ m_{k=1}\overline\Omega_ k\), where \(\Omega_ k\) are disconnected contractible regions in \(R^ N\), contained in a bounded domain \(\Omega_ 0\subset R^ N\). The boundary conditions are \[ u=0,\quad{\partial u\over\partial n}=a_ 0=\text{const}\quad \text{ on }\partial\Omega_ 0, \] \[ u=c_ k,\quad{\partial u\over\partial n}=ak=\text{const}\quad\text{ on } \partial\Omega_ k,\quad k=1,\dots,m. \] It is shown that the problem is solvable iff \(\Omega\) is spherically symmetric.
Mathematics Subject Classification ID: 31B35
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 57715
zbMATH Keywords: solvability theorems
zbMATH Keywords: Dirichlet and Neumann conditions
author: L. E. Payne
author: G. A. Philippin
MaRDI profile type: Publication
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full work available at URL: 0022-247x(91)90333-u
OpenAlex ID: W2043053491
description: scientific article; zbMATH DE number 57715