On the Neumann problems for certain degenerate elliptic operators
type: Item
instance of: scholarly article
title: On the Neumann problems for certain degenerate elliptic operators
zbMATH Open document ID: 0802.35053
author: Toshio Horiuchi
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-08-08T00:00:00Z
review text: The aim of this paper is to study the homogeneous Dirichlet and Neumann problems for the degenerate elliptic equation \[ -\text{div} (\varphi^ \alpha \nabla u) + c(x) \varphi^{\alpha - 1} u = f \quad \text{ in } \quad \Omega, \] where \(\alpha \in C^ 2 (\Omega)\), and \(\varphi (x)\) is a given nonnegative smooth function on \(\Omega\) equivalent to a distance to the boundary: \(\Omega = \{\varphi (x) > 0\}\), \(\partial \Omega = \{\varphi (x) = 0\}\), \(\alpha (x) \in C^ 2 (\Omega)\). It is proved that if \(\alpha (x) < 0\) and \(\partial \alpha/ \partial n = 0\) on \(\partial \Omega\) and \(\inf_{x \in \Omega} c(x) > 0\), then for every \(f \in C^ \mu (\Omega)\) there exists a unique solution \(u = \varphi^{1 - \alpha} v\) to the Dirichlet problem and \(\varphi v \in C^{2 + \mu} (\overline \Omega)\). If \(\alpha (x) > - 1\) and \(\max \{-1 + \mu,- 1 + \mu-\alpha (x), - 1 - \alpha (x)/2\} < \tau < 0\) on \(\partial \Omega\), then the Neumann problem has a unique solution in the space \(\Lambda^{2 + \mu}_{2 + \tau} (\Omega)\) for every \(f\) such that \(f/ \varphi^ \alpha \subset \Lambda^ \mu_ \tau (\Omega)\). The proof of the theorem is based upon a priori estimates for solutions, the method of continuity and the Green function for the approximate problem. An example of an unbounded solution to the Neumann problem is given.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 597342
zbMATH Keywords: uniqueness
zbMATH Keywords: homogeneous Dirichlet and Neumann problems
zbMATH Keywords: Green function
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
full work available at URL: pjaa.69.372
OpenAlex ID: W2042829165
cites work: Q3315885
cites work: The dirichlet problem for the bezgman laplacian. II
cites work: Existence and uniqueness of classical solutions for certain degenerated elliptic equations of the second order
DOI: 10.3792/PJAA.69.372
description: scientific article; zbMATH DE number 597342