On steady boundaries under the scheme of viscosity solutions
type: Item
instance of: scholarly article
title: On steady boundaries under the scheme of viscosity solutions
zbMATH Open document ID: 0884.35043
author: Zhongdan Huan
published in: Journal of Differential Equations
publication date: 1997-09-29T00:00:00Z
review text: The possible boundary values of viscosity solutions on removable boundaries are studied. On such boundaries the values of a solution are in general determined by the boundary conditions on the other part of the boundary and there is little information on those values. This work is concerned with the situation where the values depend only on the removable boundary itself, such removable boundaries will be called steady boundaries. Removable boundaries are related to degeneracy of elliptic equations. The main result of this paper shows that, in some assumptions, for any subsolution \(u\in USC(\overline\Omega)\) and supersolution \(\nu\in LSC(\overline \Omega)\), \[ u_b(x)= \limsup_{\Omega \ni y\to x} u(y) \quad \text{and} \quad v_b(x) =\liminf_{\Omega \ni y\to x} \nu(y) \] are subsolutions, resp. supersolutions of \(F(x,w(x), Dw(x), D^2w(x)) =0\) in \(B_r(\widehat x) \cap\partial \Omega\). Applications of the main theorem on studying asymptotic behavior are given in the last section.
Mathematics Subject Classification ID: 35J67
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 1067764
zbMATH Keywords: removable boundary
zbMATH Keywords: subsolution
zbMATH Keywords: supersolution
zbMATH Keywords: asymptotic behavior
reviewed by: Vasile Iftode
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3290
OpenAlex ID: W1999852628
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DOI: 10.1006/JDEQ.1997.3290
description: scientific article; zbMATH DE number 1067764