Viscosity solutions of nonlinear second order elliptic PDEs involving nonlocal operators
type: Item
instance of: scholarly article
title: Viscosity solutions of nonlinear second order elliptic PDEs involving nonlocal operators
zbMATH Open document ID: 0814.35142
author: Katsuyuki Ishii
author: N. Yamada
published in: Osaka Journal of Mathematics
publication date: 1994-07-07T00:00:00Z
review text: Nonlinear partial differential equations of the form \(F(x,u, Du, D^ 2u, u - Mu) = 0\) in \(\Omega\) and \(\beta (x,u,u - Mu) = 0\) on \(\partial \Omega\), where \(M\) is a nonlocal operator, are studied in terms of viscosity solutions. Using monotonicity, ellipticity and local Lipschitz conditions on \(F\) it is shown that if there is an u.s.c. (viscosity) subsolution and a l.s.c. (viscosity) supersolution, then a unique solution exists. The results are applied to two model problems \[ \max \{Lu - f,u - \varphi \} = 0 \quad \text{in} \quad \Omega, \quad u(x) = \int_ \Omega u(y) Q(dy,x),\;x \in \partial \Omega, \] where \(L\) is an integro-differential operator and \[ \max \{Lu - f,u - Mu\} = 0 \quad \text{in} \quad \Omega, \quad \max \{u - g,u - Mu\} = 0 \quad \text{on} \quad \partial \Omega, \] where \(L\) is a second order elliptic operator.
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 579206
zbMATH Keywords: nonlocal operators
zbMATH Keywords: viscosity solutions
zbMATH Keywords: subsolution
zbMATH Keywords: supersolution
zbMATH Keywords: unique solution
zbMATH Keywords: integro-differential operator
zbMATH Keywords: elliptic operator
reviewed by: S. P. Banks
MaRDI profile type: Publication
Recommended article: Q4321159
Recommended article: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations
Recommended article: Local property of viscosity solutions of fully nonlinear second order elliptic partial differential equations
Recommended article: Q4384601
Recommended article: Viscosity solutions of nonlinear second-order partial differential equations in hilbert spaces
Recommended article: Viscosity solutions of nonlinear systems of degenerated elliptic equations of second order
Recommended article: Q3979536
Recommended article: Q3475669
Recommended article: On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE's
Recommended article: Unbounded viscosity solutions of nonlinear second order PDE's
description: scientific article; zbMATH DE number 579206