Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions
type: Item
instance of: scholarly article
title: Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions
zbMATH Open document ID: 1160.35340
author: M. Motta
author: C. Sartori
published in: Discrete and Continuous Dynamical Systems
publication date: 2008-09-29T00:00:00Z
review text: In this paper, the authors study the following mixed boundary value problem \[ \begin{aligned} G(x,u,Du,D^2u)=0 &\quad\text{in } {\mathcal O},\\ u=\psi\;\text{ or }\;G(x,u,Du,D^2u)=0 &\quad\text{on } \Gamma_1,\\ G(x,u,Du,D^2u)\leq0 &\quad\text{on } \Gamma_2,\\ u=\psi &\quad\text{on } \Gamma_3, \end{aligned} \] where \(\mathcal O\) is an open set of \({\mathbb R}^n\), which is not necessarily bounded, \(\Gamma_1,\Gamma_2,\Gamma_3\) are pairwise disjoint and \(\Gamma_1\cup\Gamma_2\cup\Gamma_3=\partial{\mathcal O}\), and finally \(G\) is a second order degenerate elliptic operator of Bellman-Isaacs type. The equation and the boundary conditions have to be interpreted in the viscosity sense. Such a problem may arise from exit or stopping time stochastic differential games or optimal control problems with constraints. For such a problem, the authors give a comparison and uniqueness result as well as an existence result. The novelty of those results is due to the presence of the three different boundary conditions in connection with the fact that \(G\) is a second order operator.
reviewed by: Fabio Bagagiolo
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60G40
zbMATH DE Number: 5347944
zbMATH Keywords: viscosity solutions
zbMATH Keywords: stochastic control problems
zbMATH Keywords: stochastic differential games
MaRDI profile type: Publication
full work available at URL: dcds.2008.20.739
OpenAlex ID: W2000249442
DOI: 10.3934/DCDS.2008.20.739
description: scientific article; zbMATH DE number 5347944