Problèmes de Neumann quasilinéaires. (Quasilinear Neumann problems)
type: Item
instance of: scholarly article
title: Problèmes de Neumann quasilinéaires. (Quasilinear Neumann problems)
zbMATH Open document ID: 0579.35028
DOI: 10.1016/0022-1236(85)90019-9
author: Francis Gimbert
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: The main result is the existence and uniqueness of a solution (u,\(\lambda)\) satisfying \[ (1)\quad -a_{ij}u_{x_ ix_ j}(x)+\phi (x,Du(x))=\lambda \quad in\quad \partial \Omega;\quad \frac{\partial u}{\partial \gamma}=0\quad on\quad \partial \Omega,\quad \int_{\Omega}udx=0. \] Problem (1) is approximated by \[ - a_{ij}u^{\alpha}_{x_ ix_ j}(x)+\phi (x,Du^{\alpha}(x))+\alpha u^{\alpha}(x)=0\quad in\quad \partial \Omega;\quad \frac{\partial u^{\alpha}}{\partial \gamma}=0\quad on\quad \partial \Omega. \] A priori estimates on \(Du^{\alpha}\) are obtained by use of Bernstein-type comparison functions. Then the convergence results are as follows: \[ u^{\alpha}-\frac{1}{| \Omega |}\int_{\Omega}u^{\alpha}dx\to u\quad and\quad \frac{\alpha}{| \Omega |}\int_{\Omega}u^{\alpha}dx\to -\lambda. \]
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 3927472
zbMATH Keywords: ergodic control
zbMATH Keywords: reflected diffusion
zbMATH Keywords: quasilinear partial
zbMATH Keywords: differential equation
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: A priori estimates
zbMATH Keywords: Bernstein-type comparison functions
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90019-9
OpenAlex ID: W2068361250
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description: scientific article; zbMATH DE number 3927472