Potential estimates in Orlicz spaces
type: Item
instance of: scholarly article
title: Potential estimates in Orlicz spaces
zbMATH Open document ID: 0686.31002
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Using the capacity method, the author establishes inequalities useful in solving quasilinear Dirichlet problems of the form \[ (1-\Delta)^ mu=f(x,u)\quad in\quad \Omega \subset {\mathbb{R}}^ n,\quad u(x)=0\quad on\quad \partial \Omega, \] by means of topological methods. More precisely, he considers continuous even functions \(M_ 0\), M: \({\mathbb{R}}\to {\mathbb{R}}\) for which there are constants \(C_ 1\), \(C_ 2\) such that (a) \(M_ 0\) is strictly increasing in \(| t|\) and M is convex with \(M(2t)\leq C_ 1M(t),\) \(t>0;\) (b) \(M_ 0^{-1}(\int^{\infty}_{0}M_ 0(g(t))dM_ 0(t))\leq C_ 2M^{-1}(\int^{\infty}_{0}M(g(t))dM(t))\) for all decreasing functions g(t)\(\geq 0;\) (c) \(M_ 0(t)\to \infty\), M(t)\(\to \infty\) as \(t\to \infty\) and \(M(0)=0.\) There are interesting examples for which these conditions hold. Defining \(\rho_{\nu}(u,M)=\int M(u)d\nu,\) the author determines sufficient conditions on \(M_ 0\), M, \(\nu\) and \(\mu\) under which \[ M_ 0^{- 1}(\rho_{\nu}(u,M_ 0))\leq CM^{-1}(\rho_{\nu}((1-\Delta)^ mu,M)) \] holds for all \(u\in C^{\infty}({\mathbb{R}}^ n)\) and \[ M_ 0^{- 1}(\rho_{\mu}(u,M_ 0))\leq CM^{-1}(\rho_{\nu}(\Delta^ mu,M)) \] holds for all \(u\in C^{\infty}_ 0({\mathbb{R}}^ n).\) The author also offers applications and comments.
reviewed by: Patrizia Pucci
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 31B30
zbMATH DE Number: 4124068
zbMATH Keywords: potential estimates, Orlicz spaces, Dirichlet problems
author: Martin Schechter
MaRDI profile type: Publication
full work available at URL: pjm.1988.133.381
OpenAlex ID: W2005777811
DOI: 10.2140/PJM.1988.133.381
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description: scientific article; zbMATH DE number 4124068