Regularity of the solution of a variational inequality for two second order elliptic operators
type: Item
instance of: scholarly article
title: Regularity of the solution of a variational inequality for two second order elliptic operators
zbMATH Open document ID: 0583.49006
author: Raffaele Toscano
author: Angela Gallo
published in: Ricerche di Matematica
publication date: 1984-01-01T00:00:00Z
review text: The authors consider a Lipschitz bounded open subset \(\Omega\) of \({\mathbb{R}}^ n\) and two uniformly elliptic operators \(A_ k\) \((k=1,2)\) given by (with the usual summation convention) \[ A_ k=-D_ j(a^ k_{ij}(x)D_ i)+b^ k_ i(x)D_ i+c^ k(x) \] where (for \(n\geq 3)\) \(a^ k_{ij}\in L^{\infty}(\Omega)\), \(b^ k_ i\in L^ n(\Omega)\), \(c^ k\in L^{n/2}(\Omega)\). Under the assumption that the bilinear forms \[ <A_ ku,v>=\int_{\Omega}[a^ k_{ij}(x)D_ iuD_ jv+b^ k_ i(x)D_ iuv+c^ k(x)uv]dx \] are coercive in \(H^ 1_ 0(\Omega)\), the variational inequality \[ (*)\quad <A_ 1u_ 1,v_ 1- u_ 1>+<A_ 2u_ 2,v_ 2-u_ 2>\geq <f_ 1,v_ 1-u_ 1>+<f_ 2,v_ 2-u_ 2>,\quad (u_ 1,u_ 2)\in K,\quad (v_ 1,v_ 2)\in K \] is considered, where \(K=\{(v_ 1,v_ 2)\in (H^ 1_ 0(\Omega))^ 2: v_ 1+\psi \geq v_ 2\) a.e. on \(\Omega\) \(\}\), \(f_ i\in H^{- 1}(\Omega)\), and \(\psi \in H^ 1(\Omega)\) with \(\psi\geq 0\) on \(\partial \Omega\). The paper is devoted to the study of the regularity properties of the solution of (*).
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 3933870
zbMATH Keywords: two uniformly elliptic operators
zbMATH Keywords: variational inequality
zbMATH Keywords: regularity properties
reviewed by: Giuseppe Buttazzo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3933870