On hyperbolic variational inequalities of first order and some applications
type: Item
instance of: scholarly article
title: On hyperbolic variational inequalities of first order and some applications
zbMATH Open document ID: 1058.35171
published in: Monatshefte für Mathematik
publication date: 2004-10-28T00:00:00Z
review text: The author studies strong solutions and their coincidence sets to the obstacle problem for linear transport operators. He starts with the elliptic variational inequality \[ \quad u_\delta\in \mathbb K^1_-:\delta\int_Q \nabla u_\delta\cdot \nabla(v-u_\delta)+(Hu_\delta-f)(v-u_\delta)\geq 0,\;\forall v\in \mathbb K^1_-, \] where \(\delta>0,\;f\in L^2(Q),\;Q\subset \mathbb R^N\;\) is an open domain with a Lipschitz boundary and the first order operator \(H\) is given by \(\;Hv=\mathbf b\cdot\nabla v+cv,\;\mathbf b=(b_i(x)).\;\) The convex set \(\mathbb K^1_-\) has the form \(\;\mathbb K^1_-=\mathbb K\cap H^1_{\Sigma_-}(Q),\) \( H^1_{\Sigma_-}(Q)=\{v\in H^1(Q):v=0\;\text{in}\;\Sigma_-\}, \;\Sigma_-=\{x\in \partial Q:\mathbf b\cdot\mathbf n<0\}. \) The existence and uniqueness of a solution is an immediate consequence of the general theory for the second order obstacle problem. The strong convergence of the solutions and the characteristic functions of their coincidence sets is obtained in for \(\;\delta\to 0\;\) in the passage to the first order problem \(\quad u\in \mathbb K_-:\int_Q (Hu_\delta-f)(v-u_\delta)\geq 0,\;\forall v\in \mathbb K.\) An application to a steady state chemotaxis problem is given. The extension to the two obstacles problem of first order is also presented.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35L85
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35F30
Mathematics Subject Classification ID: 92C17
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 2111479
zbMATH Keywords: hyperbolic variational inequalities
zbMATH Keywords: obstacle problem
zbMATH Keywords: linear transport operator
zbMATH Keywords: free boundary problems
zbMATH Keywords: chemotaxis problem
author: José Francisco Rodrigues
MaRDI profile type: Publication
full work available at URL: s00605-004-0238-3
OpenAlex ID: W2010574340
DOI: 10.1007/S00605-004-0238-3
description: scientific article; zbMATH DE number 2111479