Variational problems with topological constraints
type: Item
instance of: scholarly article
title: Variational problems with topological constraints
zbMATH Open document ID: 0961.49023
author: Peter Laurence
author: E. Stredulinsky
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-05-06T00:00:00Z
review text: The authors are motivated by the goal of characterizing the topological characteristics of level sets of Sobolev functions that are preserved under weak limits. For example, they prove the following: If \(m>0\), \(\Omega\subset {\mathbb R}^2\) is open and bounded, \(g:{\mathbb R}\to{\mathbb R}\) is monotone increasing and left continuous with \(g=0\) on \((-\infty,0]\) and \(g>0\) on \((0,\infty)\), then \[ \begin{aligned} H^1_0(\Omega)&\cap C({\mathbb R}^2) \cap \Big\{ u : u\geq 0,\;\{u\leq t\} \text{\;and\;} \{u>t\} \text{\;are connected for\;}t\geq 0\Big\} \\ &\cap \Big\{u: \text{max} u =m,\;|\{u>t\}|\geq g(m-t) \text{\;for\;}t\geq 0\Big\} \end{aligned} \] is weakly closed in \(H_0^1(\Omega)\). This result can be applied to variational problems. In particular, the authors apply it to the problem of minimizing the Dirichlet integral in \[ H^1_0(\Omega)\cap C({\mathbb R}^2) \cap \Big\{ u : u\geq 0,\;\{u\leq t\} \text{\;and\;} \{u>t\} \text{\;are connected for\;}t\geq 0\Big\} \] among functions constrained to have a given decreasing rearrangement. Additional results and applications extending those described above are given. These are based on decompositions of Sobolev functions in the form \(\sum_{i=1}^{N}f_i(u_i)\) with the \(f_i\) Lipschitzian and the \(u_i\) chosen from spaces such as those above.
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 1455251
zbMATH Keywords: Sobolev space
zbMATH Keywords: decreasing rearrangement
zbMATH Keywords: topological rearrangement
zbMATH Keywords: minimization
zbMATH Keywords: Sobolev functions
zbMATH Keywords: Dirichlet integral
reviewed by: Harold Parks
MaRDI profile type: Publication
full work available at URL: s005260050150
OpenAlex ID: W2023106764
DOI: 10.1007/S005260050150
description: scientific article; zbMATH DE number 1455251