Uniqueness results for optimzation problems with prescribed rearrangement
type: Item
instance of: scholarly article
title: Uniqueness results for optimzation problems with prescribed rearrangement
zbMATH Open document ID: 0858.35007
DOI: 10.1007/BF00396778
published in: Potential Analysis
publication date: 1996-09-25T00:00:00Z
review text: The paper is concerned with comparison results for the solutions of elliptic Dirichlet problems as well as Hamilton-Jacobi equations in a domain \(\Omega\) in \(\mathbb{R}^n\) and the solutions of the corresponding symmetric problems. In particular, the authors investigate the case when equality occurs for \(L^\infty\) or \(L^1\) norms in such comparison. A typical result is the following. Let \(f\in L^p(\Omega)\), \(\nu>0\), \(\nu^{-1}\in L^t\), \(1/t+1/p<2/n\), and let \(u\in W_0^{1,p}(\nu,\Omega)\) satisfy the equation \(-(a_{ij}u_i)_j=f\) in \(\Omega\), where \(a_{ij}(x)\xi_i\xi_j\geq \nu|\xi|^2\) for all \(\xi\in\mathbb{R}^n\). Let \(\nu_\#\) be the increasing spherically symmetric rearrangement of \(\nu\), let \(f^\#\) be the decreasing spherically symmetric rearrangement of \(f\), let \(\Omega^\#\) be a ball such that \(|\Omega^\#|= |\Omega|\), and let \(v\in W^{1,2}(\nu_\#,\Omega^\#)\) satisfy the equation \(-(\nu_\# v_i)_i=f^\#\) in \(\Omega^\#\). If \(|u|_\infty=|v|_\infty\) then it must be \(\Omega=\Omega^\#\), \(u=\mp u^\#\), \(f=\mp f^\#\), \(\nu=\nu_\#\) and \(a_{ij}x_j=\nu x_i\) modulo translations. A similar result is given when \(n=2\), assuming \(|u|_1=|v|_1\). Furthermore, the torsion problem, the capacity problem and the Hamilton-Jacobi equation are discussed. In all cases the authors prove that equality in the comparison of certain norms is achieved only in spherically symmetric situations.
reviewed by: Giovanni Porru
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 49K30
Mathematics Subject Classification ID: 49L99
zbMATH DE Number: 903386
zbMATH Keywords: Hamilton-Jacobi equations
zbMATH Keywords: spherically symmetric rearrangement
zbMATH Keywords: torsion problem
zbMATH Keywords: capacity problem
author: Maria Francesca Betta
author: A. Mercaldo
MaRDI profile type: Publication
cites work: On optimization problems with prescribed rearrangements
cites work: Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri
cites work: Isoperimetric inequalities in the torsion problem for multiply connected domains
cites work: Rearrangements of functions, maximization of convex functionals, and vortex rings
cites work: Symmetrization results in electrostatic problems
cites work: A remark on a comparison theorem
cites work: Symmetrization in a class of first-order Hamilton- Jacobi equations
cites work: Q3961832
cites work: Boundary value problems for some degenerate-elliptic operators
cites work: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization
cites work: On the torsional rigidity of multiply connected cross-sections
description: scientific article; zbMATH DE number 903386