Milton's conjecture on the regularity of solutions to isotropic equations
type: Item
instance of: scholarly article
title: Milton's conjecture on the regularity of solutions to isotropic equations
zbMATH Open document ID: 1029.30012
DOI: 10.1016/S0294-1449(03)00014-3
author: Daniel Faraco
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-09-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78601
review text: The paper deals with the solutions of the isotropic equation \(\operatorname {div}p(z)\nabla u = 0 \) in a square \(Q\) of the plane. The main result is the following. Let \(K > 1\). There exist sequences of functions \(\{p_j \} \in L^\infty (Q,\{K,1/K\}) \) and \(\{u_j\} \in W^{1,2}(Q,R)\) with \(\|u_j\|_{W^{1,2} } \leq 1,\) such that \(\operatorname {div}((p_j(z) \nabla u_j(z))= 0 \) in \(Q,\) and for every compact set \(R\) of positive measure contained in \(Q\) \[ \lim_{j \to \infty }\int_R|\nabla u_j(z)|^{2K/(K-1)}dz = \infty. \]
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 1975938
zbMATH Keywords: isotropic equations
reviewed by: Bodo Dittmar
MaRDI profile type: Publication
Wikidata QID: Q122889935
cites work: Q5512337
cites work: Area distortion of quasiconformal mappings
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cites work: Quasiregular mappings and Young measures
cites work: Composites and quasiconformal mappings: new optimal bounds in two dimensions
cites work: Q4389324
cites work: Q3252854
cites work: Implicit partial differential equations
cites work: On the Area Distortion by Quasiconformal Mappings
cites work: Quasiharmonic fields.
cites work: Q4552394
cites work: The Degree of Regularity of a Quasiconformal Mapping
cites work: Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton
cites work: Q5579201
cites work: Q5334555
cites work: Q3768459
cites work: On the Solutions of Quasi-Linear Elliptic Partial Differential Equations
cites work: Unexpected solutions of first and second order partial differential equations
cites work: Laminates and microstructure
cites work: Parametrized measures and variational principles
cites work: Q5647335
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description: scientific article; zbMATH DE number 1975938