Quasiharmonic fields

Tadeusz Iwaniec, Carlo Sbordone

Research output: Contribution to journalArticle

86 Scopus citations

Abstract

To every solution of an elliptic PDE there corresponds a quasiharmonic field F = [B, E] - a pair of vector fields with div B = 0 and curl E = 0 which are coupled by a distortion inequality. Quasiharmonic fields capture all the analytic spirit of quasiconformal mappings in the complex plane. Among the many desirable properties, we give dimension free and nearly optimal Lp-estimates for the gradient of the solutions to the divergence type elliptic PDEs with measurable coefficients. However, the core of the paper deals with quasiharmonic fields of unbounded distortion, which have far reaching applications to the non-uniformly elliptic PDEs. As far as we are aware this is the first time non-isotropic PDEs have been successfully treated. The right spaces for such equations are the Orlicz-Zygmund classes L2 logα L. Examples we give here indicate that one cannot go far beyond these classes.

Original languageEnglish (US)
Pages (from-to)519-572
Number of pages54
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume18
Issue number5
DOIs
StatePublished - Jan 1 2001

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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