Quasiharmonic fields

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 L2logαL. Examples we give here indicate that one cannot go far beyond these classes.