Quasiharmonic fields

To every solution of an elliptic PDE there corresponds a quasiharmonic fiel d 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 th e analytic spirit of quasiconformal mappings in the complex plane. Among th e many desirable properties, we give dimension free and nearly optimal L-p- estimates for the gradient of the solutions to the divergence type elliptic PDEs with measurable coefficients. However, the core of the paper deals wi th quasiharmonic fields of unbounded distortion, which have far reaching ap plications 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 ri ght spaces for such equations are the Orlicz-Zygmund classes L-2 log(alpha) L. Examples we give here indicate that one cannot go far beyond these clas ses. (C) 2001 Editions scientifiques et medicales Elsevier SAS.