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Quasiharmonic fields. (English) Zbl 1068.30011

Summary: To every solution of an elliptic PDE there corresponds a quasiharmonic field \(\mathcal F=[B,E]\), a pair of vector fields with \(\text{div }B=0\) and \(\text{curl }E=0\) 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 \(L^p\)-estimates for the gradient of the solutions to the divergence type elliptic PDEs with measurable coefficients. However, the core of the paper deals 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 \(L^2\log^{\alpha}L\). Examples we give here indicate that one cannot go far beyond these classes.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
35J60 Nonlinear elliptic equations
46N20 Applications of functional analysis to differential and integral equations
47N20 Applications of operator theory to differential and integral equations
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