Published Paper

Inserted: 10 jan 2002

Journal: Ann. I. H. Poincaré - AN
Volume: 18
Number: 5
Pages: 519-572
Year: 2001

Abstract:

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\centerline{\Huge \bf Quasiharmonic Fields}

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\begin{center} {\sc Tadeusz IWANIEC$^*$} and {\sc Carlo SBORDONE$^{**}$}

$^{*}$ Dept. of Mathematics, Syracuse University

{\small Syracuse, NY 13210}

$^{**}$ Dipartimento di Matematica e Applicazioni ``R. Caccioppoli"

{\small Università, Via Cintia - 80126 Napoli} \end{center}

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\centerlineAbstract

\begin{quote} To every solution of an elliptic PDE there corresponds a quasiharmonic field $\cF = [B,E] -a$ 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 $L^p$-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 $L^2\log^\ap L$. Examples we give here indicate that one cannot go far beyond these classes. \end{quote} \end{document}