TY - JOUR
T1 - Quasiharmonic fields
AU - Iwaniec, Tadeusz
AU - Sbordone, Carlo
N1 - Funding Information:
Research was performed during T. Iwaniec’s visit to the Dipartimento di Matematica e Applicazioni “R. Caccioppoli” dell’Università degli Studi di Napoli “Federico II”. It was supported by GNAFA-CNR, “Progetto di Ricerca MURST 97012260402” su Equazioni Differenziali e Calcolo delle Variazioni and by NSF Grant DMS-9706611. Both authors are grateful to Zoltan Balogh for his careful reading the first version of the manuscript.
PY - 2001
Y1 - 2001
N2 - 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.
AB - 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.
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U2 - 10.1016/S0294-1449(00)00058-5
DO - 10.1016/S0294-1449(00)00058-5
M3 - Article
AN - SCOPUS:0000709953
SN - 0294-1449
VL - 18
SP - 519
EP - 572
JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
IS - 5
ER -