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1 March 2001 Beltrami operators in the plane
Kari Astala, Tadeusz Iwaniec, Eero Saksman
Duke Math. J. 107(1): 27-56 (1 March 2001). DOI: 10.1215/S0012-7094-01-10713-8

Abstract

We determine optimal Lp-properties for the solutions of the general nonlinear elliptic system in the plane of the form

f$\overline{z}$=H(z, fz), hLp(C),

where H is a measurable function satisfying |H(z,w1)−H(z,w2)|≤ k|w1w2| and k is a constant k<1.

We also establish the precise invertibility and spectral properties in Lp(C) for the operators

I, IμT, and Tμ,

where T is the Beurling transform. These operators are basic in the theory of quasi-conformal mappings and in linear and nonlinear elliptic partial differential equations (PDEs) in two dimensions. In particular, we prove invertibility in Lp(C) whenever 1+||μ|| <p<1+1/||μ||.

We also prove related results with applications to the regularity of weakly quasiconformal mappings.

Citation

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Kari Astala. Tadeusz Iwaniec. Eero Saksman. "Beltrami operators in the plane." Duke Math. J. 107 (1) 27 - 56, 1 March 2001. https://doi.org/10.1215/S0012-7094-01-10713-8

Information

Published: 1 March 2001
First available in Project Euclid: 5 August 2004

zbMATH: 1009.30015
MathSciNet: MR1815249
Digital Object Identifier: 10.1215/S0012-7094-01-10713-8

Subjects:
Primary: 30C62
Secondary: 35J60, 42B20

Rights: Copyright © 2001 Duke University Press

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Vol.107 • No. 1 • 1 March 2001
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