Duke Mathematical Journal

Beltrami operators in the plane

Kari Astala, Tadeusz Iwaniec, and Eero Saksman

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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.

Article information

Duke Math. J., Volume 107, Number 1 (2001), 27-56.

First available in Project Euclid: 5 August 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 35J60: Nonlinear elliptic equations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Astala, Kari; Iwaniec, Tadeusz; Saksman, Eero. Beltrami operators in the plane. Duke Math. J. 107 (2001), no. 1, 27--56. doi:10.1215/S0012-7094-01-10713-8. https://projecteuclid.org/euclid.dmj/1091736135

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