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), h∈ Lp(C),
where H is a measurable function satisfying |H(z,w1)−H(z,w2)|≤ k|w1−w2| and k is a constant k<1.
We also establish the precise invertibility and spectral properties in Lp(C) for the operators
I−Tμ, I−μT, and T−μ,
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
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
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