Beltrami equations in the plane and Sobolev regularity

Martí Prats

doi:10.3934/cpaa.2018018

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2018, Volume 17, Issue 2: 319-332. Doi: 10.3934/cpaa.2018018

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Beltrami equations in the plane and Sobolev regularity

Martí Prats

Departamento de Matemáticas, Universidad Autónoma de Madrid -ICMAT ,Ciudad Universitaria de Cantoblanco -28049 Madrid, Spain

The author was funded by the European Research Council under the grant agreement 307179-GFTIPFD and MTM2011-28198 and he acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the "Severo Ochoa" Programme for Centres of Excellence in R&D (SEV-2015-0554). He was partially funded by AGAUR -Generalitat de Catalunya (2014 SGR 75) as well

Received: December 31, 2016

Revised: December 31, 2016

Published: June 2018

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Abstract

New results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation $\bar{\partial} f = μ \partial f + ν \overline{\partial f}$ for discontinuous Beltrami coefficients $μ$ and $ν$ are obtained, using Kato-Ponce commutators, obtaining that $\overline \partial f$ belongs to a Sobolev space with the same smoothness as the coefficients but some loss in the integrability parameter. A conjecture on the cases where the limitations of the method do not work is raised.

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Mathematics Subject Classification: Primary:35J46, 30C62, 46E35.

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Figure 1. General embeddings for Sobolev spaces (and Triebel-Lizorkin spaces with $q$ fixed) in dimension $d=3$ (see (7), (80 and subsequent embeddings)

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Figure 2. Embeddings for compactly supported or bounded functions

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Figure 3. Regularity of the principal quasiconformal solution to (2) when the coefficients satisfy a-priori conditions.

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Figure 4. Regularity of the principal quasiconformal solution to (2) when the coefficients satisfy a-priori conditions

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