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Sobolev regularity of quasiconformal mappings on domains

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Abstract

Consider a Lipschitz domain Ω and a measurable function μ supported in \(\overline{\Omega}\)with ‖μ‖L∞ < 1. Then the derivatives of a quasiconformal solution of the Beltrami equation \(\overline{\partial}f=\mu\;\partial{f}\) inherit the Sobolev regularity Wn,p(Ω) of the Beltrami coefficient μ as long as Ω is regular enough. The condition obtained is that the outward unit normal vector N of the boundary of the domain is in the trace space, that is, \(N\in{B}_{p,p}^{n-1/p}(\partial\Omega)\).

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References

  1. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.

    MATH  Google Scholar 

  2. K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, NJ, 2009.

    MATH  Google Scholar 

  3. K. Astala, T. Iwaniec and E. Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), 27–56.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 7–60.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Citti and F. Ferrari, A sharp regularity result of solutions of a transmission problem, Proc. Amer. Math. Soc. 140 (2012), 615–620.

    Article  MathSciNet  MATH  Google Scholar 

  6. [CFM+09]_A. Clop, D. Faraco, J. Mateu, J. Orobitg and X. Zhong, Beltrami equations with coefficient in the Sobolev space W 1,p, Publ. Mat. 53 (2009), 197–230.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Clop, D. Faraco and A. Ruiz, Stability of Calderón’s inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging 4 (2010), 49–91.

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Cruz, J. Mateu and J. Orobitg, Beltrami equation with coefficient in Sobolev and Besov spaces, Canad. J. Math. 65 (2013), 1217–1235.

    Article  MathSciNet  MATH  Google Scholar 

  9. V. Cruz and X. Tolsa, Smoothness of the Beurling transform in Lipschitz domains, J. Funct. Anal. 262 (2012), 4423–4457.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. C. Evans, Partial Differential Equations, Oxford University Press, Oxford, 1998.

    MATH  Google Scholar 

  11. T. Iwaniec, L p-theory of quasiregular mappings, in Quasiconformal Space Mappings, Springer, Berlin-Heidelberg, 1992, pp. 39–64.

    Chapter  Google Scholar 

  12. P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71–88.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Mateu, J. Orobitg, and J. Verdera, Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings, J. Math. Pures Appl. 91 (2009), 402–431.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Prats, Sobolev regularity of the Beurling transform on planar domains, Publ. Mat. 2 (2017), 291–336.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Prats and X. Tolsa, A T(P) theorem for Sobolev spaces on domains, J. Funct. Anal. 268 (2015), 2946–2989.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter; Berlin-New York, 1996.

    Book  MATH  Google Scholar 

  17. M. Schechter, Principles of Functional Analysis, American Mathematical Society, Providence, RI, 2002.

    MATH  Google Scholar 

  18. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

  19. X. Tolsa, Regularity of C1 and Lipschitz domains in terms of the Beurling transform, J. Math. Pures Appl. 100 (2013), 137–165.

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  21. H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.

    Book  MATH  Google Scholar 

  22. J. Verdera, L 2 boundedness of the Cauchy integral and Menger curvature in Harmonic Analysis and Boundary Value Problems, American Mathematical Society, Providence, RI, 2001, pp. 139–158.

    Book  MATH  Google Scholar 

Download references

Acknowledgement

The author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 320501. Also, he was partially supported by grants 2014-SGR-75 (Generalitat de Catalunya), MTM-2010-16232 and MTM-2013-44304-P (Spanish government) and by a FI-DGR grant from the Generalitat de Catalunya, (2014FI-B2 00107).

The author would like to thank Xavier Tolsa for advice on his Ph.D. thesis, which gave rise to this work; Cruz,Mateu, Orobitg andVerdera for their advice and interest; and the editor and referee for their patient work and valuable comments.

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  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Catalonia, Spain

    Martí Prats

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  1. Martí Prats
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Prats, M. Sobolev regularity of quasiconformal mappings on domains. JAMA 138, 513–562 (2019). https://doi.org/10.1007/s11854-019-0031-9

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  • DOI: https://doi.org/10.1007/s11854-019-0031-9

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