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March  2018, 17(2): 319-332. doi: 10.3934/cpaa.2018018

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  January 2017 Revised  January 2017 Published  March 2018

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.

Keywords: Sobolev spaces, fractional derivatives, Beltrami equation, quasiconformal mappings, Kato-Ponce commutator.
Mathematics Subject Classification: Primary:35J46, 30C62, 46E35.
Citation: Martí Prats. Beltrami equations in the plane and Sobolev regularity. Communications on Pure & Applied Analysis, 2018, 17 (2) : 319-332. doi: 10.3934/cpaa.2018018
References:
[1]

K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173 (1994), 37-60.   Google Scholar

[2]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, vol. 48 of Princeton Mathematical Series, Princeton University Press, 2009.  Google Scholar

[3]

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

[4]

A. L. Baisón, A. Clop, R. Giova, J. Orobitg and A. P. di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal, (), 1-28.   Google Scholar

[5]

A. L. Baisón, La Ecuación de Beltrami Generalizada y Otras Ecuaciones Elípticas, PhD thesis, Universitat Autónoma de Barcelona, 2016. Google Scholar

[6]

A. L. Baisón, A. Clop and J. Orobitg, Beltrami equations with coefficient in the fractional Sobolev space $W^{θ, \frac2θ}$, Proc. Amer. Math. Soc., 145 (2017), 139-149.   Google Scholar

[7]

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.   Google Scholar

[8]

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.   Google Scholar

[9]

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

[10]

M. Frazier, R. H. Torres and G. Weiss, The boundedness of Calderón-Zygmund Operators on the spaces $F^{α, q}_p$., Rev. Mat. Iberoam., 4 (1988), 41-72.   Google Scholar

[11]

L. Grafakos, Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics, 2nd edition, New York: Springer, 2008.  Google Scholar

[12]

S. Hofmann, An off-diagonal T1 Theorem and applications, J. Funct. Anal., 160 (1998), 581-622.   Google Scholar

[13]

T. Iwaniec, $L^p$-theory of quasiregular mappings, in Quasiconformal space mappings, Springer Berlin Heidelberg, 1992, 39–64.  Google Scholar

[14]

M. Prats, Sobolev regularity of quasiconformal mappings on domains, J. Anal. Math. , to appear, arXiv: 1507.04332 [math. CA]. Google Scholar

[15]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of De Gruyter series in nonlinear analysis and applications, Walter de Gruyter; Berlin; New York, 1996.  Google Scholar

[16]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, vol. 30 of Princeton Mathematical Series, Princeton University Press, 1970.  Google Scholar

[17]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of North-Holland Mathematical Library, North-Holland, 1978.  Google Scholar

[18]

H. Triebel, Theory of Function Spaces, Reprint (2010) edition, Birkhäuser, 1983.  Google Scholar

[19]

H. Triebel, Theory of Function Spaces III, vol. 100 of Monographs in Mathematics, Birkhäuser, 2006.  Google Scholar

show all references

References:
[1]

K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173 (1994), 37-60.   Google Scholar

[2]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, vol. 48 of Princeton Mathematical Series, Princeton University Press, 2009.  Google Scholar

[3]

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

[4]

A. L. Baisón, A. Clop, R. Giova, J. Orobitg and A. P. di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal, (), 1-28.   Google Scholar

[5]

A. L. Baisón, La Ecuación de Beltrami Generalizada y Otras Ecuaciones Elípticas, PhD thesis, Universitat Autónoma de Barcelona, 2016. Google Scholar

[6]

A. L. Baisón, A. Clop and J. Orobitg, Beltrami equations with coefficient in the fractional Sobolev space $W^{θ, \frac2θ}$, Proc. Amer. Math. Soc., 145 (2017), 139-149.   Google Scholar

[7]

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.   Google Scholar

[8]

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.   Google Scholar

[9]

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

[10]

M. Frazier, R. H. Torres and G. Weiss, The boundedness of Calderón-Zygmund Operators on the spaces $F^{α, q}_p$., Rev. Mat. Iberoam., 4 (1988), 41-72.   Google Scholar

[11]

L. Grafakos, Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics, 2nd edition, New York: Springer, 2008.  Google Scholar

[12]

S. Hofmann, An off-diagonal T1 Theorem and applications, J. Funct. Anal., 160 (1998), 581-622.   Google Scholar

[13]

T. Iwaniec, $L^p$-theory of quasiregular mappings, in Quasiconformal space mappings, Springer Berlin Heidelberg, 1992, 39–64.  Google Scholar

[14]

M. Prats, Sobolev regularity of quasiconformal mappings on domains, J. Anal. Math. , to appear, arXiv: 1507.04332 [math. CA]. Google Scholar

[15]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of De Gruyter series in nonlinear analysis and applications, Walter de Gruyter; Berlin; New York, 1996.  Google Scholar

[16]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, vol. 30 of Princeton Mathematical Series, Princeton University Press, 1970.  Google Scholar

[17]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of North-Holland Mathematical Library, North-Holland, 1978.  Google Scholar

[18]

H. Triebel, Theory of Function Spaces, Reprint (2010) edition, Birkhäuser, 1983.  Google Scholar

[19]

H. Triebel, Theory of Function Spaces III, vol. 100 of Monographs in Mathematics, Birkhäuser, 2006.  Google Scholar

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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