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

Beltrami equations in the plane and Sobolev regularity

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.

Citation: Martí Prats. Beltrami equations in the plane and Sobolev regularity. Communications on Pure and 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. 

[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.

[3]

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

[4]

A. L. BaisónA. ClopR. GiovaJ. Orobitg and A. P. di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal, (), 1-28. 

[5]

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

[6]

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

[7]

A. ClopD. FaracoJ. MateuJ. Orobitg and X. Zhong, Beltrami equations with coefficient in the Sobolev space $W^{1, p}$, Publ. Mat., 53 (2009), 197-230. 

[8]

A. ClopD. Faraco and A. Ruiz, Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging, 4 (2010), 49-91. 

[9]

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

[10]

M. FrazierR. 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. 

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

show all references

References:
[1]

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

[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.

[3]

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

[4]

A. L. BaisónA. ClopR. GiovaJ. Orobitg and A. P. di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal, (), 1-28. 

[5]

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

[6]

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

[7]

A. ClopD. FaracoJ. MateuJ. Orobitg and X. Zhong, Beltrami equations with coefficient in the Sobolev space $W^{1, p}$, Publ. Mat., 53 (2009), 197-230. 

[8]

A. ClopD. Faraco and A. Ruiz, Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging, 4 (2010), 49-91. 

[9]

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

[10]

M. FrazierR. 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. 

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

Figure 1.  General embeddings for Sobolev spaces (and Triebel-Lizorkin spaces with $q$ fixed) in dimension $d=3$ (see (7), (80 and subsequent embeddings)
Figure 2.  Embeddings for compactly supported or bounded functions
Figure 3.  Regularity of the principal quasiconformal solution to (2) when the coefficients satisfy a-priori conditions.
Figure 4.  Regularity of the principal quasiconformal solution to (2) when the coefficients satisfy a-priori conditions
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