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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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 |
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 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. Astala, T. Iwaniec and E. Saksman,
Beltrami operators in the plane, Duke Math. J., 107 (2001), 27-56.
|
| [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.
|
| [5] |
A. L. Baisón,
La Ecuación de Beltrami Generalizada y Otras Ecuaciones Elípticas, PhD thesis, Universitat Autónoma de Barcelona, 2016. |
| [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.
|
| [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.
|
| [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.
|
| [9] |
V. Cruz, J. Mateu and J. Orobitg,
Beltrami equation with coefficient in Sobolev and Besov spaces, Canad. J. Math., 65 (2013), 1217-1235.
|
| [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.
|
| [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. Astala, T. Iwaniec and E. Saksman,
Beltrami operators in the plane, Duke Math. J., 107 (2001), 27-56.
|
| [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.
|
| [5] |
A. L. Baisón,
La Ecuación de Beltrami Generalizada y Otras Ecuaciones Elípticas, PhD thesis, Universitat Autónoma de Barcelona, 2016. |
| [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.
|
| [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.
|
| [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.
|
| [9] |
V. Cruz, J. Mateu and J. Orobitg,
Beltrami equation with coefficient in Sobolev and Besov spaces, Canad. J. Math., 65 (2013), 1217-1235.
|
| [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.
|
| [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 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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