Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru
Dear authors!
Submission of all materials is carried out only electronically through Online Submission System in
personal account.
DOI: 10.23671/VNC.2014.4.10258
Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. II
Egorov A. A.
Vladikavkaz Mathematical Journal 2014. Vol. 16. Issue 4.
Abstract: The aim of this paper is to establish a result on removability of singularities for solutions of the differential inequality with a null Lagrangian. Also, we obtain integral estimates for wedge products of closed differential forms and for minors of a Jacobian matrix.
Keywords: null Lagrangian, removability of singularities, integral estimates, closed differential forms, minors of a Jacobian matrix
For citation: Egorov A. A. Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. II // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 16, no. 4, pp. 41-48. DOI 10.23671/VNC.2014.4.10258
1. Astala K. Area distortion of quasiconformal mappings. Acta
Math., 1994, vol. 173, no. 1, pp. 37-60.
2. Astala K., Clop A., Mateu J., Orobitg J., Uriarte-Tuero I.
Distortion of Hausdorff measures and improved Painlev'e removability
for quasiregular mappings. Duke Math. J., 2008, vol. 141, no. 3, pp.
539-571.
3. Astala K., Iwaniec T., Martin G. Elliptic Partial Differential
Equations and Quasiconformal Mappings in the Plane, Princeton,
Princeton Univ. Press, 2009, xvi+677 p. (Princeton Math. Ser. Vol.
48).
4. Dairbekov N. S. Removable singularities of locally
quasiconformal maps. Sib. Math. J., 1992, vol. 33, no. 1, pp.
159-161.
5. Dairbekov N. S. On removable singularities of solutions to first
order elliptic systems with irregular coefficients. Sib. Math. J.,
1993, vol. 34, no. 1, pp. 55-58.
6. Dairbekov N. S. Stability of Classes of Mappings, Beltrami
Equations, and Quasiregular Mappings of Several Variables. Doctoral
Dissertation, Novosibirsk, 1995 (Russian).
7. Egorov A. A. Solutions of the Differential Inequality with a
Null Lagrangian: Regularity and Removability of Singularities, 2010,
URL: http://arxiv.org/abs/1005.3459.
8. Egorov A. A. Solutions of the Differential Inequality with a
Null Lagrangian: Higher Integrability and Removability of
Singularities. I. Vladikavkaz Math. J., 2014, vol. 16, no. 3, pp.
22-37.
9. Gol'dshtein V. M., Reshetnyak Yu. G. Introduction to the Theory
of Functions with Generalized Derivatives, and Quasiconformal
Mappings, Moskva, Nauka, 1983, 285 p. (Russian).
10. Gol'dshtein V. M., Reshetnyak Yu. G. Quasiconformal Mappings and
Sobolev Spaces, Dordrecht etc., Kluwer Acad. Publ., 1990, xix+371 p.
(Math. and its Appl. Soviet Ser. Vol. 54).
11. Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami
Equation. A Geometric Approach, Berlin, Springer, 2012, xiii+301 p.
(Developments in Math. Vol. 26.).
12. Iwaniec T. On Lp-integrability in PDE's and quasiregular
mappings for large exponents. Ann. Acad. Sci. Fenn. Ser. AI., 1982,
vol. 7, pp. 301-322.
13. Iwaniec T. p-Harmonic tensors and quasiregular mappings. Ann.
Math., 1992, vol. 136, pp. 651-685.
14. Iwaniec T., Martin G. Quasiregular mappings in even dimensions.
Acta Math., 1993, vol. 170, pp. 29-81.
15. Iwaniec T., Martin G. Geometric Function Theory and Non-Linear
Analysis. Oxford Math. Monogr, Oxford, Oxford Univ. Press, 2001.
16. Iwaniec T., Migliaccio L., Nania L., Sbordone C. Integrability
and removability results for quasiregular mappings in high
dimensions. Math. Scand., 1994, vol. 75, no. 2, pp. 263-279.
17. Martio O. Modern Tools in the Theory of Quasiconformal Maps.
Textos de Matematica. Serie B. Vol. 27, Coimbra, Universidade de
Coimbra, Departamento de Matematica, 2000, 43 p.
18. Martio O., Ryazanov V., Srebro U., Yakubov E. Moduli in Modern
Mapping Theory, N.Y., Springer, 2009, xii+367 p.
19. Miklyukov V. M. Removable singularities of quasi-conformal
mappings in space. Dokl. Akad. Nauk SSSR, 1969, vol. 188, pp.
525-527 (Russian).
20. Pesin I. N. Metric properties of Q-quasiconformal mappings. Mat.
Sb. (N.S.), 1956, vol. 40(82), no. 3, pp. 281-294 (Russian).
21. Poletskii E. A. On the removal of singularities of
quasiconformal mappings. Math. USSR Sb., 1973, vol. 21, no. 2, pp.
240-254.
22. Reshetnyak Yu. G. Space Mappings with Bounded Distortion,
Novosibirsk, Nauka, 1982, 288 p. (Russian).
23. Reshetnyak Yu. G. Space Mappings with Bounded Distortion, Providence (R.I.), Amer. Math. Soc., 1989, 362 p. (Transl. of Math. Monogr. Vol. 73).
24. Rickman S. Quasiregular Mappings, Berlin, Springer-Verlag, 1993,
x+213 p. (Results in Math. and Related Areas (3). Vol. 26).
25. Vodop'yanov S. K., Gol'dshtein V. M. A criterion for the
possibility of eliminating sets for the spaces Wp1 of quasiconformal
and quasi-isometric mappings. Sov. Math. Dokl., 1975, vol. 16, pp.
139-142.
26. Vodop'yanov S. K., Gol'dshtein V. M. Criteria for the
removability of sets in spaces of Lp1, quasiconformal and
quasi-isometric mappings. Sib. Math. J., 1977, vol. 18, no. 1, pp.
35-50.
27. Vodop'yanov S. K., Gol'dshtein V. M., Reshetnyak Yu. G. On
geometric properties of functions with generalized first
derivatives. Russ. Math. Surv., 1979, vol. 34, no. 1, pp. 19-74.
28. Vuorinen M. Conformal Geometry and Quasiregular Mappings, Berlin
etc., Springer-Verlag, 1988, xix+209 p. (Lect. Notes Math. Vol.
1319).
29. Zorich V. A. An isolated singularity of mappings with bounded
distortion. Math. USSR Sb., 1970, vol. 10, no. 4, pp. 581-583.
