Address: Markusa st. 22, Vladikavkaz,
362027, RNO-A, Russia
Phone: (8672)50-18-06
E-mail: rio@smath.ru
DOI: 10.23671/VNC.2014.3.10232
Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. I
Egorov A. A.
Vladikavkaz Mathematical Journal 2014. Vol. 16. Issue 3.
Abstract: The aim of this paper is to derive the self-improving property of integrability for derivatives of solutions of the differential inequality with a null Lagrangian. More precisely, we prove that the solution of the Sobolev class with some Sobolev exponent slightly smaller than the natural one determined by the structural assumption on the involved null Lagrangian actually belongs to the Sobolev class with some Sobolev exponent slightly larger than this natural
exponent. We also apply this property to improve Holder regularity and stability theorems of [19].
Keywords: null Lagrangian, higher integrability, self-improving regularity, Holder regularity, stability of classes of mappings
For citation: Egorov A. A. Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. I // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 16, no. 3, pp. 22-37. DOI 10.23671/VNC.2014.3.10232
1. Astala K. Area distortion of quasiconformal mappings. Acta Math.,
1994, vol. 173, no. 1, pp. 37-60.
2. 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 Mathematical
Series. Vol. 48.).
3. Ball J. M. Convexity conditions and existence theorems in
nonlinear elasticity. Arch. Ration. Mech. Anal., 1977, vol. 63, P.
337-403.
4. Ball J. M. Constitutive inequalities and existence theorems in
nonlinear elastostatics. Nonlinear Analysis and Mechanics:
Heriot-Watt Symposium (Edinburgh, 1976), London: Pitman, 1977, vol.
1, p. 187-241.
5. Ball J. M, Currie J. C, Olver P. J. Null Lagrangians, weak
continuity, and variational problems of arbitrary order. J. Funct.
Anal., 1981, vol. 41, pp. 135-174.
6. Belinskii P. P. General Properties of Quasiconformal Mappings,
Novosibirsk: Nauka, 1974, [Russian].
7. Bezrukova O. L, Dairbekov N. S, Kopylov A. P. On mappings which
are close in the C-norm to classes of solutions of linear elliptic
partial differential equations. Tr. Inst. Math, 1987, vol. 7, pp.
19-30, [Russian].
8. Bojarski B. Homeomorphic solutions of Beltrami systems. Dokl.
Akad. Nauk SSSR, 1955, vol. 102, pp. 661-664 [Russian].
9. Bojarski B. Generalized solutions of a system of first-order
differential equations of elliptic type with discontinuous coefficients.
Math. Sb., 1957, Vol. 43 (85), no. 4, pp. 451-503, [Russian].
10. Bojarski B, Gutlyanskii V, Martio O, Ryazanov V. Infinitesimal
Geometry of Quasiconformal and bi-Lipschitz Mappings in the Plane,
Zurich: European Math. Soc., 2013, ix+205 p.
11. Dairbekov N. S. Stability in the C-Norm of Classes of
Solutions of Linear Elliptic Partial Differential Equations.
Candidate's dissertation, Novosibirsk, 1986 [Russian].
12. Dairbekov N. S. The concept of a quasiregular mapping of several
n-dimensional variables. Russ. Acad.
Sci. Dokl. Math., 1992, vol. 45, no. 3, pp. 578-582.
13. Dairbekov N. S. Quasiregular mappings of several n-dimensional
variables. Sib. Math. J., 1993, vol. 34, no. 4, pp. 669-682.
14. Dairbekov N. S. Stability of Classes of Mappings, Beltrami
Equations, and Quasiregular Mappings of Several Variables. Doctoral
dissertation, Novosibirsk, 1995 [Russian].
15. Edelen D. G. B. Non Local Variations and Local Invariance of
Fields, N.Y.: Elsivier, 1969 (Modern Anal. and Comp. Methods in Sci.
and Engineering. Vol. 19.).
16. Egorov A. A. Stability of classes of solutions to partial
differential relations constructed by convex and quasiaffine
functions. Proc. on Geometry and Analysis, Novosibirsk: Izdatel'stvo
Instituta Matematiki SO RAN, 2003, pp. 275-288 [Russian].
17. Egorov A. A. Stability of classes of solutions to partial
differential relations constructed by quasiconvex functions and
null Lagrangians. Equadiff 2003, Hackensack: World Sci. Publ, 2005,
pp. 1065-1067.
18. Egorov A. A. Stability of classes of mappings, quasiconvexity,
and null Lagrangians. Dokl. Math., 2007, vol. 76, no. 1, pp.
599-602.
19. Egorov A. A. Quasiconvex functions and null Lagrangians in the
stability problems of classes of mappings. Sib. Math. J., 2008,
vol. 49, no. 4, pp. 637-649.
20. 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.
21. Egorov A. A. One integral inequality for solutions of the
differential inequality with a null Lagrangian and its application.
The collection of the scientific articles of the International
school-seminar "Lomonosov's readings in Altai 2012" (Barnaul, 20-23
November, 2012), Barnaul: ASPA, 2012, vol. 1, P. 284-289.
22. Egorov A. A. Solutions of the Differential Inequality with a
Null Lagrangian: Higher Integrability and Removability of
Singularities. II (To appear.).
23. Faraco D, Zhong X. A short proof of the self-improving
regularity of quasiregular mappings. Proc. Amer. Math. Soc., 2006,
vol. 134, no. 1, pp. 187-192.
24. Gao H. Regularity for weakly (K1,K2)-quasiregular mappings.
Science in China. Ser. A, 2003, vol. 46, no. 4, pp. 499-505.
25. Gao H, Huang Q, Qian F. Regularity for weakly
(K1,K2(x))-quasiregular mappings of several n-dimensional
variables. Front. Math. China, 2011, vol. 6, no. 2, pp. 241-251.
26. Gao H, Li T. On degenerate weakly (K1,K2)-quasiregular mappings.
Acta Math. Sci. Ser. B, 2008, vol. 28, no. 1, pp. 163-170.
27. Gao H, Zhou S, Meng Y. A new inequality for weakly
(K1,K2))-quasiregular mappings. Acta Math. Sin. Engl. Ser., 2007,
vol. 23, no. 12, pp. 2241-2246.
28. Gehring F. W. The Lp-integrability of the partial derivatives of
a quasiconformal mapping. Bull. Am. Math. Soc., 1973, vol. 79, pp.
465-466.
29. Gol'dshtein V. M. The behavior of mappings with bounded
distortion when the coefficient of distortion is close to unity.
Sib. Math. J., 1971, vol. 17, no. 6, pp. 900-906.
30. Gol'dshtein V. M, Reshetnyak Yu. G. Introduction to the Theory
of Functions with Generalized Derivatives, and Quasiconformal
Mappings, Moscow, Nauka, 1983, 285 p. [Russian].
31. 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.).
32. 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).
33. 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.
34. Iwaniec T. p-Harmonic tensors and quasiregular mappings. Ann.
Math., 1992, vol. 136, pp. 651-685.
35. Iwaniec T, Martin G. Quasiregular mappings in even dimensions.
Acta Math., 1993, vol. 170, pp. 29-81.
36. Iwaniec T, Martin G. Geometric Function Theory and Non-Linear
Analysis. Oxford Math. Monogr, Oxford: Oxford Univ. Press, 2001.
37. 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.
38. Knops R. J, Stuart C. A. Quasiconvexity and uniqueness of
equilibrium solutions in nonlinear elasticity. Arch. Ration. Mech.
Anal, 1984, vol. 86, no. 3, pp. 233-249.
39. Kopylov A. P. Stability of classes of multidimensional
holomorphic mappings. III. Properties of mappings that are close to
holomorphic mappings. Sib. Math. J., 1983, vol. 24, no. 3, pp.
373-391.
40. Kopylov A. P. Stability of Classes of Multidimensional
Holomorphic Mappings. Doctoral dissertation, Novosibirsk, 1984,
[Russian].
41. Kopylov A. P. Stability in the C-Norm of Classes of Mappings,
Novosibirsk: Nauka, 1990, [Russian].
42. Kopylov A. P. On stability of classes of conformal mappings. I.
Sib. Math. J., 1995, vol. 36, no. 2, pp. 305-323.
43. Kopylov A. P. On stability of classes of conformal mappings.
III. Sib. Math. J., 1997, vol. 38, no. 4, pp. 715-729.
44. Kristensen J. Finite functionals and Young measures generated by
gradients of Sobolev functions. Math. Inst. Technical Univ. of
Denmark Mat-Report no. 1994-34, Denmark: Lyngby, 1994, 58 p.
45. Krushkal' S. L. Quasiconformal Mappings and Riemann Surfaces,
Novosibirsk: Nauka, 1975, 195 p. [Russian].
46. Krushkal' S. L. Quasiconformal Mappings and Riemann Surfaces.
Scripta Series in Math. A Halsted Press Book, Washington: V. H.
Winston & Sons, N.Y. etc.: John Wiley & Sons, 1979, xii+319 p.
47. Landers A. W. Invariant multiple integrals in the calculus of
variations. Contributions to the Calculus of Variations, 1938-1941,
Chicago: Univ. of Chicago Press, 1942, pp. 184-189.
48. Lehto O. Remarks on the integrability of the derivatives of
quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. AI, 1965, vol.
371, pp. 1-8.
49. Lehto O, Virtanen K. I. Quasiconformal Mappings in the Plane.
2nd ed. Grundlehren der Math. Wissenschaften. Vol. 126,
Berlin-Heidelberg-N.Y.: Springer-Verlag, 1973, viii+258 p.
50. Martio O, Rickman S, Vaisala J. Topological and metric
properties of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. AI,
1971, vol. 488, 31 p.
51. Martio O. On the integrability of the derivative of a
quasiregular mapping. Math. Scand., 1974, vol. 35, pp. 43-48.
52. 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.
53. Martio O, Ryazanov V, Srebro U, Yakubov E. Moduli in Modern
Mapping Theory, N. Y.: Springer, 2009, xii+367 p.
54. Meyers N. G, Elcrat A. Some results on regularity for solutions
of non-linear elliptic systems and quasi-regular functions. Duke
Math. J., 1975, vol. 42, pp. 121-136.
55. Morrey Ch. B. Multiple Integrals in the Calculus of Variations.
Grundlehren der Math., Wiss. Vol. 130, Berlin etc.: Springer-Verlag,
1966.
56. Muller S. Variational models for microstructure and phase
transitions Calculus of variations and geometric evolution
problems. Lectures given at the 2nd session of the Centro
Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 15-22,
1996, Berlin: Springer, 1999, pp. 85-210 (Lect. Notes Math. Vol.
1713).
57. Reshetnyak Yu. G. Stability estimates in Liouville's theorem
and Lp-integrability of the derivatives of quasiconformal mappings.
Sib. Math. J., 1976, vol. 17, no. 4, pp. 653-674.
58. Reshetnyak Yu. G. Space Mappings with Bounded Distortion,
Novosibirsk: Nauka, 1982, 288 p. [Russian].
59. Reshetnyak Yu. G. Stability Theorems in Geometry and Analysis,
Novosibirsk: Nauka, 1982, 232 p. [Russian].
60. Reshetnyak Yu. G. Space Mappings with Bounded Distortion,
Providence (R.I.): Amer. Math. Soc, 1989, 362 p. (Transl. of Math.
Monogr. Vol. 73).
61. Reshetnyak Yu. G. Stability Theorems in Geometry and Analysis,
Dordrecht: Kluwer Acad. Publ, 1994, xi+394 p. (Math. and its Appl.
Vol. 304).
62. Reshetnyak Yu. G. Stability Theorems in Geometry and Analysis.
2nd rev. ed, Novosibirsk: Izdatel'stvo Instituta Matematiki SO RAN,
1996, 424 p. [Russian].
63. Rickman S. Quasiregular Mappings, Berlin: Springer-Verlag, 1993,
x+213 p. (Results in Math. and Related Areas (3). Vol. 26).
64. Sokolova T. V. Behavior of nearly homothetic mappings. Math.
Notes, 1991, vol. 50, no. 4, pp. 1089-1090.
65. Sokolova T. V. Stability in the Space W1p of Homothety
Transformations. Candidate's dissertation, Novosibirsk, 1991,
[Russian].
66. Sychev M. Young measure approach to characterization of
behaviour of integral functionals on weakly convergent sequences by
means of their integrands. Ann. Inst. H. Poincare Anal. Non
Lineaire, 1998, vol. 15, pp. 755-782.
67. Tong Y, Gu J. Li Y. A new inequality for weakly
(K1,K2)-quasiregular mappings. J. Inequal. Pure Appl. Math., 2007,
vol. 8, no. 3, paper no. 91, 4 p.
68. Vaisala J. Lectures on n-Dimensional Quasiconformal Mappings,
Berlin-Heidelberg-N.Y.: Springer-Verlag, 1971, xiv+144 p.
69. Vuorinen M. Conformal Geometry and Quasiregular Mappings, Berlin
etc.: Springer-Verlag, 1988, xix+209 p, (Lect. Notes Math. Vol.
1319.).
