Решения дифференциального неравенства с нуль-лагранжианом: повышающаяся интегрируемость и устранимость особенностей. I

Решения дифференциального неравенства с нуль-лагранжианом: повышающаяся интегрируемость и устранимость особенностей. I

Автор: Егоров Александр Анатольевич

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 3 т.16, 2014 года.

Бесплатный доступ

Целью настоящей статьи является установление свойства самоулучшающейся интегрируемости производных решений дифференциального неравенства с нуль-лагранжианом. Более точно, мы доказываем, что решение класса Соболева с показателем суммирумости, немного меньшим естественно определенного структурными предположениями на нуль-лагранжиан показателя, фактически принадлежит пространству Соболева с показателем суммируемости, немного большим естественного показателя. Мы также применяем это свойство, чтобы улучшить теоремы о гельдеровой регулярности и об устойчивости из статьи [19].

Короткий адрес: https://sciup.org/14318466

IDR: 14318466   |   УДК: 517.957:517.548

Solutions of the differential inequality with a null lagrangian: higher integrability and removability of singularities. I

The aim of this paper is to derive the self-improving property of integrability for derivatives of solutions of the differential inequality with a 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 Lagrangian actually belongs to the Sobolev class with some Sobolev exponent slightly larger than this natural exponent. We also apply this property to improve H\"older regularity and stability theorems of [19].

Список литературы Решения дифференциального неравенства с нуль-лагранжианом: повышающаяся интегрируемость и устранимость особенностей. I

  • Astala K. Area distortion of quasiconformal mappings//Acta Math.-1994.-Vol. 173, № 1.-P. 37-60.
  • 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.).
  • Ball J. M. Convexity conditions and existence theorems in nonlinear elasticity//Arch. Ration. Mech. Anal.-1977.-Vol. 63.-P. 337-403.
  • 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.
  • 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.-P. 135-174.
  • Belinskii P. P. General Properties of Quasiconformal Mappings.-Novosibirsk: Nauka, 1974.-[Russian].
  • 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.-P. 19-30.-[Russian].
  • Bojarski B. Homeomorphic solutions of Beltrami systems//Dokl. Akad. Nauk SSSR.-1955.-Vol. 102.-P. 661-664.-[Russian].
  • Bojarski B. Generalized solutions of a system of first-order differential equations of elliptic type with discontinuous coefficients//Math. Sb.-1957.-Vol. 43\,(85), № 4.-P. 451-503.-[Russian].
  • 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.
  • Dairbekov N. S. Stability in the $C$-Norm of Classes of Solutions of Linear Elliptic Partial Differential Equations. Candidate's dissertation.-Novosibirsk, 1986.-[Russian].
  • Dairbekov N. S. The concept of a quasiregular mapping of several $n$-dimensional variables//Dokl. Akad. Nauk, Ross. Akad. Nauk.-1992.-Vol. 324, № 3.-P. 511-514. [Russian]; Engl. transl.: Russ. Acad. Sci., Dokl., Math.-1992.-Vol. 45, № 3.-P. 578-582.
  • Dairbekov N. S. Quasiregular mappings of several $n$-dimensional variables//Sib. Mat. Zh.-1993.-Vol. 34, № 4.-P. 87-102.-[Russian]; Engl. transl.: Sib. Math. J.-1993.-Vol. 34, № 4.-P. 669-682.
  • Dairbekov N. S. Stability of Classes of Mappings, Beltrami Equations, and Quasiregular Mappings of Several Variables. Doctoral dissertation.-Novosibirsk, 1995.-[Russian].
  • 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.).
  • 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.-P. 275-288.-[Russian].
  • 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.-P. 1065-1067.
  • Egorov A. A. Stability of classes of mappings, quasiconvexity, and null Lagrangians//Dokl. Akad. Nauk, Ross. Akad. Nauk.-2007.-Vol. 415, № 6.-P. 730-733.-[Russian]; Engl. transl.: Dokl. Math. 2007.-Vol. 76, № 1.-P. 599-602.
  • Egorov A. A. Quasiconvex functions and null Lagrangians in the stability problems of classes of mappings//Sib. Mat. Zh.-2008.-Vol. 49, № 4.-P. 796-812.-[Russian]; English transl.: Sib. Math. J.-2008.-Vol. 49, № 4.-P. 637-649.
  • 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.
  • 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.
  • Egorov A. A. Solutions of the Differential Inequality with a Null Lagrangian: Higher Integrability and Removability of Singularities. II.-(To appear.).
  • Faraco D., Zhong X. A short proof of the self-improving regularity of quasiregular mappings//Proc. Amer. Math. Soc.-2006.-Vol. 134, № 1.-P. 187-192.
  • Gao H. Regularity for weakly $(K_1,K_2)$-quasiregular mappings//Science in China. Ser. A.-2003.-Vol. 46, № 4.-P. 499-505.
  • Gao H. Regularity for weakly $(K_1,K_2)$-quasiregular mappings//Science in China. Ser. A.-2003.-Vol. 46, № 4.-P. 499-505.
  • Gao H., Huang Q., Qian F. Regularity for weakly $(K_1,K_2(x))$-quasiregular mappings of several $n$-di\-men\-sional variables//Front. Math. China.-2011.-Vol. 6, № 2.-P. 241-251.
  • Gao H., Li T. On degenerate weakly $(K_1,K_2)$-quasiregular mappings//Acta Math. Sci. Ser. B.-2008.-Vol. 28, № 1.-P. 163-170.
  • Gao H., Zhou S., Meng Y. A new inequality for weakly $(K_1,K_2))$-quasiregular mappings//Acta Math. Sin. Engl. Ser.-2007.-Vol. 23, № 12.-P. 2241-2246.
  • Gehring F. W. The $L^p$-integrability of the partial derivatives of a quasiconformal mapping//Bull. Am. Math. Soc.-1973.-Vol. 79.-P. 465-466.
  • Gol'dshtein V. M. The behavior of mappings with bounded distortion when the coefficient of distortion is close to unity//Sib. Mat. Zh.-1971.-Vol. 17, № 6.-P. 1250-1258.-[Russian]; English transl.: Sib. Math. J.-1971.-Vol. 17, № 6.-P. 900-906.
  • Gol'dshtein V. M., Reshetnyak Yu. G. Introduction to the Theory of Functions with Generalized Derivatives, and Quasiconformal Mappings.-M.: Nauka, 1983.-285 p.-[Russian].
  • 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.).
  • 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.).
  • Iwaniec T. On $L^p$-integrability in PDE's and quasiregular mappings for large exponents//Ann. Acad. Sci. Fenn. Ser. AI.-1982.-Vol. 7.-P. 301-322.
  • Iwaniec T. $p$-Harmonic tensors and quasiregular mappings//Ann. Math.-1992.-Vol. 136.-P. 651-685.
  • Iwaniec T., Martin G. Quasiregular mappings in even dimensions//Acta Math.-1993.-Vol. 170.-P. 29-81.
  • Iwaniec T., Martin G. Geometric Function Theory and Non-Linear Analysis. Oxford Math. Monogr.-Oxford: Oxford Univ. Press, 2001.
  • Iwaniec T., Migliaccio L., Nania L., Sbordone C. Integrability and removability results for quasiregular mappings in high dimensions//Math. Scand.-1994.-Vol. 75, № 2.-P. 263-279.
  • Knops R. J., Stuart C. A. Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity//Arch. Ration. Mech. Anal.-1984.-Vol. 86, № 3.-P. 233-249.
  • Kopylov A. P. Stability of classes of multidimensional holomorphic mappings. III. Properties of mappings that are close to holomorphic mappings//Sib. Mat. Zh.-1983.-Vol. 24, № 3.-P. 70-91.-[Russian]; English transl.: Sib. Math. J.-1983.-Vol. 24, № 3.-P. 373-391.
  • Kopylov A. P. Stability of Classes of Multidimensional Holomorphic Mappings. Doctoral dissertation.-Novosibirsk, 1984.-[Russian].
  • Kopylov A. P. Stability in the $C$-Norm of Classes of Mappings.-Novosibirsk: Nauka, 1990.-[Russian].
  • Kopylov A. P. On stability of classes of conformal mappings. I//Sib. Mat. Zh.-1995.-Vol. 36, № 2.-P. 348-369.-[Russian]; English transl.: Sib. Math. J.-1995.-Vol. 36, № 2.-P. 305-323.
  • Kopylov A. P. On stability of classes of conformal mappings. III//Sib. Mat. Zh.-1997.-Vol. 38, № 4.-P. 825-842.-[Russian]; English transl.: Sib. Math. J.-1997.-Vol. 38, № 4.-P. 715-729.
  • Kristensen J. Finite functionals and Young measures generated by gradients of Sobolev functions. Math. Inst. Technical Univ. of Denmark Mat-Report № 1994-34.-Denmark: Lyngby, 1994.-58 p.
  • Krushkal' S. L. Quasiconformal Mappings and Riemann Surfaces.-Novosibirsk: Nauka, 1975.-195 p.-[Russian].
  • 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.
  • 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.-P. 184-189.
  • Lehto O. Remarks on the integrability of the derivatives of quasiconformal mappings//Ann. Acad. Sci. Fenn. Ser. AI.-1965.-Vol. 371.-P. 1-8.
  • 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.
  • Martio O., Rickman S., V\"ais\"al\"a J. Topological and metric properties of quasiregular mappings//Ann. Acad. Sci. Fenn. Ser. AI.-1971.-Vol. 488.-31 p.
  • Martio O. On the integrability of the derivative of a quasiregular mapping//Math. Scand.-1974.-Vol. 35.-P. 43-48.
  • 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.
  • Martio O., Ryazanov V., Srebro U., Yakubov E. Moduli in Modern Mapping Theory.-N.\,Y.: Springer, 2009.-xii+367 p.
  • 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.-P. 121-136.
  • Morrey Ch. B. Multiple Integrals in the Calculus of Variations. Grundlehren der Math, Wiss. Vol. 130.-Berlin etc.: Springer-Verlag, 1966.
  • 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.-P. 85-210.-(Lect. Notes Math. Vol. 1713).
  • Reshetnyak Yu. G. Stability estimates in Liouville's theorem and $L_p$-integrability of the derivatives of quasiconformal mappings//Sib. Mat. Zh.-1976.-Vol. 17, № 4.-P. 868-896.-[Russian]; English transl.: Sib.\ Math. J.-1976.-Vol. 17, № 4.-P. 653-674.
  • Reshetnyak Yu. G. Space Mappings with Bounded Distortion.-Novosibirsk: Nauka, 1982.-288 p.-[Russian].
  • Reshetnyak Yu. G. Stability Theorems in Geometry and Analysis.-Novosibirsk: Nauka, 1982.-232 p.-[Russian].
  • Reshetnyak Yu. G. Space Mappings with Bounded Distortion.-Providence (R.\,I.): Amer. Math. Soc., 1989.-362 p.-(Transl. of Math. Monogr. Vol. 73.).
  • Reshetnyak Yu. G. Stability Theorems in Geometry and Analysis.-Dordrecht: Kluwer Acad. Publ., 1994.-xi+394 p.-(Math. and its Appl. Vol. 304.).
  • Reshetnyak Yu. G. Stability Theorems in Geometry and Analysis. 2nd rev. ed.-Novosibirsk: Izdatel'stvo Instituta Matematiki SO RAN, 1996.-424 p.-[Russian].
  • Rickman S. Quasiregular Mappings.-Berlin: Springer-Verlag, 1993.-x+213 p.-(Results in Math. and Related Areas (3). Vol. 26.).
  • Sokolova T. V. Behavior of nearly homothetic mappings//Mat. Zametki.-1991.-Vol. 50, № 4.-P. 154-156.-[Russian]; English transl.: Math. Notes.-1991.-Vol. 50, № 4.-P. 1089-1090.
  • Sokolova T. V. Stability in the Space $W^1_p$ of Homothety Transformations. Candidate's dissertation.-Novosibirsk, 1991.-[Russian].
  • Sychev M. Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands//Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire.-1998.-Vol. 15.-P. 755-782.
  • Tong Y., Gu J. Li Y. A new inequality for weakly $(K_1,K_2)$-quasiregular mappings//J. Inequal. Pure Appl. Math.-2007.-Vol. 8, № 3, Paper № 91.-4 p.
  • Vaisala J. Lectures on $n$-Dimensional Quasiconformal Mappings.-Berlin-Heidelberg-N.\,Y.: Springer-Verlag, 1971.-xiv+144 p.
  • Vuorinen M. Conformal Geometry and Quasiregular Mappings.-Berlin etc.: Springer-Verlag, 1988.-xix+209 p.-(Lect. Notes Math. Vol. 1319.).
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