On removable singularities of bounded solutions to quasielliptic equations. (English. Russian original) Zbl 0791.35023

Sib. Math. J. 33, No. 4, 543-556 (1992); translation from Sib. Mat. Zh. 33, No. 4, 3-14 (1992).
The authors consider a class of quasilinear quasielliptic differential equations with measurable coefficients. They give sufficient conditions in terms of capacity for subsets of the given domain which allow removability of the singularities on such subsets of bounded solutions.


35H10 Hypoelliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35B60 Continuation and prolongation of solutions to PDEs
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[1] L. Carleson, Selected Problems on Exceptional Sets [Russian translation], Mir, Moscow (1971). · Zbl 0224.31001
[2] J. Wermer, Potential Theory [Russian translation], Mir, Moscow (1980). · Zbl 0545.31001
[3] G. Choquet, ?Forme abstraite de th?or?me de capacitabilite,? Ann. Inst. Fourier,9, 83-89 (1959). · Zbl 0093.29701
[4] T. Serrin, ?Local behavior of solutions of quasilinear equations,? Acta Math.,111, No. 3/4, 247-302 (1964). · Zbl 0128.09101
[5] D. G. Aronson, ?Removable singularities for linear parabolic equations,? Arch. Ration. Mech. and Anal.,17, No. 1, 79-84 (1964). · Zbl 0128.09402
[6] V. G. Maz’ya, ?Removable singularities of bounded solutions of quasilinear elliptic equations of any order,? J. Sov. Math., 3, No. 4 (1975).
[7] V. G. Maz’ya and V. P. Khavin, ?Nonlinear potential theory,? Usp. Mat. Nauk,27, No. 6, 67-138 (1972).
[8] W. Littman, ?Polar sets and removable singularities of partial differential equations,? Ark. Math.,7, No. 1, 1-9 (1965). · Zbl 0158.11004
[9] R. Harvey and F. Polking, ?Removable singularities of solutions of linear partial differential equations,? Acta Math.,125, No. 1/2, 39-56 (1970). · Zbl 0214.10001
[10] Yu. V. Egorov, ?on removable singularities under boundary conditions for the solutions of linear partial differential equations,? Dokl. Akad. Nauk SSSR,289, No. 1, 27-29 (1986).
[11] S. M. Nikol’skii, ?A variational problem,? Mat. Sb.,62, No. 9, 25-32 (1963).
[12] L. D. Kudryavtsev, ?Direct and inverse embedding theorems. Application to solving elliptic solutions by variational methods,? Tr. Mat. Inst. im. V. A. Steklova AN SSSR,55 (1959).
[13] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1964).
[14] G. N. Yakovlev. ?On a variational problem,? Differents. Uravn.,5, No. 7, 1303-1312 (1969).
[15] A. D. Djabrailov, ?Research of some classes of quasilinear elliptic equations of the second order. I,? Differents. Uravn.,5, No. 12, 2245-2247 (1969).
[16] Yu. G. Reshetnyak, Spatial Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982). · Zbl 0487.30011
[17] M. S. Alborova, ?On removable singularities of extremals for one class of functions in calculus of variations,? Novosibirsk (1986), submitted to VINITI on Sept. 23, 1986, No. 6800.
[18] L. H?rmander, Linear Partial Differential Operators [Russian translation], Mir, Moscow (1965).
[19] L. R. Volevich, ?Local properties of solutions of quasielliptic systems,? Mat. Sb.,59, 3-52 (1962).
[20] L. R. Volevich and S. G. Gindikin, ?On a class of hypoelliptic polynomials, Mat. Sb.,75, No. 3, 400-416 (1968).
[21] E. Giusti, ?Equzioni quasi elliptich e spaziZ p,? (?, ?)(I),? Ann. Mat. Pure ed Appl.,75, No. 3, 400-416 (1967).
[22] P. S. Filatov, ?On differential properties of the solutions to quasi-elliptic-type equations in unbounded domains,? Dokl. Akad. Nauk SSSR,205, No. 5, 1050-1059 (1972). · Zbl 0261.35017
[23] G. G. Kazaryan, ?Estimates for differential operators and hypoelliptic operators,? Tr. Mat. Inst. im. V. A. Steklova AN SSSR,140, 130-161 (1976). · Zbl 0399.35025
[24] S. V. Uspenski? and B. N. Chistyakov, ?On exit to a polynomial of solutions to pseudodifferential equations of one class, as |x|??,? Proceedings of S. L. Sobolev’s Seminar [in Russian], No. 1, Novosibirsk (1979), pp. 119-135.
[25] S. M. Nikol’ski?, Approximation of Functions of Several Variables and the Embedding Theorems, [in Russian], Nauka, Moscow (1977).
[26] O. V. Besov, V. P. Il’in, and S. M. Nikol’ski?, Integral Representations of Functions and the Embedding Theorems [in Russian], Nauka, Moscow (1975).
[27] S. K. Vodop’yanov, ?Geometric properties of domains that satisfy the extension condition for spaces of differentiable functions,? in: Some Applications of Functional Analysis to Problems of Mathematical Physics: Proceedings of S. L. Sobolev’s Seminar [in Russian] No. 2, Novosibirsk (1984), pp. 65-95.
[28] S. K. Vodop’yanov, ?Geometric properties of maps and domains; lower bounds for the norm of an extension operator,? in: Issledovanya v Geometrii i Matematicheskom Analize: Proceedings of the Institute of Mathematics, the Academy of Sciences of the USSR, Siberian Branch,7, Novosibirsk (1987), pp. 70-101.
[29] S. K. Vodop’yanov, ?The maximum principle in the potential theory and imbedding theorems for anisotropic spaces of differentiable functions,? Sib. Mat. Zh.,29, No. 2, 17-33 (1988). · Zbl 0699.31019
[30] S. K. Vodop’yanov, ?Potential theory on homogeneous groups,? Mat. Sb.,180, No. 1, 57-77 (1989).
[31] S. K. Vodop’yanov, ?L p potential theory and quasiconformal mappings on homogeneous groups,? in: Problemi Geometrii i Analiza: Proceedings of the Institute of Mathematics, the Academy of Sciences of the USSR, Siberian Branch, Vol. 14, Nauka, Novosibirsk (1989), pp. 45-89.
[32] Yu. G. Reshetnyak, ?Capacity in the theory of functions with generalized derivatives,? Sib. Mat. Zh.,10, No. 5, 1109-1138 (1969).
[33] V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad State Univ. (1985).
[34] M. S. Alborova and S. K. Vodop’yanov, ?On removable singularities of bounded solutions for a class of quasielliptic equations,? in: Abstracts of Reports to the Eleventh All-Union School on Operator Theory in Function Spaces (Chelyabinsk, May, 1986) [in Russian], Part 3, Chelyabinsk (1986), p. 10.
[35] M. S. Alborova and S. K. Vodop’yanov, ?Removable singularities of solutions to quasielliptic equations,? submitted to VINITI on Feb. 4, 1987, Rept. No. 804-B87.
[36] F. Kral, ?Potentials and removability of singularities,? in: Nonlinear Evolution Equations and Potential Theory, Prague (1975), pp. 95-106.
[37] P. I. Lizorkin, ?(L p, Lq)-multipliers of the Fourier integrals,? Dokl. Akad. Nauk SSSR,152, No. 4, 808-811 (1963).
[38] A. A. Davtyan, ?The spaces of anisotropic potentials. Applications.? Tr. Mat. Inst. im. V. A. Steklova AN SSSR,173, 113-124 (1986).
[39] N. G. Meyers, ?A theory of capacities for potentials of functions in Lebesgue classes,? Mat. Scand.,26, No. 2, 255-292 (1970). · Zbl 0242.31006
[40] K. T. Mynbaev ?Anisotropic seminorm of Strichartz type and capacities,? Tr. Mat. Inst. im. V. A. Steklova AN SSSR,181, 200-212 (1988). · Zbl 0666.46035
[41] V. A. Solonnikov, ?Inequalities for functions from the classesW p 1 (R n),?, J. Sov. Math., 3, No. 4 (1975).
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