On Removable Sets of Solutions of Neuman Problem for Quasilinear Elliptic Equations of Divergent Form

ABSTRACT

In this paper we consider a nondivergent elliptic equation of second order whose leading coefficients are from some weight space. The sufficient condition of removability of a compact with respect to this equation in the weight space of Holder functions was found.

Cite this paper

T. Gadjiev and O. Aliyev, "On Removable Sets of Solutions of Neuman Problem for Quasilinear Elliptic Equations of Divergent Form,"*Applied Mathematics*, Vol. 4 No. 2, 2013, pp. 290-298. doi: 10.4236/am.2013.42044.

T. Gadjiev and O. Aliyev, "On Removable Sets of Solutions of Neuman Problem for Quasilinear Elliptic Equations of Divergent Form,"

References

[1] L. Carleson, “Selected Problems on Exceptional Sets,” D. Van Nostrand Company, Toronto, 1967, 126 p.

[2] E. I. Moiseev, “On Neumann Problems in Piecewise Smooth Domains,” Differentsial’nye Uravneniya, Vol. 7, No. 9, 1971, pp. 1655-1656.

[3] E. I. Moiseev, “On Existence and Non-Existence Boundary Sets of Neumann Problem,” Differentsial’nye Uravneniya, Vol. 9, No. 5, 1973, pp. 901-911.

[4] E. M. Landis, “To Question on Uniqueness of Solution of the First Boundary Value Problem for Elliptic and Parabolic Equations of the Second Order,” Uspekhi Matematicheskikh Nauk, Vol. 33, No. 3, 1978, p. 151.

[5] V. A. Kondratyev and E. M. Landis, “Qualitative Theory of Linear Partial Differential Equations of Second Order,” Modern Problems of Mathematics, Fundamental Directions, Partial Differential Equations, Vol. 3, Itogi Nauki i Tekhniki, Seriya, VINITI, 1988, pp. 99-212.

[6] I. T. Mamedov, “On Exceptional Sets of Solutions of Dirichlet Problem for Elliptic Equations of Second Order with Discontinuous Coefficients,” Proceedings of Institute of Mathematics and Mechanics of Academy of Science of Azerbaijan, Vol. 3, No. 16, 1998, pp. 137-149.

[7] M. L. Gerver and E. M. Landis, “One Generalization of a Theorem on Mean Value for Multivariable Functions,” Doklady Akademii Nauk USSR, Vol. 146, No. 4, 1962, pp. 761-764.

[8] E. M. Landis, “Second Order Equations of Elliptic and Parabolic Types,” Nauka, 1971, 288 p.

[9] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order,” Springer-Verlag, Berlin, 1977, 401 p. doi:10.1007/978-3-642-96379-7

[10] V. Kayzer and B. Muller, “Removable Sets for Heat Conduction,” Vestnik, Moscow University, Moscow, 1973, pp. 26-32.

[11] V. A. Mamedova, “On Removable Sets of Solutions of Boundary Value Problems for Elliptic Equations of Second Order,” Transactions of the NAS of Azerbaijan, Vol. 25, No. 1, 2005, pp. 101-106.

[12] T. S. Gadjiev and V. A. Mamedova, “On Removable Sets of Solutions of Second Order Elliptic and Parabolic Equations in Nondivergent Form,” Ukrainian Mathematical Journal, Vol. 61, No. 11, 2009, pp. 1485-1496.

[13] T. Kilpelainen and X. Zhong, “Removable Sets for Continuous Solutions of Quasilinear Elliptic Equations,” Proceedings of the American Mathmatical Society, Vol. 130, No. 6, 2002, pp. 1681-1688. doi:10.1090/S0002-9939-01-06237-2

[14] J. Diederich, “Removable Sets for Pointwise Solutions of Elliptic Partial Differential Equations,” Transactions of the American Mathmatical Society, Vol. 165, 1972, pp. 333-352. doi:10.1090/S0002-9947-1972-0293235-X

[15] R. Harvey and J. Polking, “Removable Singularities of Solutions of Linear Partial Differential Equations,” Acta Mathematica, Vol. 125, No. 1, 1970, pp. 39-56. doi:10.1007/BF02838327

[16] B. E. J. Dahlberg, “On Exceptional Sets at the Boundary for Subharmonic Functions,” Arkiv f?r Matematik, Vol. 15, No. 2, 1977, pp. 305-312.

[17] A. V. Pokrovskii, “Removable Singularities of Solutions of Linear Uniformly Elliptic Second Order Equations,” Funktsionalnyj Analiz i Prilozhenija, Vol. 42, No. 2, 2008, pp. 44-55.

[18] A. V. Pokrovskii, “Removable Singularities for Solutions of Second-Order Linear Uniformly Elliptic Equations in Non-Divergence Form,” Mathematics Sbornik, Vol. 199, No. 6, 2008, pp. 137-160.

[1] L. Carleson, “Selected Problems on Exceptional Sets,” D. Van Nostrand Company, Toronto, 1967, 126 p.

[2] E. I. Moiseev, “On Neumann Problems in Piecewise Smooth Domains,” Differentsial’nye Uravneniya, Vol. 7, No. 9, 1971, pp. 1655-1656.

[3] E. I. Moiseev, “On Existence and Non-Existence Boundary Sets of Neumann Problem,” Differentsial’nye Uravneniya, Vol. 9, No. 5, 1973, pp. 901-911.

[4] E. M. Landis, “To Question on Uniqueness of Solution of the First Boundary Value Problem for Elliptic and Parabolic Equations of the Second Order,” Uspekhi Matematicheskikh Nauk, Vol. 33, No. 3, 1978, p. 151.

[5] V. A. Kondratyev and E. M. Landis, “Qualitative Theory of Linear Partial Differential Equations of Second Order,” Modern Problems of Mathematics, Fundamental Directions, Partial Differential Equations, Vol. 3, Itogi Nauki i Tekhniki, Seriya, VINITI, 1988, pp. 99-212.

[6] I. T. Mamedov, “On Exceptional Sets of Solutions of Dirichlet Problem for Elliptic Equations of Second Order with Discontinuous Coefficients,” Proceedings of Institute of Mathematics and Mechanics of Academy of Science of Azerbaijan, Vol. 3, No. 16, 1998, pp. 137-149.

[7] M. L. Gerver and E. M. Landis, “One Generalization of a Theorem on Mean Value for Multivariable Functions,” Doklady Akademii Nauk USSR, Vol. 146, No. 4, 1962, pp. 761-764.

[8] E. M. Landis, “Second Order Equations of Elliptic and Parabolic Types,” Nauka, 1971, 288 p.

[9] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order,” Springer-Verlag, Berlin, 1977, 401 p. doi:10.1007/978-3-642-96379-7

[10] V. Kayzer and B. Muller, “Removable Sets for Heat Conduction,” Vestnik, Moscow University, Moscow, 1973, pp. 26-32.

[11] V. A. Mamedova, “On Removable Sets of Solutions of Boundary Value Problems for Elliptic Equations of Second Order,” Transactions of the NAS of Azerbaijan, Vol. 25, No. 1, 2005, pp. 101-106.

[12] T. S. Gadjiev and V. A. Mamedova, “On Removable Sets of Solutions of Second Order Elliptic and Parabolic Equations in Nondivergent Form,” Ukrainian Mathematical Journal, Vol. 61, No. 11, 2009, pp. 1485-1496.

[13] T. Kilpelainen and X. Zhong, “Removable Sets for Continuous Solutions of Quasilinear Elliptic Equations,” Proceedings of the American Mathmatical Society, Vol. 130, No. 6, 2002, pp. 1681-1688. doi:10.1090/S0002-9939-01-06237-2

[14] J. Diederich, “Removable Sets for Pointwise Solutions of Elliptic Partial Differential Equations,” Transactions of the American Mathmatical Society, Vol. 165, 1972, pp. 333-352. doi:10.1090/S0002-9947-1972-0293235-X

[15] R. Harvey and J. Polking, “Removable Singularities of Solutions of Linear Partial Differential Equations,” Acta Mathematica, Vol. 125, No. 1, 1970, pp. 39-56. doi:10.1007/BF02838327

[16] B. E. J. Dahlberg, “On Exceptional Sets at the Boundary for Subharmonic Functions,” Arkiv f?r Matematik, Vol. 15, No. 2, 1977, pp. 305-312.

[17] A. V. Pokrovskii, “Removable Singularities of Solutions of Linear Uniformly Elliptic Second Order Equations,” Funktsionalnyj Analiz i Prilozhenija, Vol. 42, No. 2, 2008, pp. 44-55.

[18] A. V. Pokrovskii, “Removable Singularities for Solutions of Second-Order Linear Uniformly Elliptic Equations in Non-Divergence Form,” Mathematics Sbornik, Vol. 199, No. 6, 2008, pp. 137-160.