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The Dirichlet Problem in Weight Spaces


The solvability of the Dirichlet problem for quasilinear elliptic second-order equations of nondivergence form is studied in weight spaces. Bibliography: 10 titles.

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  1. 1.

    O.A. Ladyzhenskaya and N.N. Uraltseva, “A survey of results on the solvability of boundary value problems for uniformly elliptic and parabolic second-order quasilinear equations having unbounded singularities,” Usp. Mat. Nauk, 41, No. 5, 59-83 (1986).

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  2. 2.

    A.D. Aleksandrov, “Uniqueness conditions and bounds for the solution of the Dirichlet problem,” Vestn. Leningrad. Univ. Mat. Mekh. Astronom., 18, No. 13, 5-29 (1963).

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    A. I. Nazarov, “Estimates for the maximumof solutions of elliptic and parabolic equations in terms of weighted norms of the right-hand side,” Algebra Analiz, 13, No. 2, 151-164 (2001).

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    D.E. Apushkinskaya and A. I. Nazarov, “The Dirichlet problem for quasilinear elliptic equations in domains with smooth closed edges,” Probl. Mat. Anal., No. 21, 3-29 (2000).

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    O.A. Ladyzhenskaya and N.N. Uraltseva, “Estimates of max |u x| for solutions of quasilinear elliptic and parabolic equations of general type and some existence theorems,” Zap. Nauchn. Semin. LOMI, 138, 90-107 (1984).

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  6. 6.

    D. E. Apushkinskaya and A. I. Nazarov, “Boundary estimates for the first-order derivatives of a solution to a nondivergent parabolic equation with composite right-hand side and coeficients of lower-order derivatives,” Probl. Mat. Anal., No. 14, 3-27 (1995).

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  7. 7.

    D. E. Apushkinskaya and A. I. Nazarov, “The initial boundary-value problem for nondivergent parabolic equations with Venttsel' boundary condition,” Algebra Analiz, 6, No. 6, 1-29.

  8. 8.

    O.V. Besov, V.P. Il'in, and S.M. Nikol'skii, Integral Representations of Functions and Embedding Theorems [in Russian], Moscow (1996).

  9. 9.

    A. I. Nazarov, “L p-estimates for a solution to the Dirichlet problem and to the Neumann problem for the heat equation in a wedge with edge of arbitrary codimension,” Probl. Mat. Anal., No. 22, 126-159 (2001).

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  10. 10.

    O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations [in Russian], Moscow (1973).

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Apushkinskaya, D.E., Nazarov, A.I. The Dirichlet Problem in Weight Spaces. Journal of Mathematical Sciences 123, 4527–4538 (2004).

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  • Dirichlet Problem
  • Weight Space
  • Nondivergence Form