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On the Relation between Weak Solutions of Elliptic Dirichlet and Neumann Boundary-Value Problems for Plane Simply Connected Domains

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Abstract

For any (possibly unbounded) simply connected domain Ω ⊂ ℝ2 whose complement has nonempty interior, we establish an explicit relation between the solving operators of the elliptic Dirichlet and Neumann boundary-value problems for classes of weak solutions with first derivatives from Lp(Ω). It is assumed that the uniformly elliptic operators are of divergence form with essentially bounded matrix coefficients and with functionals on the right-hand side which are bounded on the spaces of corresponding weak solutions. The relation between the solving operators is established under the necessary and sufficient solvability conditions for the Neumann problem, i.e., when the given functional does vanish on the subspace of constants.

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References

  1. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., (Princeton Univ. Press, Princeton, NJ, 1970; Mir, Moscow, 1973).

    MATH  Google Scholar 

  2. R. Adams and J. J. F. Fournier, Sobolev Spaces, in Pure Appl. Math. (Amst.) (Elsevier, Amsterdam, 2003), Vol. 140.

    Google Scholar 

  3. S. Agmon, A. Douglis, and J. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II,” Comm. Pure Appl. Math. 17, 35–92 (1964).

    Article  MathSciNet  Google Scholar 

  4. W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973; Mir, Moscow, 1975).

    MATH  Google Scholar 

  5. P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications (SIAM, Philadelphia, PA, 2013).

    MATH  Google Scholar 

  6. L. Beznea, M. N. Pascu, and N. R. Pascu, “An equivalence between the Dirichlet and the Neumann problem for the Laplace operator,” Potential Anal. 44 (4), 655–672 (2016).

    Article  MathSciNet  Google Scholar 

  7. L. Beznea, M. N. Pascu, and N. R. Pascu, “Between the Dirichlet and the Neumann problem for continuous and integrable boundarydata,” in Stochastic Analysis and Related Topics, Progr. Probab. (Springer, Cham, 2017), Vol. 72, pp.85–97.

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    V. N. Denisov & A. M. Bogovskii

Authors
  1. V. N. Denisov
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  2. A. M. Bogovskii
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Correspondence to V. N. Denisov or A. M. Bogovskii.

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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 1, pp. 32–48.

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Denisov, V.N., Bogovskii, A.M. On the Relation between Weak Solutions of Elliptic Dirichlet and Neumann Boundary-Value Problems for Plane Simply Connected Domains. Math Notes 107, 27–41 (2020). https://doi.org/10.1134/S0001434620010046

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  • DOI: https://doi.org/10.1134/S0001434620010046

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