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Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts

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Abstract

We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts.

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References

  1. Krutitskii, P.A. and Prozorov, K.V., On a Problem for the Helmholtz Equation Outside Cuts on the Plane with the Dirichlet Conditions and the Oblique Derivative Condition on Opposite Sides of the Cuts, Differential Equations, 2011, vol. 47, no. 9, pp. 1281–1296.

    Article  MathSciNet  MATH  Google Scholar 

  2. Krutitskii, P.A., The Dirichlet problem for the Helmholtz Equation in the Exterior of Cuts in the Plane, Comput. Maths. Math. Phys., 1994, vol. 34, no. 8–9, pp. 1073–1090.

    MathSciNet  MATH  Google Scholar 

  3. Krutitskii, P.A., The Neumann Problem for the Helmholtz Equation in the Exterior of Cuts in the Plane, Comput. Maths. Math. Phys., 1994, vol. 34, no. 11, pp. 1421–1431.

    MathSciNet  MATH  Google Scholar 

  4. Krutitskii, P.A., A Mixed Problem for the Helmholtz Equation in a Multiply Connected Domain, Comput. Maths. Math. Phys., 1996, vol. 36, no. 8, pp. 1087–1095.

    MathSciNet  MATH  Google Scholar 

  5. Krutitskii, P.A., A Mixed Problem for the Helmholtz Equation Outside Cuts on the Plane, Differential Equations, 1996, vol. 32, no. 9, pp. 1204–1212.

    MathSciNet  MATH  Google Scholar 

  6. Krutitskii, P.A. and Prozorov, K.V., The Dirichlet-Neumann Problem for the Helmholtz Equation Outside Cuts on the Plane, Doklady Mathematics, 2004, vol. 70, no. 2, pp. 765–769.

    Google Scholar 

  7. Krutitskii, P.A. and Sgibnev, A.I., Mixed Problem for the Laplace Equation Outside Cuts in a Plane with Setting Dirichlet and Skew Derivative Conditions on Different Sides of the Cuts, Quart. Appl. Math., 2006, vol. 64, no. 1, pp. 105–136.

    MathSciNet  MATH  Google Scholar 

  8. Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1981 (in Russian). Engl. transl.: Moscow: Mir, 1984.

    Google Scholar 

  9. Nikiforov, A.F. and Uvarov, V.B., Spetsial’nye funktsii matematicheskoi fiziki (Special Functions of Mathematical Physics), Moscow: Nauka, 1984 (in Russian). Engl. transl.: Basel: Birkhäuser, 1988.

    Google Scholar 

  10. Muskhelishvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Nauka, 1968 (in Russian). Engl. transl.: Groningen: Nordhoff, 1972.

    MATH  Google Scholar 

  11. Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Function Theory and Functional Analysis), Moscow, 1972 (in Russian). Engl. transl.: New York: Dover, 1999.

  12. Funktsional’nyi analiz (Functional Analysis), Krein, S.G., Ed., Moscow: Nauka, 1964 (in Russian). Engl. transl.: Groningen: Nordhoff, 1972.

    Google Scholar 

  13. Kantorovich, A.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1984 (in Russian). Engl. transl.: Oxford: Pergamon, 1982.

    Google Scholar 

  14. Trenogin, V.A., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1980 (in Russian). French transl.: Moscow: Mir, 1985.

    Google Scholar 

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Authors and Affiliations

  1. Institute for Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    P. A. Krutitskii & K. V. Prozorov

  2. University of the Russian Academy of Education, Moscow, Russia

    P. A. Krutitskii & K. V. Prozorov

Authors
  1. P. A. Krutitskii
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  2. K. V. Prozorov
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Additional information

Original Russian Text © P.A. Krutitskii, K.V. Prozorov, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 9, pp. 1219–1233.

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Krutitskii, P.A., Prozorov, K.V. Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts. Diff Equat 48, 1197–1211 (2012). https://doi.org/10.1134/S0012266112090017

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  • DOI: https://doi.org/10.1134/S0012266112090017

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