# On the boundary integral equation method of solving boundary value problems for the two dimensional Laplace equation

### Abstract

We consider approach based on the integral representation of solutions

in domain which consists of bounded and unbounded parts that

gives us opportunity to reduce different transmission type problems to

connected with them equivalent boundary equations of the first and the second kind.

We suppose also that solutions of some of these boundary problems are unbounded at infinity.

Interior and exterior Dirichlet and Neumann boundary value

problems for Laplace equation are restrictions of the solutions

os more general this transmission problems.

Interior Neumann and exterior Dirichlet boundary value problems we also can solve using

integral equation of the second kind that have not unique solution.

Corresponding modified equations are constructed in this case and solutions of obtained equations are unique.

We also show correctness of all obtained boundary equations of the second type

given on closed Lipschitz curve in some Hilbert spaces

without compactness of corresponding integral operators.

### References

M.S. Agranovich, Sobolev spaces, their generalizations and elliptic boundary value problems in domains with smooth and Lipschitz boundary, M., 2013, 379 p. (in Russian)

J.-P. Aubin, Approximation of elliptic boundary-value problems, Wiley-Interscience, New York, 1972.

M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal., 19 (1988), 613–626.

E. Gagliardo, Caratterizazioni delle trace sullo frontiera relative ad alcune classi de funzioni in „n” variabili, Rendiconti del Seminario Matematico della Universita di Padova, 27 1957, 284–305.

G.C. Hsiao, W.L. Wendland, Boundary integral equations, Springer-Verlag, Berlin, 2008, 640 p.

R. Kress, Linear integral equations, Springer–Verlag, Berlin, 1998.

V.G. Maz‘ya, Boundary integral equations, In Encyclopaedia of Mathematical Sciences. V.27, Analysis IV. Maz’ya, V.G., Nikolskii eds. Springer-Verlag, Berlin, 1991, 127–228.

W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, 2000, 357 p.

S.E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Analysis Appl., 378 (2011), 324–342.

S.G Mikhlin, Сourse of mathematical phisycs, M., 1968, 575 p. (in Russian)

T. von Petersdorff, Boundary integral equations for the mixed, Dirichlet, Neumann and transmission problems, Math. Meth. in Appl. Sci., 11 (1989), 185–213.

Yu. Sybil, On some problems of solving linear equations that have a non-unique solution, Herald of LNU, ser. Applied mathematics and informatics, 28 (2020), 73–81. (in Ukrainian)

V.S. Vladimirov, Equations of mathematical physics, M., 1981, 512 p. (in Russian)

W.L.Wendland, E.P. Stephan, G.C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. Appl. Sci., 1 (1979), 265–321.

*Matematychni Studii*,

*62*(1), 60-76. https://doi.org/10.30970/ms.62.1.60-76

Copyright (c) 2024 Yu. Sybil

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.