On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

Krutitskii, P.

doi:10.1090/S0033-569X-09-01130-1

Author: P. A. Krutitskii
Journal: Quart. Appl. Math. 67 (2009), 379-399
MSC (2000): Primary 35J05, 35J25
DOI: https://doi.org/10.1090/S0033-569X-09-01130-1
Published electronically: March 26, 2009
MathSciNet review: 2514640
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Dirichlet problem for the Laplacian in a planar multiply connected interior domain bounded by smooth closed curves is considered in the case when the boundary data is piecewise continuous; i.e., it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may not be continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists.

References

N. E. Tovmasjan, On existence and uniqueness of the solution of the Dirichlet problem for the Laplace equation in the class of functions having singularities on the boundary of the domain, Sibirsk. Mat. Ž. 2 (1961), 290–312 (Russian). MR 0124518
V. A. Oganyan, The Dirichlet problem for elliptic systems of differential equations with discontinuous boundary conditions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 16 (1981), no. 6, 465–477, 504 (Russian, with English and Armenian summaries). MR 659978
P. A. Krutitskii, The Dirichlet problem for the two-dimensional Laplace equation in a multiply connected domain with cuts, Proc. Edinburgh Math. Soc. (2) 43 (2000), no. 2, 325–341. MR 1763054, DOI https://doi.org/10.1017/S0013091500020952

Krutitskii P.A. On the harmonic Dirichlet problem on a two-dimensional domain with cuts. Doklady Akademii Nauk, 2007, v.415, No.1, pp.21-25. (in Russian). English translation in Doklady Mathematics, 2007, v.76, No.1, pp.497-501.

P. A. Krutitskii, On existence of a classical solution and non-existence of a weak solution to the Dirichlet problem in a planar domain with slits for Laplacian, Quart. Appl. Math. 66 (2008), no. 1, 177–190. MR 2396656, DOI https://doi.org/10.1090/S0033-569X-07-01067-3

P. A. Krutitskii, The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks, J. Differential Equations 198 (2004), no. 2, 422–441. MR 2039149, DOI https://doi.org/10.1016/j.jde.2003.09.007

N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494

A. N. Tikhonov and A. A. Samarskii, Equations of mathematical physics, The Macmillan Co., New York, 1963. Translated by A. R. M. Robson and P. Basu; translation edited by D. M. Brink; A Pergamon Press Book. MR 0165209

V. S. Vladimirov, Equations of mathematical physics, “Mir”, Moscow, 1984. Translated from the Russian by Eugene Yankovsky [E. Yankovskiĭ]. MR 764399

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR 0473443

Krein S.G. (editor). Functional analysis. Nauka, Moscow, 1964 (in Russian). English translation: Wolters-Noordhoff Publishing, Groningen, 1972.

L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597

V. Trénoguine, Analyse fonctionnelle, Traduit du Russe: Mathématiques. [Translations of Russian Works: Mathematics], “Mir”, Moscow, 1985 (French). Translated from the Russian by V. Kotliar. MR 836334

Christine Bernardi and Andréas Karageorghis, L’équation de Laplace avec conditions aux limites discontinues: convergence d’une discrétisation par éléments spectraux, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 10, 1161–1168 (French, with English and French summaries). MR 1451941, DOI https://doi.org/10.1016/S0764-4442%2897%2987905-0

Boshko S. Ĭovanovich, The difference method for solving the Dirichlet problem for the Laplace equation in the disc in the case of discontinuous boundary conditions, Mat. Vesnik 5(18)(33) (1981), no. 1, 69–79 (Russian). MR 681241

G. K. Berikelašvili, A difference scheme of high-order accuracy for the solution of the Dirichlet problem for the Laplace equation with discontinuous boundary conditions, Soobshch. Akad. Nauk Gruzin. SSR 92 (1978), no. 1, 29–32 (Russian, with English and Georgian summaries). MR 540085

E. Ju. Arhipova, A difference Dirichlet problem with discontinuous boundary conditions, Ž. Vyčisl. Mat i Mat. Fiz. 16 (1976), no. 1, 224–228, 278 (Russian). MR 418479

P. A. Krutitskiĭ, The Dirichlet problem for the Helmholtz equation in the exterior of cuts in the plane, Zh. Vychisl. Mat. i Mat. Fiz. 34 (1994), no. 8-9, 1237–1258 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 34 (1994), no. 8-9, 1073–1090. MR 1300397

S. L. Sobolev, Nekotorye primeneniya funktsional′nogo analiza v matematicheskoĭ fizike, 3rd ed., “Nauka”, Moscow, 1988 (Russian). MR 986735