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

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

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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.

**1.**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****2.**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****3.**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**, https://doi.org/10.1017/S0013091500020952**4.**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.**5.**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**, https://doi.org/10.1090/S0033-569X-07-01067-3**6.**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**, https://doi.org/10.1016/j.jde.2003.09.007**7.**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****8.**A. N. Tikhonov and A. A. Samarskii,*Equations of mathematical physics*, Translated by A. R. M. Robson and P. Basu; translation edited by D. M. Brink. A Pergamon Press Book, The Macmillan Co., New York, 1963. MR**0165209****9.**V. S. Vladimirov,*Equations of mathematical physics*, “Mir”, Moscow, 1984. Translated from the Russian by Eugene Yankovsky [E. Yankovskiĭ]. MR**764399****10.**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****11.**Krein S.G. (editor). Functional analysis. Nauka, Moscow, 1964 (in Russian). English translation: Wolters-Noordhoff Publishing, Groningen, 1972.**12.**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****13.**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****14.**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**, https://doi.org/10.1016/S0764-4442(97)87905-0**15.**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****16.**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****17.**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**0418479****18.**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****19.**S. L. Sobolev,*\cyr Nekotorye primeneniya funktsional′nogo analiza v matematicheskoĭ fizike*, 3rd ed., “Nauka”, Moscow, 1988 (Russian). MR**986735**

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Additional Information

**P. A. Krutitskii**

Affiliation:
KIAM, Miusskaya Sq. 4, Moscow 125047, Russia

DOI:
https://doi.org/10.1090/S0033-569X-09-01130-1

Keywords:
Laplace equation,
Dirichlet problem,
discontinuous boundary data.

Received by editor(s):
February 14, 2008

Published electronically:
March 26, 2009

Article copyright:
© Copyright 2009
Brown University