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

Author:
P. A. Krutitskii

Journal:
Quart. Appl. Math. **66** (2008), 177-190

MSC (2000):
Primary 35J05, 35J25

DOI:
https://doi.org/10.1090/S0033-569X-07-01067-3

Published electronically:
December 7, 2007

MathSciNet review:
2396656

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Dirichlet problem for the Laplacian in a planar domain bounded by smooth closed curves and smooth double-sided open arcs (slits) is considered in the case when the solution is not continuous at the ends of the slits. The cases of both interior and exterior domains are considered. 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 solution is obtained. It is shown that a weak solution of the problem does not typically exist, though the classical solution exists.

**1.**I. K. Lifanov,*Singular integral equations and discrete vortices*, VSP, Utrecht, 1996. MR**1451377****2.**P. A. Krutitskii,*The integral representation for a solution of the 2-D Dirichlet problem with boundary data on closed and open curves*, Mathematika**47**(2000), no. 1-2, 339–354 (2002). MR**1924510**, https://doi.org/10.1112/S0025579300015941**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.**P. A. Krutitskii,*The 2-dimensional Dirichlet problem in an external domain with cuts*, Z. Anal. Anwendungen**17**(1998), no. 2, 361–378. MR**1632551**, https://doi.org/10.4171/ZAA/827**5.**P. A. Krutitskiĭ,*A mixed problem for the Laplace equation outside cuts on the plane*, Differ. Uravn.**33**(1997), no. 9, 1181–1190, 1293 (Russian, with Russian summary); English transl., Differential Equations**33**(1997), no. 9, 1184–1193 (1998). MR**1638915****6.**P. A. Krutitskii,*The mixed harmonic problem in an exterior cracked domain with Dirichlet condition on cracks*, Comput. Math. Appl.**50**(2005), no. 5-6, 769–782. MR**2165638**, https://doi.org/10.1016/j.camwa.2005.03.013**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.**Smirnov V.I.*A course of higher mathematics.*V. IV, V. Pergamon Press, Oxford, 1964.**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.**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**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
35J05,
35J25

Retrieve articles in all journals with MSC (2000): 35J05, 35J25

Additional Information

**P. A. Krutitskii**

Affiliation:
KIAM, Department 25, Miusskaya Sq. 4, Moscow 125047, Russia

DOI:
https://doi.org/10.1090/S0033-569X-07-01067-3

Received by editor(s):
April 6, 2007

Published electronically:
December 7, 2007

Article copyright:
© Copyright 2007
Brown University