## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 30, Number 1 (2006), 103-129.

### The Dirichlet-Neumann problem for the dissipative Helmholtz equation in a 2-D cracked domain with the Neumann condition on cracks

V. V. Kolybasova and P. A. Krutitskii

#### Abstract

The Dirichlet-Neumann problem for the dissipative Helmholtz equation in a connected plane region bounded by closed curves and open arcs (cuts) is studied. The Dirichlet condition is specified on the closed curves, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable. An integral representation for the solution of the problem is obtained. Our approach holds for both interior and exterior domains.

#### Article information

**Source**

Tsukuba J. Math., Volume 30, Number 1 (2006), 103-129.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496165031

**Digital Object Identifier**

doi:10.21099/tkbjm/1496165031

**Mathematical Reviews number (MathSciNet)**

MR2248286

**Zentralblatt MATH identifier**

1210.35035

#### Citation

Krutitskii, P. A.; Kolybasova, V. V. The Dirichlet-Neumann problem for the dissipative Helmholtz equation in a 2-D cracked domain with the Neumann condition on cracks. Tsukuba J. Math. 30 (2006), no. 1, 103--129. doi:10.21099/tkbjm/1496165031. https://projecteuclid.org/euclid.tkbjm/1496165031