On the mixed problem for the dissipative Helmholtz equation in a 2D cracked domain with the Neumann condition on cracks§

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. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.