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Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors

Abstract. The purpose of the present research is to investigate the mixed Dirichlet–Neumann boundary value problems for the Helmholtz equation in a 2D domain with finite number of non-cuspidal angular points on the boundary. Using localization, the problem is reduced in [Proc. A. Razmadze Math. Inst. 162 (2013), 37–44] to model problems in plane angles of magnitude , . In the present paper, we apply the potential method and reduce the model mixed BVP (with Dirichlet–Neumann conditions on the boundary) to an equivalent boundary integral equation (BIE) of Mellin convolution type. Applying the recent results on Mellin convolution equations with meromorphic kernels in Bessel potential and Sobolev–Slobodeckij (Besov) spaces obtained by V. Didenko and R. Duduchava and by R. Duduchava, the unique solvability criteria (Fredholm criteria) of the above mentioned mixed BVP are obtained in classical finite energy space and also in non-classical Bessel potential spaces when . The problem has been tackled before only for angular domains of magnitude for rational angles .