Dirichlet,Neumann and Mixed boundary value problems for Helmholtz equation in a planar domains with angular points at the boundary

Author: Medea Tsaava
Keywords: Helmholtz equation, Boundary value problems, Mixed boundary condition, Potential methods, Boundary integral equations, Fredholm criteria, Symbol, Banach algebra of operators, Mellin convolution equations, Meromorphic kernels. Bessel potential spaces.
Annotation:

The purpose of the present research is to investigate the mixed Dirichlet-Neumann boundary value problems for the Helmholtz equation in a 2D domains with finite number of non-cuspidal angular points on the boundary. Using the localization the problem was reduced to model problems in plane angles. 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 recent results on Mellin convolution equations with meromorphic kernels in Bessel potential and Sobolev-Slobodeckij (Besov) spaces obtained by V. Didenko and R. Duduchava, criteria of the unique solvability (the Fredholm criteria) of the above mentioned mixed BVP in classical finite energy space and also in non-classical Bessel potential spaces. Similar mixed type problems were investigated with different methods only for rational angles in two papers by T. Ehrhardt, A.P. Nolasco and F.-O. Speck, which are eccepted for publication it the journal {\em Operators and Matrices}. Concerning the Dirichet and Neumann problems they are olso investigated with different methods only for rational angles by these authors and others.

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