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 R 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˛j 2 OE0; 2 , j D 1; : : : ; m.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 H 1 .@ ˛/ D W 1 .@ ˛/ and also in non-classical Bessel potential spaces H 1 p .@ ˛/ when 1 < p < 1. The problem has been tackled before only for angular domains of magnitude˛for rational angles˛D m=n.

Mellin convolution operators in Bessel potential spaces

“…Recall that boundary integral operators for BVPs in planar domains with corners have admissible kernels (see (2) and [15,16,18,22]).…”

“…By [22] the BVP (1) is reduced to the following equivalent system of boundary integral equations on R + ,…”

Regularized integral formulation of mixed Dirichlet-Neumann problems

“…After the initial contribution by Zaremba, early works concerning the Zaremba boundary value problem include results by Signorini [25] (1916: solution of the Zaremba problem in the upper half plane using complex variable methods); Giraud [17] (1934: existence of solution of Zaremba problems for general elliptic operators); Fichera [13, 14] (1949, 1952: regularity studies at Zaremba points, Zaremba-type problem for the elasticity equations in two spatial dimensions); Magenes [22] (1955: proof of existence and uniqueness, single layer potential representation); Lorenzi [21] (1975: Sobolev regularity around a corner which is also a Dirichlet-Neumann junction); and Wigley [29,30] (1964,1970: explicit asymptotic expansions around Dirichlet-Neumann junctions), amongst others. More recent contributions in this area include reference [28], which provides a valuable review in addition to a study of Zaremba singularities and theoretical results concerning Galerkin-based computational approaches; reference [7], which considers the Zaremba problem for the biharmonic equation; references [10,11], which study Zaremba boundary value problems for Helmholtz and Laplace-Beltrami equations; reference [8], which discusses the solvability of the Zaremba problem from the point of view of pseudo-differential calculus and Sobolev regularity theory; reference [18] which introduces a certain inverse preconditioning technique to reduce the number of linear algebra iterations for the iterative numerical solution of this problem and which gives rise to high-order convergence; and finally, reference [4], which successfully applies the method of difference potentials to the variable-coefficient Zaremba problem, with convergence order approximately equal to one.…”

Integral Equations of Diffraction Problems with Unbounded Smooth Obstacles

“…The diffraction (transmission) problems for Helmholtz and Maxwell equations on smooth bounded obstacles are classical problems of Mathematical Physics (see for instance [5,[16][17][18]27,37], and references cited there). There is an extensive literature devoted to diffraction problems on specific unbounded obstacles (see for instance [6,8,9,11,[13][14][15]20,30,[32][33][34]36,39,58], and references given there).…”