On a Class of Wedge Diffraction Problems posted by Erhard Meister

Kapanadze

2008

Proc. Amer. Math. Soc.

Abstract. Boundary value problems originated by the diffraction of an electromagnetic (or acoustic) wave by a rectangular wedge with faces of possible different kinds are analyzed in a Sobolev space framework. The boundary value problems satisfy the Helmholtz equation in the interior (Lipschitz) wedge domain, and are also subject to different combinations of boundary conditions on the faces of the wedge. Namely, the following types of boundary conditions will be under study: Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, Impedance-Dirichlet, and Impedance-Neumann. Potential theory (combined with an appropriate use of extension operators) leads to the reduction of the boundary value problems to integral equations of Fredholm type. Thus, the consideration of single and double layer potentials together with certain reflection operators originate pseudo-differential operators which allow the proof of existence and uniqueness results for the boundary value problems initially posed. Furthermore, explicit solutions are given for all the problems under consideration, and regularity results are obtained for these solutions.

Section: Introductionmentioning

confidence: 99%

Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems

Kapanadze

2008

Proc. Amer. Math. Soc.

“…Half-plane screens represent one of them, see Section 4. The possibility of applying factorization methods to other screen problems with conical configurations is not positively answered till now in the authors opinion (in contrast to wedge problems, see [8,14,15,41]). Let us consider the special geometrical case where Σ is a cone, moreover connected and convex.…”

Section: Explicit Solution Of the Bvpsmentioning

confidence: 95%

Diffraction from Polygonal-conical Screens, an Operator Approach

Operator Theory, Operator Algebras and Applications

Duduchava

Speck

2014

Self Cite

Abstract. The aim of this work is to construct explicitly resolvent operators for a class of boundary value problems in diffraction theory. These are formulated as boundary value problems for the three-dimensional Helmholtz equation with Dirichlet or Neumann conditions on a plane screen of polynomial-conical form (including unbounded and multiplyconnected screens), in weak formulation. The method is based upon operator theoretical techniques in Hilbert spaces, such as the construction of matrical coupling relations and certain orthogonal projections, which represent new techniques in this area of applications. Various cross connections are exposed, particularly considering classical WienerHopf operators in Sobolev spaces as general Wiener-Hopf operators in Hilbert spaces and studying relations between the crucial operators in game. Former results are extended, particularly to multiply-connected screens.Mathematics Subject Classification (2010). Primary 78A45; Secondary 47G30.

“…Consequently, we can identify a significant number of publications where such analysis was taken for particular cases of wedge amplitudes and/or boundary conditions (cf., e.g., [2,7,8,9,10,11,12,13,18,21,22,33,34,35,42,48,44,45,46,49,50,55,56,58,60,66,72]). However, none of these listed papers contain final solvability results for the general problems in a rigourous mathematical Sobolev space setting as is done in the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…The series of results obtained by Meister, Speck and their collaborators (cf., e.g., [10,11,12,21,22,42,48,44,45,46,58]) constitute a systematic approach to a rigourous mathematical analysis of plane wave diffraction by wedges. They obtained important conclusions on both the well-posedness of the problems and consequent closed-form solutions, in appropriate Sobolev spaces, for a large number of particular cases of aperture angles and different types of boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Wave diffraction by wedges having arbitrary aperture angle

Journal of Mathematical Analysis and Applications

Kapanadze

2015

The problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background. In fact, we may recognize significant research on this topic for more than one century. Despite this fact, up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate Sobolev space setting. In the present paper, we are considering the corresponding boundary value problems for the Helmholtz equation, with complex wave number, admitting combinations of Dirichlet and Neumann boundary conditions. The main ideas are based on a convenient combination of potential representation formulas associated with (weighted) Mellin pseudo-differential operators in appropriate Sobolev spaces, and a detailed Fredholm analysis. Thus, we prove that the problems have unique solutions (with continuous dependence on the data), which are represented by the single and double layer potentials, where the densities are solutions of derived pseudo-differential equations on the half-line.