Frank-Olme Speck

Mixed boundary value problems of the type of Sommerfeld's half-plane problem

On etudie des modifications du probleme de demi-plan de Sommerfeld gouverne par deux equations elliptiques propres dans des demi-espaces complementaires de R 3 Ω ± et on pose des conditions aux limites et de transmission sur les deux demi-plans, qui ensemble forment la frontiere commune de Ω ±

Some interior and exterior boundary-value problems for the Helmholtz equation in a quadrant

Boundary-transmission problems for two-dimensional Helmholtz equations in a quadrant Q 1 and its complement Q c 1, respectively, are considered in a Sobolev space setting. The first problem of a quadrant with Dirichlet condition on one face and transmission condition on the other is solved in closed form for the case where all quadrants are occupied by the same medium. Unique sovlability can also be shown in the case of two different media up to exceptional cases of wave numbers, while Fredholm property holds in general. In the second problem transmission conditions are prescribed on both faces. These problems are reduced to equivalent integral equations in L + 2 (ℝ) of Wiener-Hopf-Hankel type which can be solved explicitly by obtaining canonical generalized factorizations of certain non-rational 2 s 2-matrix-valued symbols.

On a Class of Wedge Diffraction Problems posted by Erhard Meister

A class of canonical wedge diffraction problems for Helmholtz equations was formulated by E. Meister in 1986 and subsequently treated by an operator theoretical approach in various publications of his research group including two of the authors. Certain subclasses of those problems, recognized of being unsolved, are subject of the present paper. Some of them are now solved explicitly by refined operator theoretical and analytical methods, others are reduced to systems of equations which contain so-called convolution type operators with symmetry. By a new factorization approach those are proved to be Fredholm in certain (fractional) Sobolev spaces, sometimes with necessary compatibility conditions. Several of the associated operators are therefore explicitly inverted and a number of new problems can be recognized reflecting the challenges of the present state-of-the-art.

Sommerfeld diffraction problems with first and second kind boundary conditions

A class of interface problems is considered where the solutions of one Helmholtz equation in the upper half-plane

Transformation techniques towards the factorization of non-rational 2×2 matrix functions

x_2 > 0

Relations between convolution type operators on intervals and on the half-line

and of another in the lower half-plane

A contribution to the quarter-plane problem in diffraction theory

x_2 0

Diffraction problems with impedance conditions

and

Sommerfeld diffraction problems with third kind boundary conditions

x_1 < 0

Operator Theoretical Methods and Applications to Mathematical Physics

, respectively. In general, there appears a coupled system of Wiener–Hopf equations. Necessary and sufficient conditions for the correctness of the problem in a Sobolev space setting are presented as well as explicit formulas for a factorization of the Fourier symbol matrix of the one-medium problem, the solution in closed form, and its asymptotics near the origin.