**Luis P. Castro, David Natroshvili**

## The Potential Method for the
Reactance Wave Diffraction Problem in a Scale of Spaces

**Abstract:**

This paper is concerned with a screen type boundary value problem arising from
the wave diffraction problem with a reactance condition. We consider the problem
in a weak formulation within Bessel potential spaces, and where both cases of a
complex and a pure real wave number are analyzed. Using the potential method,
the boundary value problem is converted into a system of integral equations. The
invertibility of the corresponding matrix pseudodifferential operator is shown
in appropriate function spaces which allows the conclusion about the existence
and uniqueness of a weak solution to the original problem. Higher regularity
properties of solutions are also proved to exist in some scale of Bessel
potential spaces, upon the corresponding smoothness improvement of given data.
In particular, the $C^{\alpha}$-smoothness of solutions in a neighbourhood of
the screen edge is established with arbitrary $\alpha<1$ in the two-dimensional
case and $\alpha<1/2$ in the three-dimensional case.

**Keywords:**

Helmholtz equation, wave diffraction, reactance condition, integral equations,
potential theory.

**MSC 2000:** 35J05, 35J25, 35P25