Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain

TL;DR

This paper investigates mixed boundary value problems for the Helmholtz equation in a 2D angular domain using advanced potential methods and Mellin convolution equations, providing explicit solvability conditions in non-classical function spaces.

Contribution

It introduces corrected and explicit conditions for the unique solvability of Helmholtz BVPs in Bessel potential spaces, extending previous work with improved accuracy.

Findings

Derived explicit solvability conditions in Bessel potential spaces.

Corrected previous errors in the analysis of Helmholtz BVPs.

Extended the potential method approach to non-classical function spaces.

Abstract

The purpose of the present research is to investigate model mixed boundary value problems for the Helmholtz equation in a planar angular domain $\Omega_\alpha\subset\mathbb{R}^2$ of magnitude $\alpha$. The BVP is considered in a non-classical setting when a solution is sought in the Bessel potential spaces $\mathbb{H}^s_p(\Omega_\alpha)$, $s>1/p$, $1<p<\infty$. The problems are investigated using the potential method by reducing them to an equivalent boun\-dary integral equation (BIE) in the Sobolev-Slobode\v{c}kii space on a semi-infinite axes $\bW^{s-1/p}_p(\bR^+)$, which is of Mellin convolution type. By applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko \& R. Duduchava in \cite{DD16}, explicit conditions of the unique solvability of this BIE in the Sobolev-Slobode\v{c}kii $\bW^r_p(\bR^+)$ and Bessel potential $\bH^r_p(\mathbb{R}^+)$ spaces for arbitrary $r$ are found and used to write explicit conditions for the Fredhoilm property and unique solvability of the initial model BVPs for the Helmholtz equation in the above mentioned non-classical setting. The same problem was investigated in the foregoing paper of the authors published in 2013, but there was made fatal errors. In the present paper we correct these results.