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

Abstract

Abstract The purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain Ω α ⊂ ℝ 2 {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces ℍ p s ⁢ ( Ω α ) {\mathbb{H}^{s}_{p}(\Omega_{\alpha})} , s > 1 p {s>\frac{1}{p}} , 1 < p < ∞ {1<p<\infty} . The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis 𝕎 p s - 1 / p ⁢ ( ℝ + ) {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})} , which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii 𝕎 p r ⁢ ( ℝ + ) {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential ℍ p r ⁢ ( ℝ + ) {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.