On the weak solution of the Neumann problem for the 2D Helmholtz equation in a convex cone and Hs regularity

Abstract

We extend previous results for the Neumann boundary value problem to the case of boundary data from the space \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$H^{-\frac{1}{2}+\varepsilon}(\Gamma), 0{<}{\varepsilon}{<}\frac{1}{2}$\end{document}, where Γ=∂Ω is the boundary of a two-dimensional cone Ω with angle β<π. We prove that for these boundary conditions the solution of the Helmholtz equation in Ω exists in the Sobolev space H1+ε(Ω), is unique and depends continuously on the boundary data. Moreover, we give the Sommerfeld representation for these solutions. This can be used to formulate explicit compatibility conditions on the data for regularity properties of the corresponding solution. Copyright © 2010 John Wiley & Sons, Ltd.