Functional Analysis and Applications Seminar 

06/11/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Anatoli Merzon, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, Mexico

On the weak solution of the Neumann problem for the 2D Helmholtz equation in a convex cone and $H^s$ regularity

We extend previous results for the Neumann boundary value problem in a convex cone $\Omega$ to the case where the boundary data are from the Sobolev trace space of order $-1/2 + \varepsilon , 0 \lt \varepsilon \lt 1/2$. We prove that for these boundary conditions the solution of the Helmholtz equation exists in the Sobolev space $H^s(\Omega)$ of order $s = 1 + \varepsilon$ ­ , 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. We extend previous results for the Neumann boundary value problem in a convex cone $\Omega$ to the case where the boundary data are from the Sobolev trace space of order $-1/2 + \varepsilon , 0 \lt \varepsilon \lt 1/2$­ . We prove that for these boundary conditions the solution of the Helmholtz equation exists in the Sobolev space $H^s(\Omega)$ of order $s = 1 + \varepsilon$ ­ , 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.