Problem of diffraction at a fine screen
Let \({\mathcal D}\subset {\mathbb{R}}^ 3\) be an open domain with smooth boundary \(\partial {\mathcal D}\) and let \(S\subset {\mathcal D}\), dim S\(=2\) be a smooth surface with boundary \(\Gamma =\partial S\). Consider the problem \[ (1)\quad (\Delta +k^ 2)U=0\quad in\quad {\mathcal D}\setminus S,\quad \lambda U=\pm (1/\sigma)\partial U/\partial n+H\quad on\quad S_{\pm},\quad \partial U/\partial \omega +\beta U=0\quad on\quad \partial {\mathcal D}. \] Here k, \(\lambda\in {\mathbb{C}}\), \(\beta\) is a smooth function, \(\omega\) is a vector field transversal to S and \(S_{\pm}\) are two sides of S. One assumes that U satisfies the so called Meixner condition near \(\Gamma\), that is \(| \nabla U|^ 2\) near \(\Gamma\) is integrable and the energy is finite. Introducing in a neighbourhood T(\(\Gamma)\) of \(\Gamma\) new coordinates \(Z=(z_ 0,z)\), \(z=(z_ 1,z_ 2)\) so that \(S\cap T(\Gamma)=\{z_ 2=0,z_ 1\leq 0\},\) \(\Gamma =\{z=0\}\), the author obtains an expression for the Laplacian near \(\Gamma\). To study the existence and the regularity of the solution, some Sobolev type spaces \(E^ s({\mathcal D}\setminus S)\), \(E^ s(S_+\cup S_-)\) are introduced. The main result states that the problem (1) has a unique solution \(U\in E^{s+2}({\mathcal D}\setminus S),\) \(s\geq 0\), provided \(H\in E^{s+}(S_+\cup S_-),\) for all \(\lambda\) with exception of a countable set. The proof is based on the analysis of second order elliptic operators with polynomial coefficients by using pseudodifferential operators.
scientific article; zbMATH DE number 3886475