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Problem of diffraction at a fine screen. (English. Russian original) Zbl 0556.35117
Let ${\cal D}\subset {\bbfR}\sp 3$ be an open domain with smooth boundary $\partial {\cal D}$ and let $S\subset {\cal D}$, dim S$=2$ be a smooth surface with boundary $\Gamma =\partial S$. Consider the problem $$(1)\quad (\Delta +k\sp 2)U=0\quad in\quad {\cal D}\setminus S,\quad \lambda U=\pm (1/\sigma)\partial U/\partial n+H\quad on\quad S\sb{\pm},\quad \partial U/\partial \omega +\beta U=0\quad on\quad \partial {\cal D}.$$ Here k, $\lambda\in {\bbfC}$, $\beta$ is a smooth function, $\omega$ is a vector field transversal to S and $S\sb{\pm}$ are two sides of S. One assumes that U satisfies the so called Meixner condition near $\Gamma$, that is $\vert \nabla U\vert\sp 2$ near $\Gamma$ is integrable and the energy is finite. Introducing in a neighbourhood T($\Gamma)$ of $\Gamma$ new coordinates $Z=(z\sb 0,z)$, $z=(z\sb 1,z\sb 2)$ so that $S\cap T(\Gamma)=\{z\sb 2=0,z\sb 1\le 0\},$ $\Gamma =\{z=0\}$, the author obtains an expression for the Laplacian near $\Gamma$. To study the existence and the regularity of the solution, some Sobolev type spaces $E\sp s({\cal D}\setminus S)$, $E\sp s(S\sb+\cup S\sb-)$ are introduced. The main result states that the problem (1) has a unique solution $U\in E\sp{s+2}({\cal D}\setminus S),$ $s\ge 0$, provided $H\in E\sp{s+}(S\sb+\cup S\sb-),$ 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.
Reviewer: V.Petkov

##### MSC:
 35Q99 PDE of mathematical physics and other areas 78A45 Diffraction, scattering (optics) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000)
Full Text:
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