Problem of diffraction at a fine screen
type: Item
instance of: scholarly article
title: Problem of diffraction at a fine screen
zbMATH Open document ID: 0556.35117
DOI: 10.1007/BF00969506
author: P. E. Berkhin
published in: Siberian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 3886475
zbMATH Keywords: Meixner condition
zbMATH Keywords: existence
zbMATH Keywords: regularity
zbMATH Keywords: Sobolev type spaces
zbMATH Keywords: polynomial coefficients
MaRDI profile type: Publication
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cites work: Q5791547
cites work: Q5648864
cites work: ON A CLASS OF ELLIPTIC PSEUDODIFFERENTIAL OPERATORS DEGENERATE ON A SUBMANIFOLD
cites work: A general calculus of pseudodifferential operators
cites work: Characterization of pseudodifferential operator and application
cites work: Spherical functions over a normed field whose residue field is infinite
cites work: On the hypoellipticity and the global analytic-hypoellipticity of pseudo- differential operators
cites work: Proprietes spectrales d'operators pseduo-differentiels
cites work: Q3897569
cites work: Q5634338
cites work: Q5605472
cites work: Q5660019
cites work: Q4159695
cites work: Q5582147
cites work: Pseudo-differential operators and non-elliptic boundary problems
cites work: ELLIPTIC PROBLEMS WITH A PARAMETER AND PARABOLIC PROBLEMS OF GENERAL TYPE
description: scientific article; zbMATH DE number 3886475