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Problem of diffraction at a fine screen. (English) Zbl 0556.35117

Let 𝒟 3 be an open domain with smooth boundary 𝒟 and let S𝒟, dim S=2 be a smooth surface with boundary Γ=S. Consider the problem

(1)(Δ+k 2 )U=0in𝒟S,λU=±(1/σ)U/n+HonS ± ,U/ω+βU=0on𝒟·

Here k, λ, β is a smooth function, ω is a vector field transversal to S and S ± are two sides of S. One assumes that U satisfies the so called Meixner condition near Γ, that is |U| 2 near Γ is integrable and the energy is finite. Introducing in a neighbourhood T(Γ) of Γ new coordinates Z=(z 0 ,z), z=(z 1 ,z 2 ) so that ST(Γ)={z 2 =0,z 1 0}, Γ={z=0}, the author obtains an expression for the Laplacian near Γ. To study the existence and the regularity of the solution, some Sobolev type spaces E s (𝒟S), E s (S + S - ) are introduced.

The main result states that the problem (1) has a unique solution UE s+2 (𝒟S), s0, provided HE s+ (S + S - ), for all λ 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:
35Q99PDE of mathematical physics and other areas
78A45Diffraction, scattering (optics)
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D05Existence of generalized solutions of PDE (MSC2000)
35D10Regularity of generalized solutions of PDE (MSC2000)
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