zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

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)
[1]N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, The Generalized Method of Characteristic Oscillations in Diffraction Theory [in Russian], Nauka, Moscow (1977).
[2]A. Ya. Povzner and I. V. Sukharevskii, ?Integral equations of second kind for problems of diffraction at an indefinitely thin screen,? Dokl. Akad. Nauk SSSR,127, No. 2, 291-294 (1959).
[3]M. S. Agranovich, The Spectral Properties of Diffraction Problems [in Russian], ?Appendix? in [1].
[4]J. von Meixner, ?Die Kantebedingung der Theorie der Beugung elektromagnetisher Wellen an vollkomen leitenden ebenen Schirme,? Ann. Phys.,6, No. 6, 1-9 (1949).
[5]S. L. Sobolev, Introduction to the Theory of Cubature Formulas [in Russian], Nauka, Moscow (1974).
[6]M. I. Vishik and V. V. Grushin, ?On a class of degenerate elliptic equations of high orders,? Mat. Sb.,79, No. 1, 3-36 (1969).
[7]V. V. Grushin, ?On a class of elliptic pseudodifferential operators that degenerate on a submanifold,? Mat. Sb.,84, No. 2, 163-195 (1971).
[8]R. Beals, ?A general calculus of pseudodifferential operators,? Duke Math. J.,42, No. 1, 1-42 (1975). · Zbl 0343.35078 · doi:10.1215/S0012-7094-75-04201-5
[9]R. Beals, ?Characterization of pseudodifferential operators and applications,? Duke Math. J.,44, No. 1, 45-58 (1977). · Zbl 0353.35088 · doi:10.1215/S0012-7094-77-04402-7
[10]V. V. Grushin, ?Pseudodifferential operators in Rn with bounded symbols,? Funkts. Anal. Prilozhen.,4, No. 3, 37-50 (1970). · Zbl 0234.43010 · doi:10.1007/BF01075618
[11]K. Taniguchi, ?On the hypoellipticity and the global analytic hypoellipticity of pseudodifferential operators,? Osaka J. Math.,11, No. 2, 221-238 (1974).
[12]V. I. Feigin, ?Two algebras of pseudodifferential operators in Rn and certain applications,? Tr. Mosk. Mat. Obshch.,36, 155-194 (1977).
[13]D. Robert, ?Properties spectral d’operateurs pseudo-differentiales,? Commun. Part. Diff. Equat.,3, 755-826 (1978). · Zbl 0392.35056 · doi:10.1080/03605307808820077
[14]M. A. Shubin, Pseudodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1978).
[15]A. I. Komech, ?Elliptic boundary-value problems for pseudodifferential operators on manifolds with conical points,? Mat. Sb.,86, No. 2, 268-298 (1971).
[16]V. V. Grushin, ?Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols,? Mat. Sb.,88, No. 4, 504-521 (1972).
[17]V. P. Maslov, ?Generalization of the notion of characteristic for A-pseudodifferential operators with operator-valued symbols,? Usp. Mat. Nauk,15, No. 6, 225-226 (1970).
[18]K. Kh. Boimatov, ?Pseudodifferential operators with an operator symbol,? Dokl. Akad. Nauk SSSR,244, No. 1, 20-23 (1979).
[19]S. Z. Levendorskii, ?Boundary-value problems in the half-space for quasielliptic pseudodifferential operators,? Mat. Sb.,111, No. 4, 483-502 (1980).
[20]V. P. Glushko, ?Estimates in L2 and solvability of general boundary-value problems for degenerate elliptic equations of second order,? Tr. Mosk. Mat. Obshch.,23, 113-178 (1970).
[21]V. A. Kondrat’ev, ?Singularities of the solution of the Dirichlet problem for an elliptic equation of second order in a neighborhood of the edge,? Differents. Uravn.,13, No. 11, 2026-2032 (1977).
[22]V. G. Maz’ya and V. A. Plamenevskii, ?On the coefficients in the asymptotic of the solutions of elliptic boundary-value problems near the edge,? Dokl. Akad. Nauk SSSR,229, No. 1, 33-36 (1976).
[23]R. Narasimhan, Analysis on Real and Complex Manifolds, 1st edn., North-Holland, Amsterdam (1968).
[24]J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, Springer-Verlag (1972).
[25]S. G. Krein and Yu. I. Petunin, ?Scales of Banach spaces,? Usp. Mat. Nauk,21, No. 2, 89-168 (1966).
[26]L. Hörmander, ?Pseudodifferential operators and nonelliptic boundary problems,? Ann. Math.,83, 129-209 (1966). · Zbl 0132.07402 · doi:10.2307/1970473
[27]M. S. Agranovich and M. I. Vishik, ?Elliptic problems with a parameter and parabolic problems of general form,? Usp. Mat. Nauk,19, No. 3, 53-161 (1964).