ON THE SMOOTHNESS OF SOLUTIONS OF ELECTRIC FIELD VOLUME SINGULAR                                INTEGRO-DIFFERENTIAL EQUATION

Smirnov Yuriy Gennad'evich, Doctor of physical and mathematical sciences, professor, head of sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), mmm@pnzgu.ru

517.3

Background. The goal of this paper is to study the smoothness of solutions of a volume singular integro-differential equation of electric field arising in the diffraction problem of an electromagnetic wave on a local bounded inhomogeneous dielectric body.
Material and methods. The main method of the research was the method of pseudodifferential operators acting in Sobolev spaces. The theory of elliptic boundary value problems and transmission problems was also used.
Results. It is proved that if the data of the problem is smooth then the squareintegrable solution of the equation will be continuous up to the boundary and smooth inside and outside the body.
Conclusions. Smoothness properties of solutions of the electric field volume singular integro-differential equation allow to investigate the equivalence of the boundary value problem and the equation.

diffraction problem of electromagnetic wave, singular integrodifferential equation, dielectric body.

1. Samokhin A. B. Integral'nye uravneniya i iteratsionnye metody v elektromagnitnom rasseyanii [Integral equations and iteration methods in electromagnetic dissipation]. Moscow: Radio i svyaz', 1998, 160 p.
2. Valovik D. V., Smirnov Yu. G. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathematical sciences ]. 2009, no. 4 (12), pp. 70–84.
3. Valovik D. V., Smirnov Yu. G. Differentsial'nye uravneniya [Differential equations]. 2012, vol. 48, no. 4, pp. 509–515.
4. Teylor M. Psevdodifferentsial'nye operatory [Pseudodifferential operators]. Moscow: Mir, 1985.
5. Mikhlin S. G. Mnogomernye singulyarnye integraly i integral'nye uravneniya [Multidimensional singular integrals and integral equations]. Moscow: Fizmatgiz, 1962.
6. Vladimirov B. C. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow: Nauka, 1981.
7. Bykhovskiy E. B., Smirnov N. V. Trudy MIAN SSSR [Proceedings of MIAN USSR]. 1960, vol. 59, pp. 5–36.
8. Costabel M. A Math. Methods Appl. Sci. 1990, vol. 12, pp. 365–368.
9. Ladyzhenskaya O. A., Ural'tseva N. N. Lineynye i kvazilineynye uravneniya ellipticheskogo tipa [Linear and quasilinear equations of elliptic type]. Moscow: Nauka, 1964.
10. Il'inskiy A. S., Smirnov Yu. G. Difraktsiya elektromagnitnykh voln na provodyashchikh tonkikh ekranakh [Diffraction of electromagnetic waves on conducting thin screens]. Moscow: IPRZhR, 1996, 177 p.
11. Lions Zh.-L., Madzhenes E. Neodnorodnye granichnye zadachi i ikh prilozheniya [Heterogeneous boundary problems and applications thereof]. Moscow: Mir, 1971.
12. Mizokhata S. Teoriya uravneniy s chastnymi proizvodnymi [Theory of equations with partial derivatives]. Moscow: Mir, 1977.
13. Kurant R., Gil'bert D. Metody matematicheskoy fiziki [Methods of mathematical physics]. Moscow: Gostekhizdat, 1951, vol. 2.