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## Archive of Issues

Russia Nizhni Novgorod
Year
2012
Issue
1
Pages
3-14
 Section Mathematics Title $L_p$-estimations of vector fields in the unbounded areas and some electromagnetic theory problems in the inhomogeneous areas Author(-s) Zhidkov A.A.a, Kalinin A.V.a, Tyukhtina A.A.a Affiliations Nizhni Novgorod State Universitya Abstract The paper is devoted to studying of estimations of scalar products of vector fields and their application in the proof of solvability for mathematical physics problems. The estimations of scalar products of vector field were proved in weighted functional spaces of summable functions. As an example of the application of such estimations there was proved the solvability for the problem of determination of stationary magnetic field in whole three-dimensional Euclidian space containing bounded conducting domain. The association between the proposed problem statement and the corresponding variational statement was shown too. There was investigated the possibility of determination of another unknown functions (electric field, volume density of electrical charge) inside the conducting domain. Keywords scalar product, vector field, Maxwell equations, solvability, functional spaces UDC 517.9 MSC 35D30, 35Q61, 46E20 DOI 10.20537/vm120101 Received 1 July 2011 Language Russian Citation Zhidkov A.A., Kalinin A.V., Tyukhtina A.A. $L_p$-estimations of vector fields in the unbounded areas and some electromagnetic theory problems in the inhomogeneous areas, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 1, pp. 3-14. References Weil H. The method of orthogonal projection in potential theory, Duke Math. Journal, 1940, pp. 411–444. Byhovskii E.B., Smirnov N.V. Orthogonal decomposition of the space of vector functions square-summable on a given domain, and the operators of vector analysis, Tr. Mat. Inst. Steklov, 1960, vol. 59, pp. 5–36. Duvaut G., Lions J.-L. Inequalities on mechanics and physics, Berlin–New York: Springer–Verlag, 1976, 397 p. Translated under the title Neravenstva v mekhanike i fizike, Moscow: Nauka, 1980, 384 p. Temam R. Navier–Stokes equations: theory and numerical analysis, Amsterdam–New York–Oxford: North–Holland Publishing Company, 1977, 408 p. Translated under the title Uravneniya Nav’e–Stoksa. Teoriya i chislennyi analiz, Moscow: Mir, 1981, 408 p. Maslennikova V.N. Estimates in $L_p$ and the asymptotic for $t\to\infty$ of Cauchy problem solution for Sobolev set, Tr. Mat. Inst. Steklov, 1968, vol. 103, pp. 107–141. Girault V., Raviart P.-A. Finite Element Approximation of the Navier–Stokes Equations, New York: Springer–Verlag, 1979, 207 p. Kalinin A.V., Morozov S.F. Stationary problems for the set of Maxwell equations on inhomogeneous domains, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 1997, vol. 20, no. 1, pp. 24–31. Kalinin A.V. Some estimations of vector field theory, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 1997, vol. 20, no. 1, pp. 32–38. Kalinin A.V., Kalinkina A.A. Estimates of vector fields and stationary set of Maxwell equations, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2002, no. 1 (25), pp. 95–107. Kalinin A.V., Kalinkina A.A. $L_p$-estimates of vector fields, Izv. Vyssh. Uchebn. Zaved. Mat., 2004, no. 3, pp. 26–35. Kalinin A.V. Estimations of scalar products of vector fields and their applications in several mathematical physics problems, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., Izhevsk, 2006, no. 3 (37), pp. 55–56. Zhidkov A.A. Estimations of scalar products of vector fields in unbounded domains, Vestn. Nizhegorod. Gos. Univ., 2007, no. 1, pp. 162–166. Kalinin A.V. Otsenki skalyarnykh proizvedenii vektornykh polei i ikh primenenie v matematicheskoi fizike (Estimations scalar products of vector fields and their applications in mathematical physics), Nizhni Novgorod: Lobachevsky State University of Nizhni Novgorod, 2007, 319 p. Coulomb J.-L., Sabonnadiere J.-K. CAO en ´electrotechnique, Paris: Hermes. Pub., 1985, 242 p. Translated under the title SAPR v electrotekhnike, Moscow: Mir, 1988, 208 p. Il’in V.P. Chislennye metody resheniya zadach elektrofiziki (Numerical methods of solution of electro-physics problems), Moscow: Nauka, 1985, 336 p. Tamm I.E. Osnovy teorii electrichestva (Fundamentals of the theory of electricity), Moscow: Nauka, 1989, 616 p. Colton D., Kress R. Integral equation methods in scattering theory, New York: John Wiley and Sons, 1983, 271 p. Translated under the title Metody integral’nykh uravnenii v teorii rasseyaniya, Moscow: Mir, 1987, 312 p. Ciarlet P. The finite element method for elliptic problems, Amsterdam–New York–Oxford: North-Holland Publishing Company, 1978, 548 p. Translated under the title Metod konechnykh elementov dlya ellipticheskikh zadach, Moscow: Mir, 1980, 512 p. Full text