Eigenvalue asymptotics for potential type operators on lipschitz surfaces of codimension greater than 1

• Autorzy: Rozenblum G. , Tashchiyan G.

Identyfikatory

DOI

10.7494/OpMath.2018.38.5.733

• Języki publikacji: EN

Abstrakty

EN

For potential type integral operators on a Lipschitz submanifold the asymptotic formula for eigenvalues is proved. The reasoning is based upon the study of the rate of operator convergence as smooth surfaces approximate the Lipschitz one.

Słowa kluczowe

EN

integral operators; potential theory; eigenvalue asymptotics

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2018
• Tom: Vol. 38, no. 5
• Strony: 733--758
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Rozenblum G.

• Department ol Mathematics Chalmers University ol Technology, Sweden
• University ol Gothenburg Eklandagatan 86, S-412 96 Gothenburg, Sweden
• Department ol Physics St. Petersburg State University Russia

autor

Tashchiyan G.

• St. Petersburg University lor Telecommunications Department ol Mathematics St. Petersburg, 198504, Russia

Bibliografia

• [1] M.S. Agranovich, Spectral properties of potential-type operators for a class of strongly elliptic systems on smooth and Lipschitz surfaces, Tr. Mosk. Mat. Obsh. 62 (2001), 3-53 [in Russian]; English translation in Trans. Moscow Math. Soc. 62 (2001), 1-47.
• [2] M.S. Agranovich, Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains, Uspehi Matem. Nauk 57 (2002) 5, 3-78 [in Russian]; English translation in Russian Mathem. Surveys 57 (2002), 847-920.
• [3] M.S. Agranovich, B.A. Amosov, Estimates of s-numbers and spectral asymptotics for integral operators of potential type on non-smooth surfaces, Funct. Anal, i Prilozh. 30 (1996) 2, 1-18 [in Russian]; English translation in: Funct. Anal, and Appl. 30 (1996) 75-89.
• [4] M.S. Agranovich, B.Z. Katsenelenbaum, A.N. Sivov, N.N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley-VCH Verlag, Berlin, 1999.
• [5] M.Sh. Birman, M.Z. Solomjak, Asymptotics of the spectrum of weakly polar integral operators, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970) 1142-1158 [in Russian]; translation in Mathematics of the USSR-Izvestiya, 4:5 (1970), 1151-1168.
• [6] M.Sh. Birman, M.Z. Solomjak, The asymptotics of the spectrum of pseudodifferential operators with anisotropic-homogeneous symbols, I, II, Vestnik Leningr. Univ. (Math.) no. 13 (1977), 13-27, no. 13 (1979), 5-10; English translation in Vestnik. Leningr. Univ. Math. 10 (1980), 12 (1982).
• [7] T. Chang, Boundary integral operator for the fractional Laplacian on the boundary of a bounded smooth domain, J. Integral Equations Appl. 28 (2016) 3, 343-372.
• [8] M. Cotlar, R. Cignoli, An Introduction to Functional Analysis, NH, 1974.
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• [10] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
• [11] M. Goffeng, Analytic formulas for the topological degree of non-smooth mappings: the odd-dimensional case, Adv. Math. 231 (2012) 1, 357-377.
• [12] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nons elf adjoint Operators, Nauka, Moscow, 1965; English transl.: Amer. Math. Soc, 1969.
• [13] G. Grubb, Singular Green operators and their spectral asymptotics, Duke Math. J. 51 (1984), 477-528.
• [14] G. Hsiao, W. Wendland, Boundary Integral Equations, Springer-Verlag, Berlin, 2008.
• [15] G.E. Karadzhov, The inclusion of integral operators in the classes Sp for p > 2, Integral and differential operators. Differential equations, Probl. Mat. Anal., vol. 3, Leningrad, 1972, pp. 28-33 [in Russian]; English transl. in J. Soviet Mathem. 1 (1973), 200-204.
• [16] G.P. Kostometov, Asymptotic behavior of the spectrum of integral operators with a singularity on the diagonal, Matemat. Sbornik. 94(136)-3(7) (1974), 444-451 [in Russian]; English translation in Math. USSR-Sb. 23:3 (1974), 417-424.
• [17] G.P. Kostometov, On the asymptotics of the spectrum of integral operators with polar kernels, Vestn. Leningr. Univ. 1977, no. 13, Mat. Mekh. Astron. 1977, no. 3, (1977) 166-167 [in Russian].
• [18] G. Rozenblum, G. Tashchiyan, Eigenvalue asymptotics for potential type operators on Lipschitz surfaces, Russian J. Math. Phys. 13 (2006) 3, 326-339.
• [19] N.N. Voitovich, B.Z. Katsenelenbaum, A.N. Sivov, Generalized Method of Eigenoscilla-tions in Diffraction Theory, Nauka, Moscow, 1977 [in Russian].
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Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).