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A regularized trace formula for a well-perturbed Laplace operator. (English. Russian original) Zbl 1325.35015
Dokl. Math. 91, No. 1, 1-4 (2015); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 460, No. 1, 7-10 (2015).
From the text: This paper essentially uses the methods of V. A. Sadovnichij and V. A. Lyubishkin’s paper [Funct. Anal. Appl. 20, 214–223 (1986; Zbl 0656.47007); translation from Funkts Anal. Prilozh. 20, No. 3, 55–65 (1986)], where finite dimensional perturbations of discrete operators were studied and formulas for the first regularized traces were derived. In [loc. cit.], the domains of the initial and the perturbed operator coincided. This paper studies perturbations of boundary conditions, i.e., the domain of the perturbed operator differs from that of the initial operator. The main result of the paper is the derivation and proof of the first regularized trace of the Laplace operator in a plane bounded domain.

MSC:
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47A55 Perturbation theory of linear operators
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[1] Sadovnichii, V A; Lyubishkin, V A, No article title, Funct. Anal. Its Appl., 20, 214-223, (1986) · Zbl 0656.47007
[2] Sadovnichii, V A; Podol’skii, V E, No article title, Differ. Equations, 45, 477-493, (2009) · Zbl 1177.47054
[3] Sadovnichii, V A; Podol’skii, V E, No article title, Sb.: Math., 193, 279-302, (2002)
[4] Murtazin, Kh Kh; Fazullin, Z Yu, No article title, Sb.: Math., 196, 1841-1874, (2005) · Zbl 1135.47009
[5] Sadovnichii, V A; Podol’skii, V E, No article title, Proc. Steklov Inst. Math. (Suppl.), 255, s161-s177, (2006) · Zbl 1135.47031
[6] Kanguzhin, B E; Aniyarov, A A, No article title, Math. Notes, 89, 819-829, (2011) · Zbl 1244.35031
[7] Kanguzhin, B E; Nurakhmetov, D B; Tokmagambetov, N E, No article title, Russian Math. (Izv. VUZ. Mat.), 58, 6-12, (2014) · Zbl 1304.35236
[8] Kanguzhin, B E; Nurakhmetov, D B; Tokmagambetov, N E, No article title, Ufimsk. Mat. Zh., 3, 80-92, (2011) · Zbl 1249.47026
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