DOI 10.1007/s10958-018-3873-2 Journal of Mathematical Sciences, Vol. 232, No. 3, July, 2018 THE NORM RESOLVENT CONVERGENCE FOR ELLIPTIC OPERATORS IN MULTI-DIMENSIONAL DOMAINS WITH SMALL HOLES D. I. Borisov Institute of Mathematics, USC RAS 112, Chernyshevskii St., Ufa 450008, Russia Bashkir State Pedagogical University 3a, October Revolution St., Ufa 450000, Russia University of Hradec Kr´ alov´e 62, Rokitansk´eho, Hradec Kr´ alov´e 50003, Czech Republic email@example.com A. I. Mukhametrakhimova Bashkir State Pedagogical University 3a, October Revolution St., Ufa 450000, Russia firstname.lastname@example.org UDC 517.956+517.958 We consider a second order elliptic operator with variable coeﬃcients in a multi- dimensional domain with a small hole and some classical boundary condition on the hole boundary. We show that the resolvent of this operator converges to the resolvent of the limit operator in the domain without holes in the sense of the norm of bounded operators acting from L to W . For the convergence rate we obtain sharp estimates relative to the smallness order. Bibliography:28 titles. Dedicated to the memory of Vasilii Vasil’evich Zhikov Boundary value problems in domains with small holes are related to the classical theory of singular perturbations. A small hole is one of the simplest examples of a geometric perturbation to
Journal of Mathematical Sciences – Springer Journals
Published: Jun 2, 2018
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