Skip to Main Content

St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Homogenization of parabolic and elliptic periodic operators in $L_2(\mathbb {R}^d)$ with the first and second correctors taken into account
HTML articles powered by AMS MathViewer

by E. S. Vasilevskaya and T. A. Suslina
Translated by: T. A. Suslina
St. Petersburg Math. J. 24 (2013), 185-261
DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
Published electronically: January 22, 2013

Abstract:

In the space $L_2(\mathbb {R}^d;{\mathbb C}^n)$, a wide class of matrix elliptic second order differential operators (DO’s) $\mathcal {A}_\varepsilon$ is studied; the $\mathcal {A}_\varepsilon$ are assumed to admit a factorization of the form $\mathcal {A}_\varepsilon = \mathcal {X}_\varepsilon ^* \mathcal {X}_\varepsilon$, where $\mathcal {X}_\varepsilon$ is a homogeneous first order DO. The coefficients of these operators are periodic and depend on $\mathbf {x}/\varepsilon$, $\varepsilon >0$. The behavior of the operator exponential $e^{-\mathcal {A}_\varepsilon \tau }$, $\tau >0$, and of the resolvent $({\mathcal {A}}_\varepsilon +I)^{-1}$ for small $\varepsilon$ is investigated. An approximation for the exponential $e^{-\mathcal {A}_\varepsilon \tau }$ in the operator norm in $L_2(\mathbb {R}^d; \mathbb {C}^n)$ with an error term of order $\tau ^{-3/2}\varepsilon ^3$ is obtained. For the resolvent $({\mathcal {A}}_\varepsilon +I)^{-1}$, approximation in the norm of operators acting from $H^1(\mathbb {R}^d; \mathbb {C}^n)$ to $L_2( \mathbb {R}^d; \mathbb {C}^n)$ is found with an error term of order $\varepsilon ^3$. In these approximations, the first and second order correctors are taken into account.
References
Similar Articles
  • Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35B27
  • Retrieve articles in all journals with MSC (2010): 35B27
Bibliographic Information
  • E. S. Vasilevskaya
  • Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
  • Email: vasilevskaya-e@yandex.ru
  • T. A. Suslina
  • Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
  • Email: suslina@list.ru
  • Received by editor(s): November 1, 2011
  • Published electronically: January 22, 2013
  • Additional Notes: The authors were supported by RFBR (grant no. 11-01-00458-a) and the Program of Support of Leading Scientific Schools (grant NSh-357.2012.1)
  • © Copyright 2013 American Mathematical Society
  • Journal: St. Petersburg Math. J. 24 (2013), 185-261
  • MSC (2010): Primary 35B27
  • DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
  • MathSciNet review: 3013323