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Transactions of the Moscow Mathematical Society

This journal, a translation of Trudy Moskovskogo Matematicheskogo Obshchestva, contains the results of original research in pure mathematics.

ISSN 1547-738X (online) ISSN 0077-1554 (print)

The 2020 MCQ for Transactions of the Moscow Mathematical Society is 0.74.

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The Vishik–Lyusternik method in the mixed problem for parabolic operators unresolved with respect to the highest time derivative
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by L. R. Volevich
Translated by: O. A. Khleborodova
Trans. Moscow Math. Soc. 2007, 67-92
DOI: https://doi.org/10.1090/S0077-1554-07-00161-6
Published electronically: November 15, 2007
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Abstract:

We consider the mixed problem for parabolic operators unresolved with respect to the highest time derivative with boundary conditions of general type and zero initial conditions. We present an analog of the Shapiro–Lopatinskii condition that allows one to obtain two-sided a priori estimates in specially constructed function spaces. In the case considered in this paper the characteristic equation in the half-space has two groups of roots with different asymptotics. Because of this, the crucial role in the study of the problem is played by the Vishik–Lyusternik method in the form presented by Volevich (2006).
References
  • M. I. Višik and L. A. Ljusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Amer. Math. Soc. Transl. (2) 20 (1962), 239–364. MR 0136861
  • L. R. Volevich, The Vishik-Lyusternik method in elliptic problems with a small parameter, Tr. Mosk. Mat. Obs. 67 (2006), 104–147 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2006), 87–125. MR 2301592, DOI 10.1090/s0077-1554-06-00154-3
  • M. S. Agranovič and M. I. Višik, Elliptic problems with a parameter and parabolic problems of general type, Uspehi Mat. Nauk 19 (1964), no. 3 (117), 53–161 (Russian). MR 0192188
  • Gennadii V. Demidenko and Stanislav V. Uspenskii, Partial differential equations and systems not solvable with respect to the highest-order derivative, Monographs and Textbooks in Pure and Applied Mathematics, vol. 256, Marcel Dekker, Inc., New York, 2003. Translated from the 1998 Russian original by Tamara Rozhkovskaya; With an appendix containing a translation of a paper by S. L. Sobolev [Izv. Akad. Nauk SSSR Ser. Math. 18 (1954), 3–50; MR0069382]. MR 2064818, DOI 10.1201/9780203911433
  • R. Denk, R. Mennicken, and L. Volevich, Boundary value problems for a class of elliptic operator pencils, Integral Equations Operator Theory 38 (2000), no. 4, 410–436. MR 1799649, DOI 10.1007/BF01228606
  • R. Denk, R. Mennicken, and L. Volevich, On elliptic operator pencils with general boundary conditions, Integral Equations Operator Theory 39 (2001), no. 1, 15–40. MR 1806842, DOI 10.1007/BF01192148
  • S. A. Nazarov, The Vishik-Lyusternik method for elliptic boundary value problems in regions with conic points. I. Problem in a cone, Sibirsk. Mat. Zh. 22 (1981), no. 4, 142–163, 231 (Russian). MR 624412
  • L. R. Volevič and B. P. Panejah, Some spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk 20 (1965), no. 1 (121), 3–74 (Russian). MR 0174970
  • Sergeĭ Nazarov, The index of an elliptic boundary value problem with a small parameter multiplying the highest derivative, Differentsial′nye Uravneniya i Primenen.—Trudy Sem. Protsessy Optimal. Upravleniya I Sektsiya 24 (1979), 75–85, 109 (Russian, with English and Lithuanian summaries). MR 566964
  • Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
  • S. G. Gindikin and L. R. Volevich, Mixed problem for partial differential equations with quasihomogeneous principal part, Translations of Mathematical Monographs, vol. 147, American Mathematical Society, Providence, RI, 1996. Translated from the Russian manuscript by V. M. Volosov. MR 1357662, DOI 10.1090/mmono/147
  • Léonid S. Frank, Coercive singular perturbations. I. A priori estimates, Ann. Mat. Pura Appl. (4) 119 (1979), 41–113. MR 551218, DOI 10.1007/BF02413170
  • S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
  • S. G. Gindikin and L. R. Volevich, Distributions and convolution equations, Gordon and Breach Science Publishers, Philadelphia, PA, 1992. Translated from the Russian by V. M. Volosov. MR 1155843
  • I. Fedotov and L. R. Volevich, The Cauchy problem for hyperbolic equations not resolved with respect to the highest time derivative, Russ. J. Math. Phys. 13 (2006), no. 3, 278–292. MR 2262830, DOI 10.1134/S1061920806030046
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Bibliographic Information
  • L. R. Volevich
  • Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4 Miusskaya Place, Moscow 125047, Russia
  • Published electronically: November 15, 2007
  • Additional Notes: The work is supported by the Russian Foundation of Fundamental Research, Grant 06-01-00096.

    • Dedicated: To Mark Iosifovich Vishik for his 80th birthday
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Moscow Math. Soc. 2007, 67-92
  • MSC (2000): Primary 35K40; Secondary 35B25
  • DOI: https://doi.org/10.1090/S0077-1554-07-00161-6
  • MathSciNet review: 2429267