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Mat. Sb. (N.S.), 1975, Volume 98(140), Number 2(10), Pages 163–184 (Mi msb3704)  

This article is cited in 27 scientific papers (total in 27 papers)

Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation

Yu. A. Dubinskii


Abstract: In the study of the Cauchy–Dirichlet problem
\begin{gather} L(u)\equiv\sum_{|\alpha|=0}^\infty(-1)^{|\alpha|}D^\alpha A_\alpha(x, D^\gamma u)=h(x),\qquad x\in G,
D^\omega u\mid_{\partial G}=0,\qquad |\omega|=0,1,…, \end{gather}
infinite order Sobolev spaces
$$ \overset\circ W ^\infty\{a_\alpha, p_\alpha\}\equiv\{u(x)\in C^\infty_0(G):\rho(u)\equiv\sum^\infty_{|\alpha|=0}a_\alpha\|D^\alpha u\|_{p_\alpha}^{p_\alpha}<\infty\}, $$
naturally arise, where $a_\alpha\geqslant0$ and $p_\alpha\geqslant1$ are numerical sequences. In this paper criteria for the nontriviality of $\overset\circ W ^\infty\{a_\alpha,p_\alpha\}$ are established and the problem (1), (2) is investigated. Further, a theorem is obtained on the existence of the limit (as $m\to\infty$) of solutions of nonlinear $2m$th order boundary value problems of elliptic and hyperbolic type, from which, in particular, follows the solvability of the mixed problem for the nonlinear hyperbolic equation $u"+L(u)=h(t,x)$, $t\in[0,T]$, where $T>0$ is arbitrary.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 27:2, 143–162

Bibliographic databases:

UDC: 517.946.9
MSC: Primary 46E35, 35J60, 35L35; Secondary 28A93
Received: 14.04.1975

Citation: Yu. A. Dubinskii, “Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation”, Mat. Sb. (N.S.), 98(140):2(10) (1975), 163–184; Math. USSR-Sb., 27:2 (1975), 143–162

Citation in format AMSBIB
\Bibitem{Dub75}
\by Yu.~A.~Dubinskii
\paper Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 98(140)
\issue 2(10)
\pages 163--184
\mathnet{http://mi.mathnet.ru/msb3704}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=412580}
\zmath{https://zbmath.org/?q=an:0324.46037}
\transl
\jour Math. USSR-Sb.
\yr 1975
\vol 27
\issue 2
\pages 143--162
\crossref{https://doi.org/10.1070/SM1975v027n02ABEH002506}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. A. Dubinskii, “Nontriviality of Sobolev spaces of infinite order for a full Euclidean space and a torus”, Math. USSR-Sb., 29:3 (1976), 393–401  mathnet  crossref  mathscinet  zmath  isi
    2. Yu. A. Dubinskii, “Traces of functions from Sobolev spaces of infinite order and inhomogeneous problems for nonlinear equations”, Math. USSR-Sb., 34:5 (1978), 627–644  mathnet  crossref  mathscinet  zmath
    3. Yu. A. Dubinskii, “Limits of Banach spaces. Imbedding theorems. Applications to Sobolev spaces of infinite order”, Math. USSR-Sb., 38:3 (1981), 395–405  mathnet  crossref  mathscinet  zmath  isi
    4. Tran Duc Van, “Elliptic equations of infinite order with arbitrary nonlinearities and corresponding function spaces”, Math. USSR-Sb., 41:2 (1982), 203–216  mathnet  crossref  mathscinet  zmath
    5. Van C., “Solvability of Boundary-Value-Problems for Degenerate Non-Linear Differential-Equations of Infinite-Order”, Differ. Equ., 16:10 (1980), 1202–1211  isi
    6. Dubinskii I., “A Method of Solving Partial-Differential Equations”, 258, no. 4, 1981, 780–784  mathscinet  isi
    7. Balashova G., “Behavior of Solutions of Certain Boundary-Value-Problems When the Order of the Equation Increases Indefinitely”, Differ. Equ., 17:2 (1981), 175–185  mathscinet  zmath  isi
    8. G. S. Balashova, “On extension theorems in spaces of infinitely differentiable functions”, Math. USSR-Sb., 46:3 (1983), 375–389  mathnet  crossref  mathscinet  zmath
    9. Kobilov A., “Non-Triviality of Some Spaces of Infinitely Differentiable Functions in Corner Domains and the Solvability of Non-Linear Elliptic-Equations”, 266, no. 5, 1982, 1040–1044  mathscinet  zmath  isi
    10. Kuchminskaya L., “Solvability of Mixed Problems for a Certain Class of Nonlinear Infinite-Order Differential-Equations”, no. 5, 1984, 8–11  mathscinet  isi
    11. Radyno Y., “Differential-Equations in a Banach-Space Scale”, Differ. Equ., 21:8 (1985), 971–979  mathnet  mathscinet  zmath  isi
    12. G. S. Balashova, “On the extension of infinitely differentiable functions”, Math. USSR-Izv., 31:3 (1988), 603–620  mathnet  crossref  mathscinet  zmath
    13. Ha Huy Bang, “On imbedding theorems in Sobolev spaces of infinite order”, Math. USSR-Sb., 64:1 (1989), 115–127  mathnet  crossref  mathscinet  zmath
    14. Nguyen Minh Chuong, Le Kuang Chung, “On limit equations for degenerate non-linear elliptic equations in Sobolev–Orlicz spaces with weight”, Russian Math. Surveys, 43:2 (1988), 181–182  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    15. Yu. A. Dubinskii, “Sobolev spaces of infinite order”, Russian Math. Surveys, 46:6 (1991), 107–147  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. P. P. Zabreiko, V. I. Nazarov, “Smoothness properties of solutions of nonlinear differential equations”, Math. USSR-Sb., 72:1 (1992), 135–150  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. Ha Huy Bang, “Properties of functions in Orlicz spaces that depend on the geometry of their spectra”, Izv. Math., 61:2 (1997), 399–434  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. Ha Huy Bang, “Separability of Sobolev-Orlicz spaces of infinite order”, Math. Notes, 61:1 (1997), 118–120  mathnet  crossref  crossref  mathscinet  zmath  isi
    19. W. Kotarski, G.M. Bahaa, “Optimal Control Problem for Infinite Order Hyperbolic System with Mixed Control-State Constraints”, European Journal of Control, 11:2 (2005), 150  crossref
    20. W. Kotarski, G.M. Bahaa, “Optimality conditions for infinite order hyperbolic control problem with non-standard functional and time delay”, Journal of Information and Optimization Sciences, 28:3 (2007), 315  crossref
    21. M. Chrif, S. El Manouni, “Anisotropic equations in weighted Sobolev spaces of higher order”, Ricerche mat, 2009  crossref
    22. Mostafa Bendahmane, Moussa Chrif, Said El Manouni, “Elliptic equations in weighted Sobolev spaces of infinite order with L <sup>1</sup> data”, Math Meth Appl Sci, 2009, n/a  crossref  isi
    23. Kowalewski A., “Optimal Control via Initial Conditions of Infinite Order Hyperbolic Systems”, 2012 17th International Conference on Methods and Models in Automation and Robotics (Mmar), IEEE, 2012, 212–215  isi
    24. B.G.aber Mohamed, “Boundary Control Problem of Infinite Order Distributed Hyperbolic Systems Involving Time Lags”, ICA, 03:03 (2012), 211  crossref
    25. M.H. Abdou, A. Benkirane, M. Chrif , S. El Manouni, “Strongly anisotropic elliptic problems of infinite order with variable exponents”, Complex Variables and Elliptic Equations, 2014, 1  crossref
    26. G.M. Bahaa, S.A.A. El-Marouf, “Pareto Optimal Control For Mixed Neumann Infinite-Order Parabolic System With State-Control Constraints”, Journal of Taibah University for Science, 2014  crossref
    27. M.H.ousseine Abdou, Moussa Chrif, Said El Manouni, “Parabolic Equations of Infinite Order withL1Data”, Abstract and Applied Analysis, 2014 (2014), 1  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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