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2000 | 73 | 1 | 1-27
Tytuł artykułu

Applications of the Carathéodory theorem to PDEs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE $ẋ {(*)\over=} ℱ(t,x)$ for the right-hand side ℱ : dom ℱ ( ⊂ ℝ × X) → X taking values in an arbitrary Banach space X, and a related result concerning an extension of x. We propose a definition of solvability of (*) admitting all connected ℐ and unifying the cases "dom ℱ is open" and "dom ℱ = ℐ × Ω for some Ω ⊂ X". We show how to use the theorems mentioned above to get approximate solutions of a nonlinear parabolic PDE and exact solutions of a linear evolution PDE with distribution data.
Rocznik
Tom
73
Numer
1
Strony
1-27
Opis fizyczny
Daty
wydano
2000
otrzymano
1997-07-08
poprawiono
1998-01-10
poprawiono
2000-02-05
Twórcy
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Bibliografia
  • [1] G. Aguaro, Sul teorema di esistenza di Carathéodory per i sistemi di equazioni differenziali ordinarie, Boll. Un. Mat. Ital. 8 (1955), 208-212.
  • [2] P. S. Bondarenko, A remark on Carathéodory's existence and uniqueness conditions, Visnik Kiiv. Univ. Ser. Mat. Meh. 14 (1972), 39-42 (in Ukrainian).
  • [3] C. Carathéodory, Vorlesungen über reelle Funktionen, Teubner, Leipzig, 1927.
  • [4] J. D. Dollard and C. N. Friedman, Product Integrals, Encyclopedia Math. Appl. 10, Addison-Wesley, London, 1979.
  • [5] W. N. Everitt and D. Race, On necessary and sufficient conditions for the existence of Carathéodory solutions of ordinary differential equations, Quaestiones Math. 2 (1977/78), 507-512.
  • [6] P. Hartman, Ordinary Differential Equations, Wiley, 1964, p. 23.
  • [7] K. Holly, Approach to an integral kernel of the evolution N-S equations in $ℝ^n$ through integration of distribution-valued curves, in preparation.
  • [8] K. Holly and M. Danielewski, Interdiffusion in solids, free boundary problem for r-component (r ≥ 2) one dimensional mixture showing constant concentration, Phys. Rev. B 50 (1994), 13336-13346.
  • [9] K. Holly and M. Wiciak, Compactness method applied to an abstract nonlinear parabolic equation, in: Selected Problems of Mathematics, Anniversary Issue, Vol. 6, Cracow Univ. of Techn., Cracow, 1995, 95-160.
  • [10] J. Liouville, Sur le développement des fonctions ou parties de fonctions en séries, etc, Second Mémoire, J. de Math. 2 (1837), 16-35.
  • [11] S. Łojasiewicz, An Introduction to the Theory of Real Functions, Wiley, 1988.
  • [12] J. Mateja, personal communication.
  • [13] Z. Opial, Sur l'équation différentielle ordinaire du premier ordre dont le second membre satisfait aux conditions de Carathéodory, Ann. Polon. Math. 8 (1960), 23-28.
  • [14] A. Pelczar and J. Szarski, Introduction to the Theory of Differential Equations, PWN, Warszawa, 1987 (in Polish).
  • [15] E. Picard, Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives, J. Math. Pures Appl. 6 (1890), 145-210.
  • [16] R. Rabczuk, Elements of Differential Inequalities, PWN, Warszawa, 1976 (in Polish).
  • [17] S. Saks, Theory of the Integral, 2nd ed., Stechert, New York, 1937.
  • [18] G. Sansone, Equazioni differenziali nel campo reale, Parte seconda, Zanichelli, Bologna, 1949.
  • [19] K. Yosida, Functional Analysis, 6th ed., Springer, 1980, Chap. V, Sec. 5, 135-136.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv73z1p1bwm
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