# Applications of the Carathéodory theorem to PDEs

• Volume: 73, Issue: 1, page 1-27
• ISSN: 0066-2216

top

## Abstract

top
We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE $ẋ\frac{\left(*\right)}{=}ℱ\left(t,x\right)$ 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.

## How to cite

top

Holly, Konstanty, and Orewczyk, Joanna. "Applications of the Carathéodory theorem to PDEs." Annales Polonici Mathematici 73.1 (2000): 1-27. <http://eudml.org/doc/262557>.

@article{Holly2000,
abstract = {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.},
author = {Holly, Konstanty, Orewczyk, Joanna},
journal = {Annales Polonici Mathematici},
keywords = {Carathéodory theorem; product integral; Galerkin method; abstract ODE; parabolic PDE; Banach space},
language = {eng},
number = {1},
pages = {1-27},
title = {Applications of the Carathéodory theorem to PDEs},
url = {http://eudml.org/doc/262557},
volume = {73},
year = {2000},
}

TY - JOUR
AU - Holly, Konstanty
AU - Orewczyk, Joanna
TI - Applications of the Carathéodory theorem to PDEs
JO - Annales Polonici Mathematici
PY - 2000
VL - 73
IS - 1
SP - 1
EP - 27
AB - 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.
LA - eng
KW - Carathéodory theorem; product integral; Galerkin method; abstract ODE; parabolic PDE; Banach space
UR - http://eudml.org/doc/262557
ER -

## References

top
1. [1] G. Aguaro, Sul teorema di esistenza di Carathéodory per i sistemi di equazioni differenziali ordinarie, Boll. Un. Mat. Ital. 8 (1955), 208-212. Zbl0064.33601
2. [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. [3] C. Carathéodory, Vorlesungen über reelle Funktionen, Teubner, Leipzig, 1927.
4. [4] J. D. Dollard and C. N. Friedman, Product Integrals, Encyclopedia Math. Appl. 10, Addison-Wesley, London, 1979.
5. [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. Zbl0392.34002
6. [6] P. Hartman, Ordinary Differential Equations, Wiley, 1964, p. 23. Zbl0125.32102
7. [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. [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. [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. [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. [11] S. Łojasiewicz, An Introduction to the Theory of Real Functions, Wiley, 1988. Zbl0653.26001
12. [12] J. Mateja, personal communication.
13. [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. Zbl0093.08404
14. [14] A. Pelczar and J. Szarski, Introduction to the Theory of Differential Equations, PWN, Warszawa, 1987 (in Polish).
15. [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. Zbl22.0357.02
16. [16] R. Rabczuk, Elements of Differential Inequalities, PWN, Warszawa, 1976 (in Polish). Zbl0351.34006
17. [17] S. Saks, Theory of the Integral, 2nd ed., Stechert, New York, 1937. Zbl0017.30004
18. [18] G. Sansone, Equazioni differenziali nel campo reale, Parte seconda, Zanichelli, Bologna, 1949. Zbl0033.36801
19. [19] K. Yosida, Functional Analysis, 6th ed., Springer, 1980, Chap. V, Sec. 5, 135-136.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.