Geodesic
Applications of the Carathéodory theorem to PDEs
Annales Polonici Mathematici, Tome 73 (2000) no. 1, pp. 1-27

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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.
DOI : 10.4064/ap-73-1-1-27
Mots-clés : Carathéodory theorem, product integral, Galerkin method 
Konstanty Holly; Joanna Orewczyk. Applications of the Carathéodory theorem to PDEs. Annales Polonici Mathematici, Tome 73 (2000) no. 1, pp. 1-27. doi: 10.4064/ap-73-1-1-27
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