PROBLEME AUX LIMITES SINGULIER A COEFFICIENTS OPERATEURS

Abstract

This work is devoted to the study of a second order abstract di¤erential equation of elliptic type : u00 (x) + 2Bu0 (x) + Au (x) = f (x) ; x 2 (0; 1) with the nonlocal boundary conditions

u (0) = u0 u (1) + Hu0 (0) = u1;0 where A;B and H are closed linear operators of respective Domains D(A);D(B) and D(H) in X (a complexe Banach space), u0,u1;0 2 X and f is a function with values in X. we are interested in the existence, the unicity and the maximum regularity of solutions of this problem when the second member f belongs to the one of the two classes of spaces of Banach of geometry di¤erent C ([0; 1];X) and LP (0; 1;X) with 0 < < 1; 1 < p < 1: In the rst framework (taking into account the regularity hölderian of the second member f), where spaces it of Banach X is unspeci ed,we will also prove, under the same assump- tions, new optimal results if and only if the data u0, u1;0, f check certain natural compati- bility conditions related to the equation it self. Here, the techniques used are based on the theory of analytic semigroups, the famous theory of operators sums : Da Prato and Grisvard and mainly on the work of Sinestrari. In the second functional framework LP (0; 1;X), when the space of Banach X has property UMD and under certain assumptions on the operators (ellipticity, commutation,. . . ), we will show new optimal results if and only if the data u0, u1;0 are in interpolation spaces. The techniques used are based on class known as BIP operators and primarily on the famous Theorem of Dore and Venni.