Etude díun problËme ‡ conditions aux limites nonlocales gÈnÈralisÈes de type Bitsadze-Samarskii dans le cadre des espaces L p

Abstract

This work is devoted to the study of General Bitsadze-Samarskii Problems of elliptic type in the framework of UMD Banach spaces. More precisely we consider the following abstract second order di§erential equation: u 00(x) + (L M)u 0 (x) LMu(x) = f(x) p.p. x 2]0; 1[; (1) with : for the Örst problem the following nonlocal generalized boundary conditions:

u(0) = u0; u(1) Hu(x0) = u1;x0 ; (2) for the second problem the following nonlocal generalized boundary conditions:

u(0) = u0; u(1) Hu0 (x0) = u1;x0 : (3) Here X is a Banach complexe space, f 2 L p (0; 1; X) where p 2]1;1[, u0; u1;x0 2 X. Moreover L; M et H are closed linear operators in X. We obtain some results about existence, uniqueness and regularity of the solution. We deÖne two types of solutions (strict and semi-strict solutions) and we give necessary and su¢ cient conditions on the data to obtain these results. The method used is based on Önding a formula to represent the solution in each case using the semi-groups and fractional powers of the operators. Then an analysis of this representation is made to Önd regularity results of the solution using the interpolation spaces and the Dore-Venni operators sums theory.