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.
