Resolution in the L p framework of some elliptical problems with a Robin-type boundary condition and two spectral parameters

Abstract

In this master thesis, we study the following abstract second order problem    u ′′(x) + Au(x) − λu(x) = f(x), x ∈ (0, +∞) u ′ (0) − Hu(0) − µu(0) = d0, (1) posed in the unbounded domain R+, such that A and H are two closed operators in a complex Banach space X, d0 is an auxiliary element given in X, and f is a function in L p (R+, X). λ and µ are two complex parameters belong to the set Ωφ0,φ1,r defined as : Ωφ0,φ1,r = {(λ, µ) ∈ Sφ0 × Sφ1 : |λ| ≥ r > 0 and |µ| 2 |λ| ≥ r > 0}, where Sφ0 , Sφ1 are defined by (1.2). For some large positive number r0, We study the existence and uniqueness of the classical solution in a complex Banach space X of type UMD of the problem (1), with the regular second condition u(+∞) = 0. By using semigroups theory [22], and fractional powers of operators [2] and [23], we prove that u ∈ W2.p (0, +∞; X) T L p (0, +∞; D(A)) and u(0) ∈ D(H).