Elliptic differential-operator with an abstract Robin boundary condition containing two spectral parameters, study in a non commutative framework

Abstract

We study the solvability of some boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +∞, X being a UMD complex Banach space. The originality of this work lies in the fact that we consider the case where two spectral complex parameters appear in the equation and in abstract Robin boundary conditions. Here, the unbounded linear operator in the equation is not commuting with the one appearing in the boundary conditions. This represents the strong novelty with respect to the existing literature. Existence, uniqueness, representation formula, maximal regularity of the solution, sharp estimates and generation of strongly continuous analytic semigroup are proved. Many concrete applications are given for which our theory applies. This paper improves, in some sense, results by the authors in [7] and it can be viewed as a continuation of the results in [1] studied only in Hilbert spaces.