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

TL;DR

This paper investigates second-order differential-operator boundary-value problems with spectral parameters and non-commuting boundary operators in a non-commutative Banach space setting, establishing solvability, regularity, and semigroup generation.

Contribution

It introduces a novel analysis of differential-operator equations with spectral parameters and non-commuting boundary conditions in a non-commutative framework, extending previous Hilbert space results.

Findings

Proved existence and uniqueness of solutions.

Derived representation formulas and regularity results.

Established generation of analytic semigroups.

Abstract

We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we have considered the case when spectral complex parameters appear in the equation and in the abstract Robin boundary condition illustrated by some unbounded operator non commuting with the one used in the equation. 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 work gives news considerations with respect to all those studied by the authors in [7] and is a continuation, in some sense, of the results in [1] studied in Hilbertian spaces.