Differential equations of elliptic type with variable operators and homogeneous Robin boundary value condition in UMD spaces

In this article, we give new results on the study of elliptic abstract second order differential equation with variable operators coefficients under the general Robin homogeneous boundary value conditions, in the framework of UMD spaces. Here, we do not assume the differentiability of the resolvent operators. However, we suppose that the family of variable operators verifies the Labbas–Terreni assumption inspired by the sum theory and similar to the Acquistapace–Terreni one. We use Dunford calculus, interpolation spaces and semigroup theory in order to obtain existence, uniqueness and maximal regularity results for the classical solution to the problem.