Study of boundary value and transmission problems governed by a class of variable operators verifying the Labbas-Terreni non commutativity assumption

The aim of this work is the study of some transmission problems, which are written as an abstract second order differential equation of elliptic type with variable operator coefficients, in the framework of Holder 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 solution of the problem.