The aim of this work is the study of complete elliptic differential equations of second order with operator coefficients in a Banach space X. A concrete application of these equations is detailed, it concerns a transmission problem of electric potential in a biological cell where the membrane is considered as a thin layer. The originality of this work is the fact that unbounded operators which are considered do not commute necessarily. A new noncommutativity hypothesis is then introduced. The analysis is performed in two distinct functional frameworks: the Hölder spaces and the Lp spaces (X being a UMD space). First, the equation is studied on the whole real line and secondly in a bounded interval with Dirichlet boundary conditions. Existence, uniqueness and maximal regularity of the classical solution are proved under some conditions on the data in interpolation spaces. The techniques used are based on semigroup theory, Dunford functional calculus and interpolation theory. All the results are applied to concrete partial differential equations of elliptic or quasi-elliptic type.
