Transmission problem for an abstract fourth-order differential equation of elliptic type in UMD spaces

Abstract

In this work, we present a new result ofexistence, uniqueness and maximal regularity for the restriction solutions of the bilaplacian transmission problem set in the juxtaposition of two rectangular bodies. The study is performed in the space $L^{p}( (-1, 0) \cup(0, \delta) ;X),$ $1 < p < \infty,$ where $\delta $ is a small parameter which is destined to tend to zero and $X$ is a UMD Banach space. The geometry of the bodies allows us to find an explicit representation of the solutions in virtue of the operational Dunford calculus. We then use essentially the famous Dore-Venni theorem among others for the analysis of the solutions.