On the justification of Koiter's model for elliptic membranes subjected to an interior normal compliance contact condition (2504.21637v1)

Abstract: The purpose of this paper is twofold. First, we rigorously justify Koiter's model for linearly elastic elliptic membrane shells in the case where the shell is subject to a geometrical constraint modelled via a normal compliance contact condition defined in the interior of the shell. To achieve this, we establish a novel density result for non-empty, closed, and convex subsets of Lebesgue spaces, which are applicable to cases not covered by the ``density property'' established in [Ciarlet, Mardare & Piersanti, \emph{Math. Mech. Solids}, 2019]. Second, we demonstrate that the solution to the two-dimensional obstacle problem for linearly elastic elliptic membrane shells, subjected to the interior normal compliance contact condition, exhibits higher regularity throughout its entire definition domain. A key feature of this result is that, while the transverse component of the solution is, in general, only of class $L^2$ and its trace is \emph{a priori} undefined, the methodology proposed here, partially based on [Ciarlet & Sanchez-Palencia, \emph{J. Math. Pures Appl.}, 1996], enables us to rigorously establish the well-posedness of the trace for the transverse component of the solution by means of an \emph{ad hoc} formula.