An Up-to-the Boundary Version of Friedrichs's Lemma and Applications to the Linear Koiter Shell Model

Abstract

In this work, we introduce a variant of the standard mollifier technique that is valid up to the boundary of a Lipschitz domain in $\mathbb{R}^n$. A version of Friedrichs's lemma is derived that gives an estimate up to the boundary for the commutator of the multiplication by a Lipschitz function and the modified mollification. We use this version of Friedrichs's lemma to prove the density of smooth functions in the new function space introduced in our earlierwork concerning the linear Koiter shell model for shells with little regularity. The density of smooth functions is in turn used to prove continuous dependence of the solution of Koiter's model on the midsurface. This provides a complete justification of our new formulation of the Koiter model.