Double forms: Regular elliptic bilaplacian operators

Abstract

Double forms are sections of the vector bundles \(\Lambda^{k}T^{\ast}{\cal{M}}\otimes\Lambda^{m}T^{\ast}\cal{M}\), where in this work (\(\cal{M},\frak{g}\)) is a compact Riemannian manifold with boundary. We study graded second-order differential operators on double forms, which are used in physical applications. A combination of these operators yields a fourth-order operator, which we call a double bilaplacian. We establish the regular ellipticity of the double bilaplacian for several sets of boundary conditions. Under additional conditions, we obtain a Hodge-like decomposition for double forms, whose components are images of the second-order operators, along with a biharmonic element. This analysis lays foundations for resolving several topics in incompatible elasticity, most prominently the existence of stress potentials and Saint-Venant compatibility.