Double forms: Regular elliptic bilaplacian operators

TL;DR

This paper investigates the properties of a fourth-order double bilaplacian operator on double forms over compact Riemannian manifolds, establishing its ellipticity and decomposition properties relevant to elasticity theory.

Contribution

It introduces and analyzes the double bilaplacian operator, proving its regular ellipticity and deriving a Hodge-like decomposition for double forms.

Findings

Proved regular ellipticity of the double bilaplacian for various boundary conditions.

Established a Hodge-like decomposition for double forms.

Provided foundational results for applications in incompatible elasticity.

Abstract

Double forms are sections of the vector bundles $\Lambda^{k}T^*\mathcal{M}\otimes \Lambda^{m}T^*\mathcal{M}$, where in this work $(\mathcal{M},\mathfrak{g})$ 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.