Title: Elliptic pre-complexes, Hodge-like decomposition, and Overdetermined boundary value problems

Abstract: We present a solution to a problem posed by Calabi more than 60 years ago, known as the Saint-Venant problem: Given a compact Riemannian manifold with boundary, find necessary and sufficient conditions for a symmetric tensor field to be a Lie derivative of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and draws upon analogies with the theory of electromagnetism. To this end, we outline a framework generalizing the theory of elliptic complexes for sequences of linear differential operators (Ak) between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions, and the order of Ak+1Ak does not exceed the order of Ak. We show that every elliptic pre-complex (Ak) can be “corrected” into a complex of pseudodifferential operators. The induced complex yieldes Hodge-like decompositions, much like in the classical theory, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. In particular, the theory applies to the so-called Calabi complex, thus resolving the Saint-Venant problem.


The talk is based on a joint work with Raz Kupferman.  

Live stream and recording: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=0daa009e-7fb3-4403-bbfc-aff000594ae6