On Fuglede's flux extensions and the point wise definition of linear partial differential operators

TL;DR

This paper surveys Fuglede's flux extensions of first order PDEs, connecting classical and modern theories, and introduces new results including a local limit formula and applications to divergence theorems.

Contribution

It revisits Fuglede's flux extensions, provides new applications, and establishes a local limit formula linking to wave cones in PDE theory.

Findings

Generalization of Morera's theorem for first order operators

New local limit formula for maximal extension of PDE operators

Connections between flux extensions and wave cones in PDEs

Abstract

In this work we provide a survey of Fuglede's flux extensions of first order partial differential operators, a concept largely forgotten today. A long the way we also survey the classical weak and strong extensions of PDE operators and the works of Friedrichs and H\"ormander. We give several applications of this theory showing its usefulness, as well as connecting it to more recent developments in connection to various sharp versions of the divergence theorem. In particular, we use it to prove a generalization of Morera's theorem valid for general first order operators. Using this theory we also prove a new local limit formula for the maximal extension of a first order operator. We initiate a study of this limit and connect it to the wave cone of the operator, a concept that first arose in the theory of compensated compactness. Hopefully, this will contribute to a rival of Fuglede's beautiful ideas.