On essential self-adjointness for first order differential operators on domains in $\mathbb{R}^d$

TL;DR

This paper establishes conditions under which first order differential operators on domains in Euclidean space are essentially self-adjoint, linking this property to the completeness of an associated Riemannian structure, with applications to wave energy confinement.

Contribution

It provides new criteria for essential self-adjointness of first order systems based on Riemannian completeness, without requiring ellipticity or high regularity of coefficients.

Findings

Essential self-adjointness linked to Riemannian completeness.

Criteria applicable to non-elliptic systems with $C^1$ coefficients.

Applications to energy confinement in wave propagation.

Abstract

We consider general symmetric systems of first order linear partial differential operators on domains $\Omega \subset \mathbb{R}^d$, and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The coefficients of the first order terms are only required to belong to $C^1(\Omega)$ and there is no ellipticity condition. Our criterion writes as the completeness of an associated Riemannian structure which encodes the propagation velocities of the system. As an application we obtain sufficient conditions for confinement of energy for some wave propagation problems of classical physics.