Lower bounds for Dirichlet Laplacians and uncertainty principles

TL;DR

This paper establishes geometric lower bounds for Dirichlet Laplacians on unbounded domains and applies these results to derive uncertainty principles for low energy functions of general elliptic operators.

Contribution

It introduces new geometric lower bounds for Dirichlet Laplacians and extends uncertainty principles to a broad class of elliptic operators.

Findings

Lower bounds depend on geometric conditions of the domain.

Uncertainty principles are derived for low energy functions.

Results apply to elliptic operators with non-continuous coefficients.

Abstract

We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence form operators with not necessarily continuous main part.