Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations

TL;DR

This paper establishes scale-free, quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations, applicable to eigenfunctions and general solutions, with uniform bounds and stability properties.

Contribution

It introduces new uniform, explicit bounds for unique continuation and sampling estimates for elliptic PDE solutions, valid across different scales and coefficient ensembles.

Findings

Derived local vanishing order estimates for solutions.

Established $L^2$-norm sampling inequalities over equidistributed regions.

Proved bounds are uniform, explicit, and stable under small shifts.

Abstract

We consider elliptic differential operators on either the entire Euclidean space $\mathbb{R}^d$ or on subsets consisting of a cube $\Lambda_L$ of integer length $L$. For eigenfunctions of the operator, and more general solutions of elliptic differential quations, we derive several quantitative unique continuation results. The first result is of local nature and estimates the vanishing order of a solution. The second is a sampling result and compares the $L^2$-norm of a solution over a union of equidistributed $\delta$-balls in space with the $L^2$-norm on the entire space. In the case where the space $\mathbb{R}^d$ is replaced by a finite cube $\Lambda_L$ we derive similar estimates. A particular feature of our bound is that they are uniform as long as the coefficients of the operator are chosen from an appropriate ensemble, they are quantitative and explicit with respect to the radius $\delta$, they are $L$-independent and stable under small shifts of the $\delta$-balls. Our proof applies to second order terms which have slowly varying coefficients on the relevant length scale. The results can be also interpreted as special cases of uncertainty relations, observability estimates, or spectral inequalities.