Elliptic unique continuation below the Lipschitz threshold

TL;DR

This paper establishes that strong unique continuation for solutions of elliptic equations holds under a modulus of continuity condition weaker than Lipschitz, specifically the Osgood condition, and identifies log-Lipschitz regularity as sharp.

Contribution

It proves strong unique continuation for elliptic equations with coefficients satisfying the Osgood condition, extending previous results to less regular coefficient matrices.

Findings

Unique continuation holds under Osgood condition on coefficients.

Log-Lipschitz regularity is shown to be sharp for unique continuation.

Counterexamples demonstrate the necessity of the regularity condition.

Abstract

In this article, we investigate unique continuation principles for solutions $u$ of uniformly elliptic equations of the form $-\mathrm{div}(A \nabla u) = 0$ when $A$ is less regular than Lipschitz. For general matrices $A$, we prove that strong unique continuation holds provided that $A$ has modulus of continuity $\omega$ satisfying the Osgood condition $\int_0^1 \omega(t)^{-1}dt = \infty$, plus some other mild hypotheses. Along with the counterexamples of Mandache, this shows that the sharp condition on $A$ that guarantees unique continuation is essentially that $A$ is log-Lipschitz.