Elliptic unique continuation below the Lipschitz threshold
Cole Jeznach
TL;DR
This paper addresses strong unique continuation for solutions to $-\mathrm{div}(A \nabla u)=0$ in $B_1$ when the coefficient matrix $A$ is less regular than Lipschitz. It develops an Almgren-frequency framework under an Osgood modulus of continuity, employing a smoothing argument and a growth-dichotomy to bound the frequency $N_u^A(r)$ and hence the vanishing order in terms of $N_u^A(1)$. The key finding is that strong unique continuation holds provided $\omega$ satisfies $\int_0^1 \frac{1}{\omega(t)} dt = \infty$, aligning the sharp threshold with log-Lipschitz regularity via Mandache’s counterexamples. Consequences include uniform doubling properties for solutions and, for Hölder coefficients, rectifiability of the critical set, contributing to a refined understanding of quantitative unique continuation in anisotropic media and informing inverse-problems analyses.
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 $ω$ satisfying the Osgood condition $\int_0^1 ω(t)^{-1}dt = \infty$, 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.