Unique continuation for elliptic operators with variable coefficients.

In this talk, we revisit some of the fundamental results on unique continuation for elliptic operators with variable coefficients. Inspired by the recent examples constructed by Mandache and Krymskii-Logunov-Pagano, we sharpen some quantitative unique continuation properties for solutions $u$ of divergence-form elliptic equations with variable coefficients $A$. In the local setting, we prove that if $A$ is log-Lipschitz, then strong unique continuation still holds for solutions, which sharpens and clarifies the Lipschitz threshold pioneered by Han-Lin. I will also discuss joint work with B. Davey, where we prove optimal Landis-type results for solutions of Schrodinger-type equations with variable coefficients in cylinders.