Elliptic unique continuation below the Lipschitz threshold

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 $\ddot{I}‰$ satisfying the Osgood condition ${\int }_0^1\ddot{I}‰(t{)}^{-1}dt=\infty$, 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.

Error found in the proof of Theorem 1.2 for isotropic equations. The main Theorem for anisotropic equations remains