Elliptic unique continuation below the Lipschitz threshold

In this article, we investigate unique continuation principles for solutions $u$ of uniformly elliptic equations of the form $-\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }u)=0$ 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 ${\int }_0^1\omega (t{)}^{-1}dt=\mathrm{\infty }$, 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.