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 $\omega$ 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. In the class of isotropic equations (i.e., $A(x) = a(x)I$ for some scalar function $a$) we show that Holder continuity of $a$ of the order $\alpha \in (2/3,1)$ is sufficient to guarantee strong unique continuation. This latter result contrasts counterexamples known for anisotropic equations, and disproves a conjecture of Miller from 1974.