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 -div(A ∇ 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 ∫₀¹ ω(t)⁻¹dt = ∞, 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 α∈ (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.