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.
Cole Jeznach (Thu,) studied this question.