On a counterexample concerning unique continuation for elliptic equations in divergence form

We construct a second order elliptic equation in divergence form in $\mathrm R^3$, with a non-zero solution which vanishes in a half-space. The coefficients are $\alpha$-Hölder continuous of any order $\alpha<1$. This improves a previous counterexample of Miller [1,2] Moreover, we obtain coefficients which belong to a finer class of smoothness, expressed in terms of the modulus of continuity.