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

Анотація

We construct a second order elliptic equation in divergence form in R³, 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.