On quantitative uniqueness for elliptic equations

We address the question of quantitative uniqueness for the equation $$\Delta u=V u$$ with either periodic or Dirichlet boundary conditions in a disk. We construct solutions u corresponding to potential functions V such that u vanishes of order $$\mathrm{const} \Vert V\Vert _{L^\infty }^{2/3}$$ . The example also shows sharpness of recently obtained bounds in the case of a parabolic equation $$u_t-\Delta u= V u$$ .