A Fast Observability for Diffusion Equations in RN

Given an equidistributed set in the whole Euclidean space, we have established in [1] that there exists a constant positive C such that the observability inequality of diffusion equations holds for all T∈]0,1[, with an observability cost being of the form CeC/T. In this paper, for any small constant ε>0, we prove that there exists a nontrivial equidistributed set (in the sense that whose complementary set is unbounded), so that the above observability cost can be improved to a fast form of Ceε/T for certain constant $C>0$. The proof is based on the strategy used in [1], as well as an interpolation inequality for gradients of solutions to elliptic equations obtained recently in [2].