The observability inequalities for heat equations with potentials

This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschitz potentials. The observability inequality for heat equations asserts that the total energy of a solution is bounded above by the energy localized in a subdomain with an observability constant. For a bounded measurable potential V = V (x, t), the factor in the observability constant arising from the Carleman estimate is best known to be exp(C|| V ||∞2/3) (even for time-independent potentials). In this paper, we show that, for Lipschtiz potentials, this factor can be replaced by exp(C(|| ∇V ||∞1/2 + || ∂tV ||∞1/3)), which improves the previous bound exp(C|| V ||∞2/3) in some typical scenarios. As a consequence, with such a Lipschitz potential, we obtain a quantitative regular control in a null controllability problem. In addition, for the one-dimensional heat equation with some time-independent bounded measurable potential V = V (x), we obtain the observability inequality with optimal constant on arbitrary measurable subsets of positive measure both in space and time.