Hamilton–Jacobi equations have attracted growing attention because of their relevance to both theory and practice. We prove a new lower bound that matches the known upper bound up to constants, and we show that the result is stable under a range of perturbations of the assumptions. We prove sharp two-sided estimates that are uniform in the dimension, and we show that the result is stable under a range of perturbations of the assumptions. Together, these results contribute to a more complete picture and open the way to further applications.