We present a new approach to deriving lower bounds for weak spatial gradients of a viscosity solution to a Hamilton-Jacobi equation when the Hamiltonian is convex. For the proof, we consider Hamiltonian systems with approximate Hamiltonians and study how the initial gradients propagate along the solutions to the Hamiltonian systems. We also apply our method to the case of homogeneous Hamiltonians, whose examples include level-set equations arising in surface evolution problems. In this case we

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... btain sharper gradient estimates. Moreover, we compare our results with the previous work [22] . We show that our results give better and optimal estimates in several senses.