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<p>In this paper, we derive the lower bounds for the gradients of viscosity solutions to the Hamilton–Jacobi equation, where the convex Hamiltonian depends on the unknown function. We obtain gradient estimates using two different methods. First, we utilize the equivalence between viscosity solutions and Barron–Jensen solutions to study the properties of the inf-convolution. Second, we examine the Lie equation to understand how initial gradients propagate along its solutions.</p>

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Lower gradient estimates for viscosity solutions to first-order Hamilton–Jacobi equations depending on the unknown function

  • 期刊论文

  • 发表日期：

    2025年2月15日

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  • Kazuya Hirose

摘要

In this paper, we derive the lower bounds for the gradients of viscosity solutions to the Hamilton–Jacobi equation, where the convex Hamiltonian depends on the unknown function. We obtain gradient estimates using two different methods. First, we utilize the equivalence between viscosity solutions and Barron–Jensen solutions to study the properties of the inf-convolution. Second, we examine the Lie equation to understand how initial gradients propagate along its solutions.