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

2024/07/05 by Kazuya Hirose, Hirose, Kazuya
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2407.04288

openalex publication_date 2024/07/05 · openalex created_date 2024/07/09 · openalex updated_date 2026/07/28

Abstract

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.

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