Quasi-convex Hamilton--Jacobi equations via limits of Finsler $p$-Laplace problems as $p\to \infty$

TL;DR

This paper demonstrates that solutions to certain quasi-convex Hamilton--Jacobi equations with boundary constraints can be obtained as limits of Finsler $p$-Laplace problems as $p$ approaches infinity, connecting PDEs and optimization.

Contribution

It introduces a novel limit approach for solving quasi-convex Hamilton--Jacobi equations using Finsler $p$-Laplace problems, extending existing techniques.

Findings

Maximal viscosity solutions obtained via $p oty$ limits.

Provides an optimal solution to a Finslerian Beckmann problem.

Recovers known results using the Evans--Gangbo technique.

Abstract

In this paper we show that the maximal viscosity solution of a class of quasi-convex Hamilton--Jacobi equations, coupled with inequality constraints on the boundary, can be recovered by taking the limit as $p\to\infty$ in a family of Finsler $p$-Laplace problems. The approach also enables us to provide an optimal solution to a Beckmann-type problem in general Finslerian setting and allows recovering a bench of known results based on the Evans--Gangbo technique.