Geometric and viscosity solutions for the Cauchy Problem of first Order

TL;DR

This paper explores the relationship between viscosity and geometric minimax solutions for the first-order Cauchy problem, demonstrating how iterative minimax procedures converge to viscosity solutions within a contact geometric framework.

Contribution

It establishes a connection between geometric minimax and viscosity solutions by showing iterative minimax procedures approximate viscosity solutions in a contact setting.

Findings

Iterating minimax solutions over shorter time intervals converges to viscosity solutions.

Extends Wei's symplectic results to the contact geometric framework.

Provides a new perspective on solution relationships for first-order PDEs.

Abstract

There are two kinds of solutions of the Cauchy problem of first order, the viscosity solution and the more geometric minimax solution and in general they are different. The aim of this article is to show how they are related: iterating the minimax procedure during shorter and shorter time intervals one approaches the viscosity solution. This can be considered as an extension to the contact framework of the result of Q.Wei in the symplectic case.