On the Hamilton-Jacobi Equation and Infimal Convolution in the Framework of Sobolev-functions

TL;DR

This paper investigates the regularity of solutions to the Hamilton-Jacobi equation and infimal convolution when initial data is in Sobolev spaces, showing convergence of derivatives in L^p and demonstrating optimality of results.

Contribution

It establishes new regularity results for Hamilton-Jacobi flows with Sobolev initial data and constructs examples confirming the optimality of these results.

Findings

Solutions' derivatives converge in L^p as time progresses.

Constructed examples demonstrate the optimality of the regularity results.

The study extends understanding of Hamilton-Jacobi equations in Sobolev spaces.

Abstract

We study the regularity properties of the Hamilton-Jacobi flow equation and infimal convolution in the case where initial datum function is continuous and lies in given Sobolev-space $W^{1,p}(\rn)$. We prove that under suitable assumptions it holds for solutions $w(x,t)$ that $D_xw(\cdot,t)\to Du(\cdot)$ in $L^p(\rn)$. Moreover, we construct examples showing that our results are essentially optimal.