Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation

TL;DR

This paper investigates the loss of boundary conditions in the diffusive Hamilton-Jacobi equation after gradient blowup, revealing that the phenomenon depends heavily on initial data and can be precisely controlled.

Contribution

It demonstrates that the loss of boundary conditions can occur everywhere or not at all, depending on initial data, and shows this set can be prescribed arbitrarily close to any boundary subset.

Findings

Loss of boundary conditions depends on initial data.

Loss can occur everywhere or not at all.

The set of boundary points losing conditions can be prescribed.

Abstract

We consider the diffusive Hamilton-Jacobi equation, with homogeneous Dirichlet conditions and regular initial data. It is known from [Barles-DaLio, 2004] that the problem admits a unique, continuous, global viscosity solution, which extends the classical solution in case gradient blowup occurs. We study the question of the possible loss of boundary conditions after gradient blowup, which seems to have remained an open problem till now. Somewhat surprisingly, our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena. For any smooth bounded domain, we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary, as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup. Actually, we show that the latter possibility is rather exceptional. More generally, we show that the set of the points where boundary conditions are lost, can be prescribed to be arbitrarily close to any given open subset of the boundary.