A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach

TL;DR

This paper establishes the uniqueness and classical nature of solutions for a class of constrained Hamilton-Jacobi equations, using a constructive approach based on the dynamic programming principle and an ODE for the solution's maximum.

Contribution

It introduces a novel method to prove uniqueness and construct solutions for Hamilton-Jacobi equations with a maximum constraint, linking PDE analysis with ODE techniques.

Findings

Proved the full problem has a unique classical viscosity solution.

Demonstrated the solution's maximum evolves according to an ODE.

Applied the approach to a selection-mutation model with concentration phenomena.

Abstract

We discuss a class of time-dependent Hamilton-Jacobi equations, where an unknown function of time is intended to keep the maximum of the solution to the constant value 0. Our main result is that the full problem has a unique viscosity solution, which is in fact classical. The motivation is a selection-mutation model which, in the limit of small diffusion, exhibits concentration on the zero level set of the solution of the Hamilton-Jacobi equation. Uniqueness is obtained by noticing that, as a consequence of the dynamic programming principle, the solution of the Hamilton-Jacobi equation is classical. It is then possible to write an ODE for the maximum of the solution, and treat the full problem as a nonstandard Cauchy problem.