Well-Posedness for Constrained Hamilton-Jacobi Equations

Abstract

The goal of this paper is to study a Hamilton-Jacobi equation

$$ \textstyle\begin{cases} u_{t}=H(Du)+R(x,I(t)) &\text{in }\mathbb{R}^{n} \times (0,\infty ), \\ \sup_{\mathbb{R}^{n}} u(\cdot ,t)=0 &\text{on }[0,\infty ), \end{cases} $$

with initial conditions $I(0)=I_{0}>0$, $u(x,0)=u_{0}(x)$ on $\mathbb{R}^{n}$. Here $(u,I)$ is a pair of unknowns and the Hamiltonian $H$ and the reaction term $R$ are given. Moreover, $I(t)$ is an unknown constraint (Lagrange multiplier) that constrains the supremum of $u$ to be always zero. We construct a solution in the viscosity setting using a fixed point argument when the reaction term $R(x,I)$ is strictly decreasing in $I$. We also discuss both uniqueness and nonuniqueness. For uniqueness, a certain structural assumption on $R(x,I)$ is needed. We also provide an example with infinitely many solutions when the reaction term is not strictly decreasing in $I$.