Well-Posedness for Constrained Hamilton-Jacobi Equations

The goal of this paper is to study a Hamilton-Jacobi equation{ut=H(Du)+R(x,I(t))in Rn×(0,∞),supRnu(⋅,t)=0on [0,∞), with initial conditions I(0) = I> 0 , u(x, 0) = u(x) on Rⁿ. 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.