
Minimax Theory and its Applications 06 (2021), No. 1, 145154 Copyright Heldermann Verlag 2021 On the Uniqueness of Solutions to OneDimensional Constrained HamiltonJacobi Equations Yeoneung Kim Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A. yeonkim@math.wisc.edu [Abstractpdf] \def\R{\mathbb{R}} The goal of this paper is to study the uniqueness of solutions to a constrained HamiltonJacobi equation \begin{equation*} \begin{cases} u_t=u_x^2+R(x,I(t)) &\text{in }\R \times (0,\infty), \\ \max_{\R} u(\cdot,t)=0 &\text{on }[0,\infty), \end{cases} \end{equation*} with an initial condition $u(x,0)=u_0(x)$ on $\R$. A reaction term $R(x,I(t))$ is given while $I(t)$ is an unknown constraint (Lagrange multiplier) that forces maximum of $u$ to be always zero. In the paper, we prove uniqueness of a pair of unknowns $(u,I)$ using the dynamic programming principle for a particular class of nonseparable reaction $R(x,I(t))$ when the space is onedimensional. Keywords: HamiltonJacobi equation with constraint, selectionmutation model. MSC: 35A02, 35F21, 35Q92. [ Fulltextpdf (112 KB)] for subscribers only. 