Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Minimax Theory and its Applications 06 (2021), No. 1, 145--154Copyright Heldermann Verlag 2021 On the Uniqueness of Solutions to One-Dimensional Constrained Hamilton-Jacobi Equations Yeoneung Kim Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A. yeonkim@math.wisc.edu [Abstract-pdf] \def\R{\mathbb{R}} The goal of this paper is to study the uniqueness of solutions to a constrained Hamilton-Jacobi 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 non-separable reaction $R(x,I(t))$ when the space is one-dimensional. Keywords: Hamilton-Jacobi equation with constraint, selection-mutation model. MSC: 35A02, 35F21, 35Q92. [ Fulltext-pdf  (112  KB)] for subscribers only.