
Minimax Theory and its Applications 04 (2019), No. 2, 341354 Copyright Heldermann Verlag 2019 Lax Formula for Obstacle Problems Emmanuel N. Barron Department of Mathematics and Statistics, Loyola University, Chicago, IL 60660, U.S.A. ebarron@luc.edu [Abstractpdf] The first order obstacle problem $\min\{u_t+H(Du)$, $g(x)u\}=0$, $u(T,x)=g(x)$ has a Hopf formula in the case when \textit{$g$ is convex}. It was first derived by A.\,I.\,Subbotin [{\it Generalized Solution of FirstOrder PDEs}, Birkh\"auser, Boston (1995)]. The case when $g$ is continuous but the \textit{Hamiltonian $H$ is convex} is considered here. The corresponding Lax formula is derived to be \begin{align*} u(t,x) & = \sup_{y \in \mathbb{R}^n}\ \inf_{t \leq \tau \leq T} \Big\{ g(y)(\tau t)H^*\Big(\frac{yx}{\tau t}\Big) \Big\} \\ & = \sup_{y \in \mathbb{R}^n}\ \inf_{t \leq \tau \leq T} \Big\{ g(x+y(\tau t))(\tau t)H^*(y) \Big\}. \end{align*} This formula is shown to provide a viscosity solution of the obstacle problem. The argument to derive and prove this is based on optimal control in L$^\infty$. Keywords: Lax formula, Hopf formula, optimal control, obstacle problem. MSC: 35C05, 49L20, 49L25 [ Fulltextpdf (120 KB)] for subscribers only. 