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Minimax Theory and its Applications 04 (2019), No. 2, 341--354
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.


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 First-Order 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{y-x}{\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

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