Lax Formula for Obstacle Problems
Keywords:
Lax formula, Hopf formula, optimal control, obstacle problem.Abstract
The first order obstacle problem min{ut + H(Du), g(x) − u} = 0, u(T,x) = g(x) has a Hopf formula in the case when g is convex. It was first derived by A.Subbotin [11]. The case when u(t, x) = sup y∈Rn = sup y∈Rn inf t≤τ≤T g(y)−(τ −t)H∗ y−x τ −t inf t≤τ≤T g(x+y(τ −t))−(τ −t)H∗(y) . g is continuous but the Hamiltonian H is convex is considered here. The corresponding Lax formula is derived to be 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∞.
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