Geodesic
Lax formula for obstacle problems
Minimax theory and its applications, Tome 4 (2019) no. 2
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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∞.
Mots-clés : Lax formula, Hopf formula, optimal control, obstacle problem 
@article{MTA_2019_4_2_a6,
     author = {E.N.Barron},
     title = {Lax formula for obstacle problems},
     journal = {Minimax theory and its applications},
     year = {2019},
     volume = {4},
     number = {2},
     zbl = {1428.35069},
     url = {http://geodesic.mathdoc.fr/item/MTA_2019_4_2_a6/}
}
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E.N.Barron. Lax formula for obstacle problems. Minimax theory and its applications, Tome 4 (2019) no. 2. http://geodesic.mathdoc.fr/item/MTA_2019_4_2_a6/