Geodesic
Hopf formulas for nonlinear obstacle problems
Minimax theory and its applications, Tome 5 (2020) no. 2
Cet article a éte moissonné depuis la source Minimax Theory and its Applications website

Voir la notice de l'article

A Hopf formula is derived for max{ut+H(Du), h(t, x)−u} = 0, u(T, x) = g(x) ≥ h(t, x), where g is assumed convex and x 7→ h(t, x) is also convex. This generalizes a formula without time dependent obstacle due to Subbotin. A Hopf formula for a concave obstacle is also derived. In addition, the Hopf formula for the obstacle problem with quasiconvex g is established. Next we consider the double obstacle problem. Assume the two obstacles g1(x) ≤ g2(x) are given functions, both convex or both concave. The nonlinear double obstacle variational inequality max{min{ut +H(Du), g2 −u}, g1 −u} = 0 on (−∞, T)×R n, with terminal data either g2 in the convex case and g1 in the concave case has a viscosity solution given by a Hopf type formula.These formulas are derived by using differential games with stopping times.
Mots-clés : Differential games, stopping times, Hopf formula, double obstacle 
@article{MTA_2020_5_2_a0,
     author = {Emmanuel N. Barron and Robert Jensen},
     title = {Hopf formulas for nonlinear obstacle problems},
     journal = {Minimax theory and its applications},
     year = {2020},
     volume = {5},
     number = {2},
     zbl = {1451.49039},
     url = {http://geodesic.mathdoc.fr/item/MTA_2020_5_2_a0/}
}
TY  - JOUR
AU  - Emmanuel N. Barron
AU  - Robert Jensen
TI  - Hopf formulas for nonlinear obstacle problems
JO  - Minimax theory and its applications
PY  - 2020
VL  - 5
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MTA_2020_5_2_a0/
ID  - MTA_2020_5_2_a0
ER  - 
%0 Journal Article
%A Emmanuel N. Barron
%A Robert Jensen
%T Hopf formulas for nonlinear obstacle problems
%J Minimax theory and its applications
%D 2020
%V 5
%N 2
%U http://geodesic.mathdoc.fr/item/MTA_2020_5_2_a0/
%F MTA_2020_5_2_a0
Emmanuel N. Barron; Robert Jensen. Hopf formulas for nonlinear obstacle problems. Minimax theory and its applications, Tome 5 (2020) no. 2. http://geodesic.mathdoc.fr/item/MTA_2020_5_2_a0/