The exterior Dirichlet problem for Hessian equations admits a unique smooth k-convex solution when the obstacle is star-shaped and strictly (k-1)-convex, not necessarily convex, for 2 <= k <= n.
Y .An extension to a theorem of J¨orgens, Calabi, and Pogorelov.Comm
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Exterior Dirichlet problem for Hessian equations on a non-convex ring
The exterior Dirichlet problem for Hessian equations admits a unique smooth k-convex solution when the obstacle is star-shaped and strictly (k-1)-convex, not necessarily convex, for 2 <= k <= n.