Exterior Dirichlet problem for Hessian equations on a non-convex ring

In this paper, we prove the existence of a solution for the exterior Dirichlet problem for Hessian equations on a non-convex ring. Moreover, the solution we obtained is smooth. This extends the result of [Bao-Li-Li, ``On the exterior Dirichlet problem for Hessian equations’’ Trans. Amer. Math. Soc.366(2014)].