Non-convexity of level sets for solutions to $k$-Hessian equations in exterior domains

Abstract

In this paper, we provide examples to show that for $1\le k\le n/2$, solutions to $k$-Hessian equations $S_k(D^{2}u)=1$ in the exterior of a strictly convex domain need not be quasiconvex, when prescribing quadratic growth at infinity. Additionally, we give a new proof for the quasiconvexity of harmonic functions in such exterior domains that decay to zero at infinity.