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

TL;DR

This paper demonstrates that solutions to certain $k$-Hessian equations in exterior domains can lack quasiconvexity, contrasting with harmonic functions which are quasiconvex under similar conditions.

Contribution

It provides counterexamples for $k$-Hessian solutions' quasiconvexity and offers a new proof for harmonic functions' quasiconvexity in exterior domains.

Findings

Solutions to $k$-Hessian equations may not be quasiconvex in exterior domains.

Harmonic functions in exterior domains are quasiconvex when decaying at infinity.

Counterexamples highlight differences between $k$-Hessian and harmonic solutions.

Abstract

In this paper, we provide examples to show that for $1 \leq k \leq n/2$, solutions to $k$-Hessian equations $S_k(D^2u)=1$ 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.