Boundary regularity of $k$-Hessian eigenfunctions

Abstract

We prove smoothness up to the boundary for $k$-Hessian eigenfunctions on smooth bounded uniformly convex domains when $2\le k<n$. This resolves an open problem in the regularity theory of degenerate fully nonlinear elliptic equations and extends the boundary smoothness theorem for Monge--Ampère eigenfunctions. In contrast to the Monge--Ampère case, the linearized $k$-Hessian operator remains uniformly elliptic up to the boundary.

arXiv comment32 pages