Boundary regularity of $k$-Hessian eigenfunctions

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.