Asymptotics for the $k$-Hessian Eigenvalue on the Unit Ball

Abstract: We compute the limit of the eigenvalue of the $k$-Hessian operator on the unit ball in $\mathbb{R}<sup>n$ when $k\rightarrow \infty$, assuming that the ratio $\frac{n}{k}$ remains fixed. When $n&lt; 2ke$, we moreover identify the limit of the corresponding eigenfunctions. We also derive monotonicity results and consider when the ratio varies in $n$.