The Dirichlet eigenvalue problems for some concave elliptic Hessian operators

Abstract

In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form \[ F(D^2u)=-Λu \quad \textrm{in} \, Ω, \qquad u=0 \quad \textrm{on} \, \partial Ω. \] These operators encompass the Monge-Ampère operator, the $k$-Hessian operators, and the $p$-Monge-Ampère operators. We impose a fairly mild constraint on the operator $F$, allowing us to demonstrate the existence of the first nonzero eigenvalue and its corresponding $\hat{I}“$-admissible eigenfunction on the smooth, strictly $\hat{I}“$-convex domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$. Furthermore, we prove that the eigenfunction $u_1$ belongs to $C^{\infty }(\hat{I}\text{c◯})\cap C^{1,1}(\overline{\hat{I}}\text{c◯})$. As an application, we prove that every invariant GÃ¥rding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established.

27 pages