Year: 2025
Type: preprint
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 $Γ$-admissible eigenfunction on the smooth, strictly $Γ$-convex domain $Ω\subset \mathbb{R}^{n}$. Furthermore, we prove that the eigenfunction $u_{1}$ belongs to $C^{\infty}(Ω) \cap C^{1,1}(\overlineΩ)$. As an application, we prove that every invariant Gårding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established.
Source: arXiv (Cornell University)
Author: Zhang, Jiaogen
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7093080985
Cites:
Cited by:
Related to: