Cristian Enache
American University of Sharjah, Sharjah, United Arab Emirates

In this talk we are going to first define the notion of a first eigenvalue for a class of Monge-Ampère type operators. To this end, we are going to exploit the property that an elliptic operator satisfies a maximum principle if a certain coefficient is less than the first eigenvalue of the operator. For instance, in the linear case, it is well known that for the operator $Lu=-\Delta u + \lambda u$ the maximum principle holds if $\lambda <\lambda _1$, where $\lambda _1$ is the first Dirichlet eigenvalue of the Laplacian. Therefore, $\lambda _1$ is the supremum of all $\lambda \in \mathbb{R}$ such that the maximum principle holds. In this talk we extend this idea to a general class of Monge-Ampère type operators. More precisely, under certain assumptions on the operator and the underlying domain $\Omega $, we show that some maximum principle hold, we establish the existence of a principal eigenvalue, as well as some Lipschitz and $\gamma $-Holder regularity results for the corresponding eigenfunction.