A. Paredes : Periodic Homogenization of the Principal Eigenvalue of Second-Order Elliptic Operators

We investigate the convergence of the solution pair of the eigenvalue problem
\begin{equation*}
F(x, \nicefrac{x}{\epsilon}, u^\epsilon, Du^\epsilon, D^2 u^\epsilon) = -\lambda^\epsilon u^\epsilon \ \mbox{in} \ \Omega, \qquad u^\epsilon = 0 \ \mbox{on} \ \partial \Omega,
\end{equation*}
analogous to the principal eigenvalue and eigenfunction. Here $H$ is a convex, positive-homogeneous and uniformly elliptic Hamiltonian, typically arising in stochastic optimal control problems. Its generalized principal eigenvalue can be defined in the spirit of the celebrated work of Berestycki, Nirenberg and Varadhan \cite{BNV}. We prove a general convergence result for this eigenvalue and its corresponding eigenfunction, and a rate of convergence for the eigenvalue in the linear case.

Joint work with E.~Topp, Universidad de Santiago de Chile, and G.~Dávila, Universidad Técnica Federico Santa María.