Son Thai Tu ,Michigan State University

https://meeting.tencent.com/dm/QSInynLhDG19

Viewingtheadditiveeigenvaluesasamapwithrespecttodomainperturbationbyscal- ing, we show that this map enjoys some regularity. Precisely, let c(λ) be the additive eigenvalue with respect to (1 + r(λ))Ω, we show that c(λ) is differentiable except at most a countable set, while one-sided derivatives exist everywhere. The regularity can be improved given more assumptions on the source of the equation. We show that c′(0) exists if and only if an invariant on viscosity Mather measures holds. Furthermore, these properties are connected to the convergence of a vanishing discount problem with respect to changing domains and also the parametrization of solutions to the corresponding ergodic problem. This is an analog of the problem of studying the principal eigenvalue of the elliptic problem with respect to domain perturbation in the linear case. This connection is in fact a one-to-one correspondence for the quadratic case via the Hopf-Cole transform. This is a joint-work with Farid Bozorgnia and Dohyun Kwon.