  We study the narrow escape problem in the disk, which consists in identifying
the first exit time and first exit point distribution of a Brownian particle
from the ball in dimension 2, with reflecting boundary conditions except on
small disjoint windows through which it can escape. This problem is motivated
by practical questions arising in various scientific fields (in particular
cellular biology and molecular dynamics). We apply the quasi-stationary
distribution approach to metastability, which requires to study the eigenvalue
problem for the Laplacian operator with Dirichlet boundary conditions on the
small absorbing part of the boundary, and Neumann boundary conditions on the
remaining reflecting part. We obtain rigorous asymptotic estimates of the first
eigenvalue and of the normal derivative of the associated eigenfunction in the
limit of infinitely small exit regions, which yield asymptotic estimates of the
first exit time and first exit point distribution starting from the
quasi-stationary distribution within the disk.
