  The narrow escape problem consists of deriving the asymptotic expansion of
the solution of a drift-diffusion equation with the Dirichlet boundary
condition on a small absorbing part of the boundary and the Neumann boundary
condition on the remaining reflecting boundaries. Using layer potential
techniques, we rigorously find high-order asymptotic expansions of such
solutions. We explicitly show the nonlinear interaction of many small absorbing
targets. Based on the asymptotic theory for eigenvalue problems developed in
\cite{book}, we also construct high-order asymptotic formulas for eigenvalues
of the Laplace and the drifted Laplace operators for mixed boundary conditions
on large and small pieces of the boundary.
