  The stochastic motion of particles in living cells is often spatially
inhomogeneous with a higher effective diffusivity in a region close to the cell
boundary due to active transport along actin filaments. As a first step to
understand the consequence of the existence of two compartments with different
diffusion constant for stochastic search problems we consider here a Brownian
particle in a circular domain with different diffusion constants in the inner
and the outer shell. We focus on the narrow escape problem and compute the mean
first passage time (MFPT) for Brownian particles starting at some pre-defined
position to find a small region on the outer reflecting boundary. For the
annulus geometry we find that the MFPT can be minimized for a specific value of
the width of the outer shell. In contrast for the two-shell geometry we show
that the MFPT depends monotonously on all model parameters, in particular on
the outer shell width. Moreover we find that the distance between the starting
point and the narrow escape region which maximizes the MFPT depends
discontinuously on the ratio between inner and outer diffusivity.
