  We consider operators $-\Delta + X$ where $X$ is a constant vector field, in
a bounded domain and show spectral instability when the domain is expanded by
scaling. More generally, we consider semiclassical elliptic boundary value
problems which exhibit spectral instability for small values of the
semiclassical parameter h, which should be thought of as the reciprocal of the
Peclet constant. This instability is due to the presence of the boundary: just
as in the case of $-\Delta + X$, some of our operators are normal when
considered on R^d. We characterize the semiclassical pseudospectrum of such
problems as well as the areas of concentration of quasimodes. As an
application, we prove a result about exit times for diffusion processes in
bounded domains. We also demonstrate instability for a class of spectrally
stable nonlinear evolution problems that are associated to these elliptic
operators.

Analysis of PDEs