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Pseudospectra of semiclassical boundary value problems

Published online by Cambridge University Press:  14 March 2014

Jeffrey Galkowski
Jeffrey Galkowski*
Affiliation:
Mathematics Department, University of California, Berkeley, CA 94720, USA([email protected])
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Abstract

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 Péclet 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 $\mathbb{R}^d$. 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 with these elliptic operators.

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 2 , April 2015 , pp. 405 - 449
Copyright
© Cambridge University Press 2014 

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