Spectral theory of partial differential equations
These lectures present highlights of spectral theory for selfadjoint partial differential operators, emphasizing problems with discrete spectrum. Spectral methods permeate the theory of partial differential equations. Linear PDEs are often solved by separation of variables, getting eigenvalues when...
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| Published in: | Spectral Theory and Applications Vol. 720; pp. 23 - 55 |
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| Main Author: | |
| Format: | Book Chapter |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
Centre de Recherches Mathematiques |
| Series: | Contemporary Mathematics |
| ISBN: | 147043556X, 9781470435561 |
| ISSN: | 0271-4132, 1098-3627 |
| Online Access: | Get full text |
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| Summary: | These lectures present highlights of spectral theory for selfadjoint partial differential operators, emphasizing problems
with discrete spectrum. Spectral methods permeate the theory of partial differential equations. Linear PDEs are often solved by
separation of variables, getting eigenvalues when the spectrum is discrete and continuous spectrum when it is not. Further,
linearized stability of a steady state or traveling wave of a nonlinear PDE depends on the sign of the first eigenvalue, or more
generally on the location of the spectrum in the complex plane.
We define eigenvalues in terms of quadratic forms on a
general Hilbert space. Particular applications include the eigenvalues of the Laplacian under Dirichlet and Neumann boundary
conditions. Rayleigh-type principles characterize the first and higher eigenvalues, and lead to a number of comparison and
domain monotonicity properties. Lastly, the role of eigenvalues in stability analysis is investigated for a reaction-diffusion
equation in one spatial dimension.
Computable examples are presented before the general theory. Some ideas are used
before being properly defined, but overall students gain a better understanding of the purpose of the theory by gaining first a
solid grounding in specific examples. |
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| ISBN: | 147043556X 9781470435561 |
| ISSN: | 0271-4132 1098-3627 |
| DOI: | 10.1090/conm/720/14521 |