# Spectral theory of partial differential equations

Research output: Chapter in Book/Report/Conference proceedingChapter

### Abstract

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.

Original language English (US) Contemporary Mathematics American Mathematical Society 23-55 33 720 https://doi.org/10.1090/conm/720/14521 Published - Jan 1 2018

### Fingerprint

Spectral Theory
Partial differential equation
Eigenvalue
Discrete Spectrum
Nonlinear PDE
Partial Differential Operators
Continuous Spectrum
First Eigenvalue
Separation of Variables
Spectral Methods
Neumann Boundary Conditions
Reaction-diffusion Equations
Rayleigh
Traveling Wave
Argand diagram
Dirichlet Boundary Conditions
Monotonicity
Stability Analysis
Hilbert space

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

Laugesen, R. S. (2018). Spectral theory of partial differential equations. In Contemporary Mathematics (Vol. 720, pp. 23-55). American Mathematical Society. https://doi.org/10.1090/conm/720/14521
Contemporary Mathematics. Vol. 720 American Mathematical Society, 2018. p. 23-55.

Research output: Chapter in Book/Report/Conference proceedingChapter

Laugesen, RS 2018, Spectral theory of partial differential equations. in Contemporary Mathematics. vol. 720, American Mathematical Society, pp. 23-55. https://doi.org/10.1090/conm/720/14521
Laugesen RS. Spectral theory of partial differential equations. In Contemporary Mathematics. Vol. 720. American Mathematical Society. 2018. p. 23-55 https://doi.org/10.1090/conm/720/14521
Laugesen, Richard S. / Spectral theory of partial differential equations. Contemporary Mathematics. Vol. 720 American Mathematical Society, 2018. pp. 23-55
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