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Some geometric bounds on eigenvalues of elliptic PDEs
Evans Harrell
Georgia Techwww.math.gatech.edu/~harrell
Spectral Theory Network 25 July, 2004 Cardiff
• Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part, preprint, 2004.
• Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, preprint, 2003.
On a (hyper) surface,what object is most like
the Laplacian?
( = the good old flat scalar Laplacian of Laplace)
• Answer #1 (Beltrami’s answer): Consider only tangential variations.
The Laplace-Beltrami operator is an intrinsic object, and as such is unaware that the surface is immersed.
• Answer #2 (The nanoanswer):
Perform a singular limit and renormalization to attain the surface as the limit of a thin domain:
- + q,
Some other answers
• In other physical situations, such as reaction-diffusion, q(x) may be other quadratic expressions in the curvature, usually q(x) ≤ 0.
• The conformal answer: q(x) is a multiple of the scalar curvature.