Timeline for Eigenvalues of an elliptic operator on shrinking domains
Current License: CC BY-SA 4.0
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| Jan 15, 2022 at 16:26 | comment | added | username | For a general operator, you could have a problem if it is not self adjoint. For self adjoint operator, You have Weyl's Lemma which tells you that thanks to the min-max, there exists a positive constant $C$ such that for all $n$, the $n$-th eigenvalue of your operator is bounded above and below by the $n$-th eigenvalue of the laplacian: $$C\lambda_n(\Delta)\leq \lambda_n(P)\leq \frac1C \lambda_n(\Delta) $$ . That gives you the order of magntitude. Once you have this, you can freeze/make asymptotics. | |
| Jan 15, 2022 at 14:07 | history | asked | Ivan | CC BY-SA 4.0 |