Motivations for the eigenvalue problem of an elliptic operator

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I was looking at the chapter 6.5 of Evans’ book about the eigenvalue problem for (anti-)symmetric elliptic operators, and I was wondering what were the motivations for such a problem.

I guess there are some physical problems motivating it, like the time-independent Schrödinger equation, or also Laplace equation.

I’ve also been told that it is linked to Lagrange multipliers, in some way that I don’t know.

Any precision or documentation would be much appreciated.