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It is well known that the Neuman eigenvalue problem has discrete spectrum and the eigen values are

nonnegative and can be arranged in a nondecreasing order of magnitude.

Do we need any smoothness condition on the boundary? Is it true for more a general Elliptic operator?

I have hard time to find a solid reference. Can anyone suggest? Thanks!

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4 Answers 4

The essential question is whether the embedding from $H^1$ to $L^2$ is compact. Without some boundary smoothness, little seems to be known.

The following reference should be of interest: http://www.math.ksu.edu/~ramm/papers/477.pdf

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There is a detailed exposition in the book by

S. G. Mikhlin, Mathematical physics, an advanced course. North-Holland, Amsterdam, 1970.

Mikhlin considers a general divergent second order elliptic operator in a domain with a piecewise smooth boundary.

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Of course, on a general domain, the question os how do you define the Neuman Laplacian. There is an excellent exposition in

W. Arendt, A.F.M. ter Elst: Sectorial forms and degenerate differential operators

suggesting methods how to do it.

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imho, the discretness of the spectra of an operator follows from the compactness of its resolvent (see the Rellich theorem and conditions on the manifold that ensure the compactness of the resolvent).

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