MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

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:

share|cite|improve this answer

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.

share|cite|improve this answer

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.

share|cite|improve this answer

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).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.