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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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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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