Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The C^r dependence problem of eigenvalues of the Laplace operator on
  domains in the plane   The C^r dependence problem of multiple Dirichlet eigenvalues on domains is
discussed for elliptic operators by regarding smooth one-parameter families of
C^1 perturbations of domains in R^n. As applications of our main theorem
(Theorem 1), we provide a fairly complete description for all eigenvalues of
the Laplace operator on disks and squares and also for its second eigenvalue on
balls in R^n for any n >= 3. The central tool used in our proof is a degenerate
implicit function theorem on Banach spaces of independent interest.

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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys