Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem

Hassannezhad, A ; Sharif University of Technology | 2011

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  1. Type of Document: Article
  2. DOI: 10.1016/j.jfa.2011.08.003
  3. Publisher: 2011
  4. Abstract:
  5. In this paper, we find upper bounds for the eigenvalues of the Laplacian in the conformal class of a compact Riemannian manifold (M,g). These upper bounds depend only on the dimension and a conformal invariant that we call "min-conformal volume". Asymptotically, these bounds are consistent with the Weyl law and improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is interesting for itself and powerful. As a further application of the method we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C1 boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the conformal invariant that we introduce
  6. Keywords:
  7. Eigenvalue ; Laplacian ; Min-conformal volume ; Steklov problem ; Upper bound
  8. Source: Journal of Functional Analysis ; Volume 261, Issue 12 , 2011 , Pages 3419-3436 ; 00221236 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0022123611002928