Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem |
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Authors: | Asma Hassannezhad |
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Institution: | Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran |
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Abstract: | 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. |
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Keywords: | Eigenvalue Upper bound Laplacian Steklov problem Min-conformal volume |
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