Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

A SHARP LOWER BOUND OF THE FIRST NEUMANN EIGENVALUE OF A COMPACT HYPERSURFACE INSIDE A CONVEX SET

  • Seo, Keomkyo (Department of Mathematics Sookmyung Women's University)
  • Received : 2010.05.12
  • Accepted : 2010.08.12
  • Published : 2010.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we provide a sharp lower bound of the first Neumann eigenvalue of a compact hypersurface $\Sigma$ inside a convex set C in a Riemannian manifold under the assumption that ${\partial}{\Sigma}$ meets ${\partial}C$ orthogonally.

Keywords

References

  1. R. Chen, Neumann eigenvalue estimate on a compact Riemannian manifold, Proc. Amer. Math. Soc. 108 (1990), no. 4, 961-970. https://doi.org/10.1090/S0002-9939-1990-0993745-X
  2. H. I. Choi and A.N. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom. 18 (1983), no. 3, 559-562. https://doi.org/10.4310/jdg/1214437788
  3. J. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math. 43 (1990), no. 7, 857-883. https://doi.org/10.1002/cpa.3160430703
  4. P. Li and S.-T. Yau, Estimates of eigenvalues of a compact Riemannian man- ifold, Proc. Sympos. Pure Math., 36, Amer. Math. Soc., Providence, R.I., (1980), 205-239.
  5. R. Reilly, Applications of the Hessian operator in a Riemannian manifold, In- diana Univ. Math. J. 26 (1977), no. 3, 459-472. https://doi.org/10.1512/iumj.1977.26.26036