Niveau: Supérieur, Doctorat, Bac+8
P -LAPLACE OPERATOR AND DIAMETER OF MANIFOLDS Jean-Franc¸ois GROSJEAN November 4, 2005 GROSJEAN Jean-Franc¸ois, Institut Elie Cartan (Mathematiques), Universite Henri Poincare Nancy I, B.P. 239, F-54506 VANDOEUVRE-LES-NANCY CEDEX, FRANCE. E-mail: Abstract Let (Mn, g) be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian ?1,p(M) and we prove that the limit of p √ ?1,p(M) when p ? ∞ is 2/d(M) where d(M) is the diameter of M . Moreover if (Mn, g) is an oriented compact hypersurface of the Euclidean space Rn+1 or Sn+1, we prove an upper bound of ?1,p(M) in term of the largest principal curvature ? over M . As applications of these results we obtain optimal lower bounds of d(M) in term of the curvature. In particular we prove that if M is a hypersurface of Rn+1 then : d(M) ≥ pi/?. Mathematics Subject Classification (2000): 53A07, 53C21. 1
- compact manifold
- eigenfunction associated
- without boundary
- associated respec- tively
- max x??
- boundary
- ∞?∞ ≤
- ∂?
- compact riemannian
- lim infpk?∞