Communications in Mathematical Physics

, Volume 208, Issue 3, pp 761–770

Second Eigenvalue of Schrödinger Operators¶and Mean Curvature

  • Ahmad  El Soufi
  • Saïd Ilias

DOI: 10.1007/s002200050009

Cite this article as:
El Soufi, A. & Ilias, S. Comm Math Phys (2000) 208: 761. doi:10.1007/s002200050009

Abstract:

Let $M$ be a compact immersed submanifold of the Euclidean space, the hyperbolic space or the standard sphere. For any continuous potential q on M, we give a sharp upper bound for the second eigenvalue of the operator −Δ+q in terms of the total mean curvature of M and the mean value of q. Moreover, we analyze the case where this bound is achieved. As a consequence of this result we obtain an alternative proof for the Alikakos–Fusco conjecture concerning the stability of the interface in the Allen–Cahn reaction diffusion model.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Ahmad  El Soufi
    • 1
  • Saïd Ilias
    • 1
  1. 1.Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France. E-mail: elsoufi@univ-tours.fr; ilias@univ-tours.frFR