, Volume 208, Issue 3, pp 761-770

Second Eigenvalue of Schrödinger Operators¶and Mean Curvature

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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.

Received: 18 June 1999 / Accepted: 6 July 1999