Second Eigenvalue of Schrödinger Operators¶and Mean Curvature
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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.
KeywordsEuclidean Space Diffusion Model Hyperbolic Space Alternative Proof Reaction Diffusion
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