The Neumann eigenvalue problem for the ∞-Laplacian

2014/05/14 by Luca Esposito, Bernd Kawohl, Esposito, L. +5
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1405.3535

Abstract

The first nontrivial eigenfunction of the Neumann eigenvalue problem for the p-Laplacian, suitable normalized, converges as p goes to ∞ to a viscosity solution of an eigenvalue problem for the ∞-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.

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