The Neumann eigenvalue problem for the ∞-Laplacian

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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.