Principal eigenvalue of a very badly degenerate operator and applications

In this paper, we define and investigate the properties of the principal eigenvalue of the singular infinity Laplace operatorΔ∞u=(D2uDu|Du|)⋅Du|Du|. This operator arises from the optimal Lipschitz extension problem and it plays the same fundamental role in the calculus of variations of L∞ functionals as the usual Laplacian does in the calculus of variations of L2 functionals. Our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of Berestycki, Nirenberg and Varadhan [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1) (1994) 47-92] in the case of uniformly elliptic linear operators. As an application, we obtain existence and uniqueness results for certain related non-homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity Laplacian.