6533b824fe1ef96bd128151c

RESEARCH PRODUCT

Principal eigenvalue of a very badly degenerate operator and applications

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Abstract In this paper, we define and investigate the properties of the principal eigenvalue of the singular infinity Laplace operator Δ ∞ u = ( D 2 u D u | D u | ) ⋅ D u | D u | . 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 L 2 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.