We review some results about the first eigenvalue of the infinity Laplacian operator and its first eigenfunctions in a general norm context. Those results are obtained in collaboration with several authors: V. Ferone, P. Juutinen and B. Kawohl (see [BFK], [BK1], [BJK] and [BK2]). In section 5 we make some remarks on the simplicity of the first eigenvalue of ∆∞: this will be the object of a join...

Let B be a Borel subset of R with finite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting B. Let A1 be the time spent by Brownian motion in a closed cone with vertex 0 until time one. We show that limu→0 logP (A1 < u)/ log u = 1/ξ where ξ is defined in terms of the first eigenvalue of the Laplacian in a compact domain. Eigenvalues of the ...

We study the global structure of the set of radial solutions of a nonlinear Dirichlet eigenvalue problem involving the p-Laplacian with p > 2, in the unit ball of RN , N > 1. We show that all non-trivial radial solutions lie on smooth curves of respectively positive and negative solutions, bifurcating from the first eigenvalue of a weighted p-linear problem. Our approach involves a local bifurc...

Nested split and double nested graphs (commonly named nested graphs) are considered. General statements regarding the signless Laplacian spectra are proven, and the nested graphs whose second largest signless Laplacian eigenvalue is bounded by a fixed integral constant are studied. Some sufficient conditions are provided and a procedure for classifying such graphs in particular cases is provide...