Principal eigenvalue, maximum principle and regularity: theory and applications

In the first part, we will show how to extend the notion of principal eigenvalue to Dirichlet problems for fullynonlinear uniformly elliptic equations using the maximum principle and some a priori regularity results, in the contest of viscosity solutions. We shall describe fundamental ideas scattering from the acclaimed work of Berestycki, Nirenberg and Varadhan and more recent results in the theory of viscosity solutions./ /In the second part we will dwell on the regularity results. Proving Holder regularity using typical viscosity technics and proving Holder regularity of the gradient a' la Caffarelli via the improvement of flatness lemma, for a class of degenerate elliptic equations.