Upper bounds for the eigenvalues of Hessian equations

We prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equationsComment: 15 pages, 1 figur

Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hölder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs

Korevaar\u2013Schoen\u2019s directional energy and Ambrosio\u2019s regular Lagrangian flows

We develop Korevaar\u2013Schoen\u2019s theory of directional energies for metric-valued Sobolev maps in the case of RCD source spaces; to do so we crucially rely on Ambrosio\u2019s concept of Regular Lagrangian Flow. Our review of Korevaar\u2013Schoen\u2019s spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of \u2018differential of a map along a vector field\u2019 and about the parallelogram identity for CAT(0) targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows