Abstract

In the classical Water Wave Problem, we construct new overhanging solitary waves by a procedure resembling desingularization of the gluing of constant mean curvature surfaces by tiny catenoid necks. The solutions here predicted have long been numerically detected. This is joint work with Juan Davila, Monica Musso, and Miles Wheeler.

Abstract

We first characterize concavity properties preserved by the Dirichlet heat flow in a convex domain of the Euclidean space. (This is a joint work with Paolo Salani and Asuka Takatsu.) Next, we show that no concavity properties are preserved  by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain. (This is a joint work with Asuka Takatsu and Haruto Tokunaga.) 

Abstract

I will explain different approaches to get Hardy inequalities and I will give some applications of such Hardy inequalities.

Abstract

We review the theory of nonlinear Dirichlet forms and nonlinear Markovian semigroups as well as the recent development of an associated nonlinear potential theory. Eventually, we emphasize the emergence of functional analytic aspects.