The area of generalised solutions for nonlinear Partial Differential Equations (PDEs) is a very active research area with challenging open problems from both the purely mathematical point of view as well as from the viewpoint of applications. Nonlinear PDEs are one of the main mathematical tools that model phenomena from physics, engineering, chemistry, biology, finance and even medical science (e.g. the recent visual cortex models). Many of the PDEs arising from these models have a nonlinear structure, and often do not admit classical solutions (i.e., continuously differentiable solutions). For instance, even the simple nonlinear equation |u’|=1 in one space dimension (known as the Eikonal equation) with vanishing Dirichlet boundary conditions is not well posed in the class of smooth solutions. This a model of particular importance in several applied sciences, including e.g. optics. It turns out that this problem is well-posed only in the so-called viscosity sense (a typical notion of generalised solutions for elliptic/parabolic problems). Such phenomena underline the importance of pure mathematical analysis as a method of study in this area. However, there is no unified approach: different types of equations require different notions of generalised solutions and different methods with one common underlying challenge, namely to deal with functions of (very) low regularity as solutions of PDEs.
The Twelfth meeting is on 9th of November 2018. Please see the Meetings pages for more information.