It is well-known that the bounded solution of the heat equation for any continuous initial condition becomes Lipschitz continuous as soon as time is positive. We aim at extending this phenomenon to nonlinear parabolic equations. In literature, this was done requiring quite restrictive structures in first order term. Here, we relax these limitations and we… Read More »
Hussien Ali Hussien Abugirda abstract
I will talk about a class of quasilinear elliptic systems of PDEs consisting of $N$ Hamilton-Jacobi-Bellman equations coupled with $N$ Fokker-Planck equations, generalising to $N>1$ populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. I will describe a wide range of sufficient conditions for the existence of solutions to these systems: … Read More »
A variational problem in L∞ involving the Laplacian
The Infinity-Laplacian is a 2nd order nonlinear PDE system with discontinuous coefficients which arises in vectorial Calculus of Variations in L-Infinity when minimising the sup-norm of the gradient over a class of Lipschitz maps. In this talk I will discuss an existence result of appropriately defined “weak” solutions to the Dirichlet problem for a generalised Infinity-Laplacian… Read More »
I will describe a method which allows to compare solutions of different elliptic and parabolic equations in different domains. As a by-product, it is possible to define a new type of rearrangement which applies to equations not necessarily in divergence form, leading to Talenti type results. In the case of variational problems, the method applies… Read More »
Recent work on a description of zero-range-processes as entropy-driven gradient-flows is presented. Special emphasis is layed on how this can be exploited for numerically finding the corresponding thermodynamic metric of the process.