In this talk we propose a model for the simulation of retinal prostheses based on the use  of organic polymer nanoparticles (NP). The model consists of a nonlinearly coupled system of elliptic partial differential equations accounting for: (1) light photoconversion into free charged carriers in the NP bulk; (2) charge transport in the NP bulk…

We improve Cheeger's lower bound for the first nonzero eigenvalue  of the Laplacian on compact Riemannian manifolds with Ricci curvature bounded from below. Zoom link: https://ncsu.zoom.us/j/8027642791?pwd=d1lNaWZyUW4zeUFvaTA5VmlsTWtjdz09

One of the major open problems in homogenization of Hamilton-Jacobi equations is to under deep properties of the effective Hamiltonian.  In this talk,  I will present some recent progress. In particular, consider the effective Hamiltonian associated with the mechanical Hamiltonian H(p,x)=(|p|^2)/2+V(x). We can show that for generic V, the effective Hamiltonian is piecewise 1d in…

On a smooth bounded Euclidean domain,  Sobolev-subcritical fast diffusion with vanishing boundary trace leads to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error,  showing the rate is either exponentially fast (with a rate constant predicted by…

In this talk, we first propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. To this end we introduce a dynamical system for which we prove that its trajectories asymptotically converge…