La settimana 13-17 Maggio si terranno tre seminari nell'ambito del Seminario di Matematica Applicata (IMATI-CNR e Dipartimento di Matematica, Pavia), http://matematica.unipv.it/it/seminari-matematica-applicata
Martedì 14 Maggio, ore 15 precise, Sala conferenze IMATI-CNR, Pavia, Juergen Sprekels (Humboldt University and WIAS, Berlin, Germany): Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources
Martedì 14 Maggio, ore 16 precise, Sala conferenze IMATI-CNR, Pavia, Leonid Berlyand (Penn State University): Mathematics of Active Gels: Stability & Traveling Waves
Mercoledì 15 Maggio, ore 15 precise, Aula Beltrami, Dipartimento di Matematica, Gian Paolo Leonardi (Università di Trento): Approximate curvatures of a varifold
Abstract Juergen Sprekels: The talk is concerned with an optimal control problem for a two-dimensional Cahn-Hilliard-Darcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality.
Abstract Leonid Berlyand: In this review talk we demonstrate that mathematical analysis can be used in the study of active gels. We first observe that out-of-equilibrium state of active matter such (e.g., active gels) leads to mathematical challenges and requires developments of novel mathematical tools. Next we discuss three models of active gels that capture key biophysical features (such as persistent & turning motions and symmetry breaking), while having a minimal set of parameters and variables. Our goal is to provide theoretical understanding of cell polarity phenomenon via mathematical analysis of stability/instability and bifurcation from steady states to traveling waves. This is done by identification of key mathematical structures behind the models such as gradient coupling in Phase-Field model, Liouville equation, Keller-Segel cross-diffusion, and nonlinearity due to free boundary conditions, e.g. Hele-Show type. We employ mathematical techniques of (i) sharp interface limit via asymptotic analysis, (ii) construction of steady states and traveling waves via Crandall-Rabinowitz bifurcation theory and (iii) topological methods such as Lerey-Schauder degree theory.
These are joint works with V. Rybalko (ILTPE, Kharkiv, Ukraine), J. Fuhrman (PSU & Mainz, Germany), M. Potomkin (PSU, USA).
Abstract Gian Paolo Leonardi: Varifolds, i.e. Radon measures on the Grassmannian bundle of (unoriented) d-tangent planes of a Riemannian n-manifold M, represent a variational generalization of d-dimensional submanifolds of M. By suitably revisiting Hutchinson's definition of generalized second fundamental form, we propose a notion of approximate second fundamental form that is well-defined for general varifolds. Rectifiability, compactness, and convergence results are proved, showing in particular the consistency and stability of approximate curvatures with respect to varifold convergence. If restricted to the case of "discrete varifolds", this theory provides a general framework for extracting geometric features from discrete datasets. Some numerical tests on point clouds, also showing the robustness with respect to noise, will be shown. This is a joint research with Blanche Buet (Univ. Paris XI - Orsay) and Simon Masnou (Univ. Lyon 1).