Mathematical Analysis Seminars @ DISIM

Title: Non-Markovian heat flows and directed hypergraphs

Abstract: On (finite) graphs, the heat flow is Markovian: positive and contractive with respect to the $\infty$-norm. Many diffusion models, however, lose these properties. This talk proposes directed hypergraphs, where a hyperedge can have several entrance and exit vertices, as a prototypical case study of non-Markovian diffusion.

After recalling the Beurling–Deny theory of positive semigroups, I will address the topic of eventually positive flows, where positivity holds only after some transient: they can be illustrated by heat equations with nonlocal boundary conditions. I will then turn to heat flows on directed hypergraphs and give simple combinatorial criteria for positivity and $\infty$-contractivity; and show how graph surgery may be performed to recover positivity for the heat flow.

Title: A variational approach to the homogenization of nonlocal composite materials: H-convergence and Γ-convergence

Abstract: In this talk we present recent results on the H-convergence of nonlocal elliptic PDEs driven by monotone operators and on the Γ-convergence of the associated energy functionals in the framework of the Riesz fractional gradient and divergence. 

The first part of the talk is devoted to the relation between local and nonlocal monotone elliptic equations. We show that, for a suitable class of nonlinear operators, H-convergence in the local setting is equivalent to the corresponding notion of H-convergence in the fractional framework. As a consequence, we obtain an H-compactness result for the class of nonlocal operators under consideration.

In the second part, we identify a natural subclass of monotone operators associated with energy functionals and investigate the relation between H- and Γ-convergence in the nonlocal setting. We prove that the Γ-convergence of the associated nonlocal energies is equivalent to the Γ-convergence of their local counterparts, yielding in particular a Γ-compactness result. Finally, we extend the classical equivalence between H- and Γ-convergence, known in the local linear theory, to both the local and nonlocal nonlinear settings.

This is a joint project in collaboration with:

Giuseppe Cosma Brusca (SISSA), Maicol Caponi (University of L'Aquila), Alessandro Carbotti (University of Salento) and Fabio Paronetto (University of Padua).