Analysis Seminar

Phase transitions in heterogeneous media

Speaker: Raghavendra VENKATRAMAN, NYU

Location: Warren Weaver Hall 312

Date: Thursday, November 4, 2021, 11 a.m.

Synopsis:

We consider a variational model for phase transitions in periodic media. In the regime where phase transitions and homogenization occur at comparable scales, we examine an asymptotic cell formula for the effective anisotropic surface tension obtained by Cristoferi, Fonseca, Hagerty and Popovici. We offer nontrivial, anisotropic bounds on the effective surface tension in terms of distance functions to hyperplanes in periodic Riemannian metrics, along with various other refined estimates on minimizers in the cell formula. Along the way, we prove a homogenization result for such distance functions in periodic Riemannian metrics to hyperplanes-- our method of proof is self-contained, and is sufficiently robust to handle almost periodic homogenization of convex Hamilton-Jacobi equations with Lipschitz dependence on the fast variable, in unbounded domains. Time permitting, we discuss associated gradient flows and their asymptotics in the small parameter. We examine a number of different parameter regimes. The central theme to the entire talk is dealing with combined effects of oscillations and concentrations.  The first half of the talk represents joint work with Rustum Choksi and Jessica Lin (McGill) and Irene Fonseca (CMU), while work on dynamics is in progress.