Date/Time
Date(s) - 07/02/2025
1:30 pm - 2:30 pm
Speaker: Antonin Monteil (Université Paris-Est)
Location: Hamilton Hall, Room 410
Title: Metric methods for the existence of heteroclinic connections and their application to the existence of entire solutions of Allen-Cahn systems
Abstract: We will discuss the classical problem of minimizing the action \int |u’|^2 + W(u) among curves u lying in a given metric space and connecting two distinct zeros of the potential W. These minimizers are usually referred to as heteroclinic connections. We will present a method to prove the existence of solutions to this problem through its classical reduction to the geodesic problem min \int sqrt(W(u))|u’|. We will begin with the case of locally compact metric spaces, including R^n, and then tackle the general case under various assumptions on the metric space and the potential W. This approach includes the case of infinite dimensional spaces, opening the possibility of using it to prove the existence of solutions to certain PDEs on unbounded domains. We will provide a direct application to the existence of solutions on an infinite cylinder, and then explain how this method applies to the existence of solutions to the Allen-Cahn system on the plane, connecting two distinct (1D) heteroclinic solutions at infinity–a problem previously solved by Alama-Bronsard-Gui, and by Schatzman.