Abstracts Spring Semester 2026

Title:  Optimising Riesz means of Laplace eigenvalues 

Abstract: In recent years, shape optimisation problems for Laplace eigenvalues in semiclassical limits have attracted considerable interest. In this talk, I will discuss recent joint work with Rupert Frank in which we consider the problem of optimizing Riesz means of the eigenvalues among convex domains of prescribed measure. A key ingredient in establishing our results is a detailed analysis of spectral asymptotics along sequences of convex domains that collapse as the spectral parameter tends to infinity. 

Title:  Flat flow weak solutions to mean curvature flow and their BV-Brakke structure 

Abstract: We present a recent result concerning a class of weak solutions to mean curvature flow known as flat flows. These solutions arise as limits of the minimizing movements scheme associated with this geometric evolution and have been extensively studied in the literature. However, whether they satisfy the appropriate limiting evolution law under general assumptions has remained an open question, as previous results typically required an additional assumption on the convergence of interfacial energies. We show that flat flows possess a BV-Brakke structure, and hence satisfy the expected evolution law, without any assumption on energy convergence. 

The talk is based on a joint work with S. Stuvard (UniMi).

Title:  Stability of extremal domains

Abstract: In this talk, we discuss the notion of stable extremal domains for the first Dirichlet eigenvalue of the Laplace-Beltrami operator on Riemannian manifolds. We classify stable extremal domains in the 2-sphere and in higher-dimensional spheres with minimal boundary. We also establish topological bounds for stable domains in compact Riemannian surfaces, assuming either nonnegative total Gaussian curvature or sufficiently small area.

This is joint work with Ivaldo Nunes.