Harmonic Functions and Laplace Eigenfunctions

Session Organizers

Eugenia Malinnikova, Stanford University

Bobby Wilson, University of Washington

Zihui Zhao, University of Chicago

8 December, 2022

Abstract

Given a bounded domain $\Omega \subset \mathbb R^n$, one says that the $L^p$-regularity problem is solvable for the Laplace equation in $\Omega$ if, given any continuous function $f$ defined in $\partial \Omega$ and the harmonic extension $u$ of $f$ to $\Omega$, the non-tangential maximal function of the gradient of $u$ can be controlled in $L^p$ norm by the tangential derivative of $f$ in $\partial\Omega$. In my talk I will review a joint result with Mourgoglou where we proved the $L^p$-regularity for more general domains, and a more recent result (with the additional collaboration of Poggi) where we extend this result to elliptic PDE’s in divergence form.

Abstract

The study of eigenfunctions of Laplacians lies at the interface of several areas of mathematics, including analysis, geometry, mathematical physics and number theory. These special functions arise in physics and in partial differential equations as modes of periodic vibration of drums and membranes.

A fundamental question surrounding Laplace-Beltrami eigenfunctions targets their concentration phenomena, via high-energy asymptotics or high-frequency behaviour. One popular approach to this question involves studying the growth of the $L^p$ norms of these eigenfunctions on the ambient manifold, or on submanifolds thereof. What can one say about the behaviour of eigenfunctions on rougher sets? We will discuss answers to this question, based on joint work with Suresh Eswarathasan.

Abstract

One guiding principle is the idea that Laplacian eigenfunctions corresponding to larger frequencies should oscillate more. The same principle should then also be true, in some form, for linear combinations of high-frequency eigenfunctions (in one dimension, this is Sturm-Liouville theory). Recent progress on this question is based on the notion of optimal transport and a very simple idea which we formalize: if it’s easy to buy milk, then there are must be many supermarkets (and, conversely, if there are only few supermarkets at least some people have to travel a large distance to buy milk). This turns into a geometric inequality that is interesting in its own right.

Abstract

An important heuristic principle in the study of eigenfunctions of the Laplace-Beltrami operator is that their properties should resemble those of polynomials. In this light, I will discuss oscillations and zeros for linear combinations of Laplace eigenfunctions on Riemannian manifolds. In particular, I will prove that zeros become dense in the manifold if not too many eigenfunctions are summed. Time permitting, I will mention related open questions on eigenfunctions sums.

Abstract

In this talk, I will discuss joint work with M. Engelstein, L. Li, and S. Mayboroda, where we introduce the notion of Dahlberg-Kenig-Pipher operators in the context of domains $\Omega$ with low dimensional boundaries. We show that when the boundary of the domain is uniformly rectifiable with small constant, then elliptic measure $\omega$ associated to this domain is an $A_\infty(d\sigma)$ weight with small constant. As consequence, we show that for $C^1$ domains, $\log (d\omega/d\sigma) \in \mathrm{VMO}$. One of the main difficulties in this context is the lack of outer graphical approximations to $\Omega$, since the exterior of $\Omega$ can be empty.

9 December, 2022

Abstract

In this talk I will present spectral features of the Dirac operator with infi nite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. A non-linear variational formulation to characterize its principal eigenvalue will be presented. This characterization allows for a simple proof of a Szegö type inequality as well as a new formulation of a Faber-Krahn type inequality for this operator. Moreover, strong numerical evidence supporting the existence of a Faber-Krahn type inequality, will be given.

This talk is based on joint work with Pedro Antunes, Vladimir Lotoreichik, and Thomas Ourmieres-Bonafos.

Abstract

In this talk, we will explore the celebrated Almgren’s monotonicity formula. This beautiful result with far-reaching consequences states that if u is harmonic in the unit ball, then a certain frequency function $N(r)$ is non-decreasing. Moreover, $N(r)=k$ for all $r<1$ if, and only if, $u$ is homogeneous of degree $k$. We will then discuss some of the many applications of this formula, and recent developments connected to it.