Schedule

All talks will be in HCB 214

The coffee will be served in HCB 220 at 9:30am

Preliminary Materials

Some preliminary materials can be found here

Abstracts of Talks

Matthew Badger, University of Connecticut

Title: Bourgain's constants for harmonic and caloric measures

Abstract: Bourgain’s constant is the largest number b_n such that the dimension of harmonic measure on any n-dimensional domain is at most n − b_n. Jones and Wolff (1988) proved that b₂ = 1. When n ≥ 3, Bourgain (1987) proved that b_n > 0 and Wolff (1995) proved that b_n < 1. In recent work, joint with Alyssa Genschaw, we prove that Bourgain’s theorem on the dimension of harmonic measure persists in the setting of the heat equation. Further, refining Bourgain’s original outline, we prove asymptotic lower bounds on Bourgain’s constant in high dimensions and explicit lower bounds on Bourgain’s constant in dimensions 3 and 4.

Ariel Barton, University of Arkansas

Title: Higher order elliptic partial differential equations with lower order terms

Abstract: Operators of the form Lu = ∇ ⋅ (A∇u), that is, second order linear differential operators in divergence form, are by now very well understood. Two important generalizations are higher order operators Lu = ∇^m ⋅ (A∇^mu), for m ≥ 2 an integer, and operators with lower order terms Lu = ∇ ⋅ (A∇u) + ∇ ⋅ (Bu) + C ⋅ ∇u + Du. In this talk we discuss some first attempts to combine the two generalizations and study higher order operators with lower order terms.

Blair Davey, Montana State University

Title: Exponential Decay Estimates for Fundamental Matrices of Generalized Schrödinger Systems

Abstract: In this work, we consider the behavior of the fundamental matrices associated to systems of generalized Schrödinger operators. The Schrödinger operators that we deal with have leading coefficients that are bounded and uniformly elliptic, and they have zeroth-order potentials that belong to certain reverse Hölder matrix classes. By exploring the properties of reverse H\"older matrices and building on the ideas of Shen, Mayboroda and Poggi, we establish the existence of fundamental matrices and prove upper and lower exponential decay estimates for them. This talk covers joint work with Josh Isralowitz.

Guy David, Université de Paris Sud

Title: An elliptic operator on the four-corner Cantor set, with elliptic measure proportional to H¹

Abstract: The lecture is about the construction, with S. Mayboroda, of an elliptic operator L =  − div(a(x)∇) on the complement of a one-dimensional Cantor set E in the plane, so that the corresponding elliptic measure on E is the product of the length measure H¹ by a function which is bounded and bounded from below. The main idea is to build the Green function by hand, deduce the function a, and use the fractal invariance to prove it is elliptic. I’ll try to explain why we care about such examples.

Max Engelstein, University of Minnesota, Twin-Cities

Title: Harmonic analysis tools for free boundary problems

Joseph Feneuil, Université Paris-Saclay 

Title: Generalized Carleson perturbations of elliptic operators.

Abstract: Let L₀=div A₀ ∇ be a uniformly elliptic operator on a domain Ω, and we assume that the elliptic measure ω_L0 is A_∞-absolutely continuous with respect to a doubling measure σ on the boundary ∂Ω. We are interested in perturbations L₁ of L₀ that preserve the A_∞-absolute continuity of the elliptic measure w.r.t. σ.

We shall discuss the literature on “Carleson perturbations of elliptic operators”, from the initial work of Fefferman, Kenig and Pipher in 1991, to the recent article of Bruno Poggi and the speaker. In the later paper, we proposed an extended notion of Carleson perturbations, and we worked on a wide range of domains and elliptic operators by assuming only a suitable elliptic theory.

John Garnett, UCLA

Title: Carleson measure estimates for bounded harmonic functions, without Ahlfors regularity assumptions.

Linhan Li, University of Minnesota, Twin-Cities

Sponsors

CBMS’22 is supported by