Search Research Symposium Abstracts

Critical Sets of Solutions of Elliptic Equations in Periodic Homogenization

In this talk I will present some of my recent work with Fanghua Lin on the geometric measure estimates of critical sets of solutions of elliptic equations with rapidly oscillating [Read More...]

Presenter: Zhongwei Shen, University of Kentucky
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 10:15 am

Besov spaces of functions on doubling metric measure spaces, and hyperbolic fillings

From the study of fractional Laplacian to the study of quasiconformal geometry, Besov spaces seem to appear as natural tools. In this talk we will give a method of connecting Besov function classes on compact doubling metric measure spaces as traces of Sobolev-type classes of functions on certain uniform domains obtained via a geometric technique called [Read More...]

Presenter: Nageswari Shanmugalingam, University of Cincinnati
Authors: Anders Bjorn, Jana Bjorn, Nageswari Shanmugalingam
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 10:40 am

Harmonic measure for Schrodinger operators in uniform domains

Let $\Omega$ be a uniform domain (a bounded domain satisfying the interior corkscrew and Harnack chain conditions). We study the time-independent Schrodinger operator Laplace $u + q u =0$ in $\Omega$, with $u =f$ on the boundary of Omega, where $q$ is a non-negative potential, or more generally, a locally finite Borel measure on $\Omega$, and $f$ is a [Read More...]

Presenter: Michael Frazier, University of Tennessee
Authors: Michael Frazier and Igor Verbitsky
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 11:05 am

Moser type log-mass weighted inequalities and exponential Choquard equations in the plane

We review some recent results on log-weighted Moser type inequalities on the whole plane. The presence of a radial non decreasing weight prevents us applying the standard symmetrization approach. Nevertheless, we are able to prove that a Moser type inequality holds on a suitable log-mass weighted Sobolev space, without restricting ourselves to radial [Read More...]

Presenter: C. Tarsi, Università degli Studi di Milano
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 11:30 am

Analysis of Minimizers for the Landau-de Gennes Q-Tensor Energy and Their Defects

We analyze minimizers for the Landau-de Gennes $Q$-tensor energy in a three-dimensional slab, $D=U x (-a,a)$, where $U$ is a bounded simply connected domain in the plane . The minimizers are subject to tangential boundary conditions on the top and bottom faces and prescribed boundary conditions on the lateral surface. Minimizers of this energy describe [Read More...]

Presenter: Patricia Bauman, Purdue University, Dept. of Mathematics, West Lafayette, IN
Authors: Patricia Bauman and Daniel Phillips
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 3:30 pm

Higher order elliptic differential equations with lower order terms

Operators of the form $Lu=\nabla\cdot (A\nabla u)$, that is, second order linear differential operators in divergence form, are by now very well understood. Two important generalizations are higher order operators $Lu=\nabla^m\cdot (A\nabla^m u)$, for $m\geq 2$ an integer, and operators with lower order terms $Lu=\nabla\cdot (A\nabla u) + \nabla\cdot (Bu) + [Read More...]

Presenter: Ariel Barton, University of Arkansas
Authors: Ariel Barton and Michael Duffy Jr
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 3:55 pm

Energy concentration, vorticity measures, and the inviscid limit problem in 2D*

The inviscid limit problem for the incompressible Navier-Stokes equations in a bounded domain is a major outstanding open problem. Formally setting viscosity to zero yields the Euler equations but, in a bounded domain, the discrepancy in boundary conditions gives rise to boundary layer phenomena, something still poorly understood and which accounts for the [Read More...]

Presenter: HELENA JUDITH NUSSENZVEIG LOPES, Universidade Federal do Rio de Janeiro
Authors: Peter Constantin, Milton C Lopes Filho, H. J. Nussenzveig Lopes and Vlad Vicol
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 4:20 pm

On Choquet integrals and Poincare-type inequalities

We discuss Poincare-type inequalities in terms of Choquet integrals with respect to Hausdorff content. Also, we consider a Trudinger-type inequality in this context. This is joint work with Petteri [Read More...]

Presenter: Professor Ritva Hurri-Syrjanen, University of Helsinki
Symposium Year: 2022
Session: Analysis of Partial Differential Equations in Memory of David R. Adams
Presentation Time: June 17, 2022; 4:45 pm