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Analysis Seminar – John-Oliver MacLellan (University of Alabama)

227 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United States

Title: "A Compact Embedding Theorem for Degenerate Sobolev Spaces" Abstract: In this talk I will prove a compact embedding theorem for degenerate Sobolev spaces into naturally associated weighted Lebesgue spaces. This is a generalization of the Rellich Kondrachov compactness theorem for classical Sobolev spaces.  These embeddings have been studied in a very general context in

Analysis Seminar – Arum Lee (University of Alabama)

227 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United States

Title: Definition of Weak Solution of Poisson Equation with Homogeneous Dirichlet Boundary Condition Abstract: In this talk, I will discuss the ideas of weak derivatives, traces of Sobolev functions and the weak solution of Poisson equation. These definitions are the very basic but important definitions in PDE. This talk will explain the motivation behind these

Analysis Seminar – Yuanzhen Shao, Georgia Southern University

227 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United States

Title: Some Applications of Singular Manifold Theory to Applied Mathematics Abstract: Many applications of applied sciences lead to differential equations with various types of singularities, including singularities of the geometry of the underlying space and singularities of the coefficients of the differential equations. The aim of this talk is to introduce the concept of singular manifolds, which can describe various kinds of singularities in a unified way, and then my recent work on the partial differential equation theory over singular manifolds will be presented. I will illustrate by several examples from applied mathematics how to use this theory to treat different types of singularities via a unified approach.

Analysis Seminar – Ollie Tapiola (University of Missouri)

227 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United States

Title: Carleson measures, uniform wrectifiability and $\varepsilon$-approximability of harmonic functions in $L^p$ Abstract: Uniform rectifiability is a geometric property that is strongly connected with harmonic analysis and elliptic PDE. Although many powerful PDE tools are not available in spaces with uniformly rectifiable boundaries, several authors have recently managed to prove positive PDE results in this

Analysis Seminar – David Cruz-Uribe (University of Alabama)

227 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United States

Title:  Poincare inequalities and Neumann problems for the p-Laplacian Abstract:  I will discuss my recent work with Scott Rodney on the following equivalence:  the existence of solutions to a degenerate p-Laplacian equation and the existence of a weighted (p,p) Poincare inequality.   Our results are in the context of degenerate Sobolev spaces, where the degeneracy is

Analysis Seminar – Shibin Dai (University of Alabama)

227 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United States

Title: An introduction to Gamma convergence Abstract: Many mathematical problems involve parameters that make those problems more and more complex or degenerate. It is of great interest to study the limiting behavior when the parameter varies. One class of such problems can be studied in a variational framework. Writing $F_\epsilpon$ as a class of functional

Analysis Seminar – Bingyuan Liu (University of California, Riverside) Geometry of the @-Neumann problem and the D{F index

227 Gordon Palmer Hall 505 Hackberry Lane, Tuscaloosa, AL, United States

Abstract. We shall rst introduce the classical works of Hormander and Kohn on the L2 estimates of the @(-Neumann) problem on bounded domains and then describe applications in complex geometry. It turns out that the boundary geometry plays the fundamental role in the Sobolev estimates of the @ solution. The Diederich{Fornss index is the geometric invariant which predicts the estimates.