Calendar

Posted March 28, 2025
Last modified October 1, 2025

Applied Analysis Seminar Questions or comments?

Wenxiong Chen , Yeshiva University
Qualitative properties of solutions to fractional elliptic and parabolic equations

In this talk, I will introduce the fractional Laplacian and other nonlocal elliptic and parabolic operators and list some recent developments in the study of qualitative properties including symmetry, asymptotic symmetry, and monotonicity of solutions for nonlinear fractional elliptic and parabolic equations such as $$ (-\triangle)^s u=f(x,u(x))$$ and $$ \frac{\partial u}{\partial t}+( -\triangle)^s u=f(x,u(x, t)) $$ I will also introduce other typical fractional parabolic operators, such as the dual fractional heat operator with Marchaud time derivative $∂^\alpha t+(-\triangle)^s$ and the master operator $(\partial_t-\triangle)^s$. The extent of their non-locality will be illustrated by simple examples with pictures. I will also mention some of our recent results on interior regularity estimates for nonnegative solutions to fractional Laplace equations and fully fractional parabolic equations.

Event contact: Jiuyi Zhu

Posted March 16, 2025
Last modified October 5, 2025

Applied Analysis Seminar Questions or comments?

Nicola Garofalo, Arizona State University Charles Wexler Professor in Mathematics,
Strichartz estimates for degenerate dispersive equations

I will discuss some new Ginibre-Velo estimates for a class of Schrodinger equations with a possibly strongly degenerate Hamiltonian. The talk will have a self-contained character and I will focus on some interesting examples

Event contact: Phuc/Zhu

Posted August 3, 2025
Last modified October 5, 2025

Applied Analysis Seminar Questions or comments?

Donatella Danielli, Arizona State University School Director and Foundation Professor
Obstacle Problems for Fractional Powers of the Laplacian

In this talk we will discuss a two-penalty boundary obstacle problem for a singular and degenerate elliptic operator naturally arising in the extension procedure for the fractional Laplacian $(-\Delta)^s$ when s between 1 and 2. Our goals are to establish regularity properties of the solution and the structure of the free boundary. To this end, we combine classical techniques from PDEs and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas. In particular, we will emphasize the striking differences with the cases s between 0 and 1 and s=$3/2$. This is joint work with A. Haj Ali (University of Michigan) and G. Gravina (Loyola University-Chicago).

Event contact: Phuc/Zhu

Posted August 27, 2025
Last modified October 15, 2025

Applied Analysis Seminar Questions or comments?

Xuenan Li, Columbia University
Soft modes in mechanism-based mechanical metamaterials: modeling, analysis, and applications

Mechanism-based mechanical metamaterials are synthetic materials that exhibit unusual microscale buckling in response to mechanical deformations. These artificial materials are like elastic composites but sometimes more degenerate since they can deform with zero elastic energy. We call such zero energy deformations mechanisms. Origami and Kirigami are typical examples of these mechanism-based mechanical metamaterials. Other than mechanisms, these metamaterials also have "soft modes" -- macroscopic deformations with very little elastic energy, some but not all of which resemble modulated mechanisms. A key question is to identity all the soft modes for a given mechanism-based metamaterial. In this talk, I will address the two-fold challenge in identifying the soft modes and our treatments: first, we establish the existence of an effective energy for a broad class of lattice metamaterials; and second, we identify soft modes as macroscopic deformations where this energy vanishes, including a complete characterization of the zero sets of the effective energy in some conformal metamaterials. Together, these results provide a rigorous link between mechanisms and soft modes, laying a mathematical foundation for future analysis and design of mechanical metamaterials. This is joint work with Robert V. Kohn.

Event contact: Stephen Shipman

Posted August 21, 2025
Last modified October 24, 2025

Applied Analysis Seminar Questions or comments?

Roy Goodman, New Jersey Institute of Technology
Leapfrogging and scattering of point vortices

The interaction among vortices is a key process in fluid motion. The n-vortex problem, which models the movement of a finite number of vortices in a two-dimensional inviscid fluid, has been studied since the late 1800s and remains relevant due to its strong link to quantum fluid dynamics. A foundational document in this area is Walter Gröbli's 1877 doctoral dissertation. We apply modern tools from dynamical systems and Hamiltonian mechanics to several problems arising from this work. First, we study the linear stability and nonlinear dynamics of the so-called leapfrogging orbit of four vortices, utilizing Hamiltonian reductions and a numerical visualization method known as Lagrangian descriptors. Second, we analyze the scattering of vortex dipoles using tools from geometric mechanics. While point vortices are typically modeled as massless particles, the final part of this talk will discuss the impact of endowing each particle with a small mass. Although some of the concepts are technical, the presentation will focus on a series of interesting and informative images and animations.

Event contact: Stephen Shipman