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Singular Integral Operators on Spaces of Vanishing Mean Oscillations

"A crowning achievement of the classical Calder\'on-Zygmund theory emergent in the 1950's was the $L^p$-boundedness, $1<p<\infty$, of Singular Integral Operators (SIO) in ${\mathbb{R}}^n$ with kernels satisfying suitable smoothness, parity, and homogeneity conditions. In particular, these include the Riesz transforms $R_j$, $1\leq j\leq [Read More...]

Presenter: Dorina Mitrea, Baylor University
Authors: Dorina Mitrea
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: September 30, 2023; 2:00 pm

A definition of fractional k-dimensional measure

I will introduce a fractional notion of k-dimensional measure, with 0≤k<n, that depends on a parameter σ that lies between 0 and 1. When k=n−1 this coincides with the fractional notions of area and perimeter, and when k=1 this coincides with the fractional notion of length. We will see that, when multiplied by the factor 1−σ, this σ-measure [Read More...]

Presenter: Cornelia Mihaila, Saint Michael's College
Authors: Cornelia Mihaila, Brian Seguin
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: September 30, 2023; 2:25 pm

A Spectral Target Signature for Thin Surfaces with Higher Order Jump Conditions

In this talk, we consider the inverse problem of determining the structural properties of a thin anisotropic and dissipative inhomogeneity from scattering data. In the asymptotic limit, as the thickness goes to zero, the thin inhomogeneity is modeled by an open m-1 manifold(here referred to as screen), and the field inside is replaced by jump conditions on [Read More...]

Presenter: Heejin Lee, Purdue University
Authors: Fioralba Cakoni, Heejin Lee, Peter Monk and Yangwen Zhang
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: September 30, 2023; 2:50 pm

Global Attractors for Suspension Bridge Models Under Unstable Flow of Gas.

The purpose of this paper is to address long-term behavior of solutions to a plate model describing a suspension bridge with mixed boundary condition in presence of wind-effect and polynomial type weak damping. We prove the wellposed of the dynamical system by the traditional theory of nonlinear semigroups and monotone operators. Existence of the global [Read More...]

Presenter: MADHUMITA ROY, Doctoral Student
Authors: Irena Lasiecka, Jose Rodrigues
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: September 30, 2023; 3:15 pm

Transonic Problems in Multidimensional Conservation Laws

We present the recent progress in the mathematical analysis of multidimensional conservation laws. Specifically, we discuss various physical configurations that give rise to transonic free boundary problems, formulations of the boundary value problems, and the analysis of understanding the solution structure of those [Read More...]

Presenter: Eun Heui Kim, California State University Long Beach
Authors: Eun Heui Kim
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: September 30, 2023; 3:40 pm

Regularity results for a class of penalized boundary obstacle problems

In this talk we will give a comprehensive overview of some classes of two-penalty boundary obstacle problems, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and optimal regularity of solutions, and establish structural properties of the free boundary. The proofs are based on tailor-made [Read More...]

Presenter: Donatella Danielli, Arizona State University
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 1, 2023; 9:45 am

Metalenses and Optimal Transport

Metalenses are ultra thin surfaces that are composed of nano structures to focus light. These nano structures manipulate light waves by abrupt phase shifts over the scale of the wavelength to bend them in unusual ways. Compared to the bulky, thick shapes of the conventional lenses, metalenses offer many advantages in optical applications due to their reduced [Read More...]

Presenter: Irem Altiner, Temple University
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 1, 2023; 10:10 am

Maximal and minimal weak solutions for elliptic coupled systems with nonlinearity on the boundary

We consider the existence of weak solutions for elliptic coupled system with quasimonotone non decreasing nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution. To prove the result, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn’s [Read More...]

Presenter: Shalmali Bandyopadhyay, University of Tennessee Martin
Authors: Nsoki Mavinga, Thomas Lewis
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 1, 2023; 10:35 am

Finite Time Blowup for the Nonlinear Schrödinger Equation with a delta potential

In this talk, we study the Cauchy problem for the nonlinear Schodinger equation with a delta potential. We show that under certain conditions, the supremum norm of the solution tends to infinity in finite time. In order to prove this, we study the associated Lagrangian and Hamiltonian, and derive an estimate of the associated variance, which extends the [Read More...]

Presenter: Sarah Raynor, Wake Forest University
Authors: Brandon Hauser, John Holmes, Eoghan O’Keefe, Sarah Raynor, and Chuanyang Yu
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 1, 2023; 11:00 am

Hypoellipticity via sums of squares

Many results on hypoellipticity of second order operators rely on the assumption that the operator can be written as a sum of squares of vector fields (e.g. Hormander's bracket condition, and Christ's hypoellipticity theorem for infinitely degenerate operators). For operators that are not subelliptic and not sums of squares, hypoellipticity has [Read More...]

Presenter: Luda Korobenko, Reed College
Authors: L. Korobenko, E. Sawyer
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 2, 2023; 8:30 am

Advances in Partial Differential Equations in a problem involving $\phi$-Laplacian operator

"We prove an existence result of positive radial solution for the problem $\Delta_\phi (u)=\lambda f(u)$ in a ball with homogeneous Dirichlet condition. In order to prove that, we use a shoothing method and our main tool is the use of Pohozaev [Read More...]

Presenter: Diana Milena Sánchez Monsalve, Universidad Nacional de Colombia
Authors: Diana Sánchez Monsalve
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 2, 2023; 8:55 am

Modeling effects of matrix heterogeneity on population persistence at the patch-level

Habitat loss and fragmentation is the largest contributing factor to species extinction and declining biodiversity. Landscapes are becoming highly spatially heterogeneous with varying degrees of human modification. Much theoretical study of habitat fragmentation has historically focused on a simple theoretical landscape with patches of habitat surrounded [Read More...]

Presenter: Keta Henderson, University of North Carolina, Greensboro
Authors: Nalin Fonseka, Jeromme Goddard II, Alketa Henderson, Dustin Nichols, Ratnasingham Shivaji
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 2, 2023; 9:20 am

Wave propagation on rotating cosmic string spacetimes

A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional [Read More...]

Presenter: Katrina Morgan, Temple University
Authors: Katrina Morgan and Jared Wunsch
Symposium Year: 2023
Session: Advances in Partial Differential Equations and Applications [Organized by Maya Chhetri, Nsoki Mavinga and Irina Mitrea]
Presentation Time: October 2, 2023; 9:45 am

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