2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023
Current contact: Irina Mitrea
The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.
Gerardo Mendoza, Temple University
Abstract: Let $M$ be a closed $n$-manifold, $H^q(M)$ its de Rham cohomology groups, which are finite dimensional vector spaces. The Lefschetz number of a smooth map $f:M\to M$ is $L_f=\sum_{q=0}^n (-1)^q\mathrm{tr}(f_q^*)$ where $f^*_q:H^q(M)\to H^q(M)$ is the linear transformation induced by $f$ and $\mathrm{tr}(f_q^*)$ is its trace. A theorem of Lefschetz asserts that if $L_f\ne 0$ then $f$ has fixed points. A theorem of Atiyah and Bott gives a formula for $L_f$ under some condition on $f$. I plan to review this, then describe work in progress with L. Hartmann in a certain setting in which $M$ has singularities and the de Rham complex is replaced by a related complex.
Irina Mitrea, Temple University
Abstract: The goal of this talk is to identify the broadest possible spectrum of radiation conditions for null-solutions of the vector Helmholtz operator. This contains, as particular cases, the Sommerfeld, Silver-Muller, and McIntosh-Mitrea radiation conditions corresponding to scattering by acoustic waves, electromagnetic waves, and null-solutions of perturbed Dirac operators, respectively. This is joint work with Dorina Mitrea and Marius Mitrea.
Farhan Abedin, Lafayette College
Abstract: I will present an iterative method for solving the Monge-Amp\`ere eigenvalue problem: given a bounded, convex domain $\Omega \subset \mathbb{R}^n$, find a convex function $u \in C^2(\Omega) \cap C(\overline{\Omega})$ and a positive number $\lambda$ satisfying $$\begin{cases} \text{det} D^2u = \lambda |u|^n & \quad \text{in } \Omega,\\ u = 0 & \quad \text{on } \partial \Omega. \end{cases}$$ By a result of P.-L. Lions, there exists a unique eigenvalue $\lambda=\lambda_{MA}(\Omega)>0$ for which this problem has a solution. Furthermore, all eigenfunctions $u$ are positive multiples of each other. In recent work with Jun Kitagawa (Michigan State University), we develop an iterative method which generates a sequence of convex functions $\{u_k\}_{k = 0}^{\infty}$ converging to a non-trivial solution of the Monge-Amp\`ere eigenvalue problem. We also show that $\lim\limits_{k \to \infty} R(u_k, \Omega) = \lambda_{MA}(\Omega)$, where the Rayleigh quotient $R(v)$ is defined as $$R(v, \Omega) := \frac{\int_{\Omega} |v| \ \text{det} D^2v}{\int_{\Omega} |v|^{n+1}}.$$ Our method converges for a large class of initial choices $u_0$ that can be constructed explicitly, and does not rely on prior knowledge of the eigenvalue $\lambda_{MA}(\Omega)$. I will also discuss other relevant iterative methods in the literature that motivated our work.
Jeongsu Kyeong, Temple University
Abstract: The poly-Cauchy operator is a natural generalization of the classical Cauchy integral, in which the salient role of the Cauchy-Riemann operator $\overline{\partial}$ is now played by $\overline{\partial}^m$, for $m\in{\mathbb{N}}$. Building on Fatou-type results for polyanalytic functions, the talk will be focused on Calderon-Zygmund theory (jump relations, higher-order boundary traces) and the study of higher-order Hardy spaces in uniformly rectifiable domains in the complex plane.
This is joint work with Irina Mitrea (Temple University), Dorina Mitrea and Marius Mitrea (Baylor University).
Elie Abdo, Temple University
Abstract: We consider an electrodiffusion model describing the time evolution of the concentrations of many ionic species, with different valences and diffusivities, in a two-dimensional incompressible fluid flowing through a porous medium. The ionic concentrations evolve according to the nonlinearly advected and nonlinearly forced Nernst-Planck equations. The velocity of the fluid obeys Darcy’s law, forced by the nonlinear electric forces occurring due to the motion of ions. The resulting Nernst-Planck-Darcy (NPD) model is a locally well-posed dissipative system of nonlinear elliptic and parabolic partial differential equations. In this talk, we address the existence of a unique global smooth solution to the NPD system and prove its spatial analyticity.
Yury Grabovsky, Temple University
Abstract: Completely monotone functions (CMF) are Laplace transforms of positive measures. I will discuss the question of extrapolation of completely monotone functions from a given interval to the entire positive semiaxis from practical point of view. Specifically, if we found a CMF that is epsilon close to a given CMF on the interval, then to what extent can we be sure that the values of our CMF approximate that of a given CMF outside of the interval? Please come to learn what CMFs are, see theorems from your Real Analysis course in action, and enjoy a cool piece of Functional Analysis.
Nsoki Mavinga, Swarthmore College
Abstract: In this talk, we will present our recent existence results concerning nonlinear elliptic equations where the nonlinearity on the boundary is nonmonotone. The iterativemethod is not applicable in this case. We use the Zorn’s lemma and a version of Kato’s inequality up to the boundary together with the surjectivity of a pseudomonotone and coercive operator to prove the existence of maximal and minimal weak solutions between an ordered pair of sub- and supersolution. We provide an application of our results to get positive solutions for a class of nolinear elliptic equations.
Irem Altiner, Temple University
Abstract: 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 thicknesses and multifunctionalities. Mathematically a metalens can be represented by a pair $(\Gamma,\Phi)$ where \Gamma is a surface in $\mathbb{R}^3$, and $\Phi$ is a $C^1$ function defined in a neighborhood of $\Gamma$, called phase discontinuity. The knowledge of $\Phi$ yields the type of arrangements of the nano structures on the surface that are needed for a specific refraction job. In this talk we are going to discuss several refraction problems starting from the existence of phase discontinuity functions that refract a ray in desired directions and conserve energy.
Nizar Bou Ezz, Temple University
Abstract: When is a sequence on the unit circle equidistribured? J. -P. Serre proved that the equidistribution of a sequence in a compact group is connected to the non-vanishing of certain L-functions associated to the sequence. The theorem has many applications to density results in number theory. In this talk we will explore the notion of equidistribution and develop the background needed to state and prove Serre’s theorem. Moreover, we will apply the theorem to deduce Dirichlet’s theorem about primes in arithmetic progressions.
Siqi Fu, Rutgers University, Camden
Abstract: We study stability of spectrum of the complex Laplacians. In particular we study spectral stability of the Kohn Laplacian under perturbation of the underlying CR manifold or CR structure and relate it to stability of embeddings of CR manifolds. This talk is based on an on-going project with Howard Jacobowitz and Weixia Zhu.
Luiz Hartmann, Universidade Federal de São Carlos, Brazil
Abstract: Given an invertible self-adjoint operator $L$ in a Hilbert space, under a certain assumption on $L$, I will describe the relation between the (regularized) Fredholm determinant, $\det_{p}(I+z\cdot L^{-1})$, and the zeta regularized determinant, $\det_{\zeta}(L+z)$. Moreover, I will discuss the asymptotic expansion of the Fredholm determinant in relation to the heat trace coefficients, showing that the constant term is the zeta-determinant of $L$.
Patrick Phelps, Temple University
Abstract: We present recent results on spatial decay and properties of non-uniqueness for the 3D Navier-Stokes equations. We show asymptotics for the ‘non-linear’ part of scaling invariant flows with data in subcritical classes. Motivated by recent work on non-uniqueness, we investigate how non-uniqueness of the velocity field would evolve in time in the local energy class. Specifically, by extending our subcritical asymptotics to approximations by Picard iterates, we may bound the rate at which two solutions, evolving from the same data, may separate pointwise. We conclude by extending this separation rate to solutions with no scaling assumption. Joint work with Zachary Bradshaw.
Zongyuan Li, Binghamton University
Abstract: In this talk, we discuss sharp conditions for Liouville-type theorems in conformally invariant elliptic PDEs. These equations, known as "nonlinear Yamabe equations", find their applications in studying conformal metrics on Riemannian manifolds. Based on recent joint work with Baozhi Chu and Yanyan Li (Rutgers).
Artur H. O. Andrade, Temple University
Abstract: A number of physical phenomena are modeled by overdetermined boundary value problems, that is, boundary problems in which one imposes both Dirichlet and Neumann type boundary conditions.
The subject of this talk is the analysis of overdetermined boundary value problems (OBVP) for 2nd order homogeneous constant complex coefficient weakly elliptic systems in non-smooth domains with boundary datum in Whitney--Lebesgue spaces with integrability index in the interval $(1,\infty)$. This analysis includes integral representation formulas, jump relations, existence and uniqueness of solutions for the OBVP in uniformly rectifiable domains, and classical Hardy spaces associated with systems.
This is joint work with Irina Mitrea (Temple University), Dorina Mitrea and Marius Mitrea (Baylor University).
Jeongsu Kyeong, Temple University
Abstract: The study of boundary value problems associated with the bi-Laplacian operator $\Delta^{2}$ plays an important role in the theory of elasticity, specifically in the Kirchhoff-Love theory of thin plates.
The goal of this talk is to investigate the solvability of the $L^{p}$ Neumann problem for the bi-Laplacian, for $p\in(1,\infty)$, in infinite sectors in two dimensions, using singular integral operators and Mellin transform techniques.
This is joint work with Irina Mitrea (Temple University) and Katharine Ott (Bates College).
Gerardo Mendoza, Temple University
Abstract: Let $\mathcal N$ be a closed $n$-manifold foliated by the orbits of a group $G$ of diffeomorphisms isomorphic to a torus, let $f:\mathcal N\to \mathcal N$ be a smooth function sending leaves to leaves. Assuming certain transversality condition on the function I'll describe how to regularize the trace of $f^*:C^\infty(\mathcal N)\to C^\infty(\mathcal N)$. The group $G$ will be the closure of the one-parameter group of isometries generated by a smooth nowhere vanishing vector field $\mathcal T$ preserving a Riemannian metric, with $f_*\mathcal T=\mathcal T$ and the $L^2$ space defined using the Riemannian measure. I plan to give a sense of what $G$ is (using an embedding of $\mathcal N$ in some $\mathbb C^N$), also review the notion of wave front set and a theorem of Hörmander on restriction of distributions. Part of the talk is based on joint work with L. Hartmann.
Maria Soria Carro, Rutgers University
Abstract: In 1940, F. Rellich introduced an integral identity while studying the Dirichlet eigenvalue problem for the Laplace operator. This identity, nowadays known as the \textit{Rellich identity}, plays fundamental roles in questions on elliptic partial differential equations. In this talk, we will discuss two of its applications within the context of the Neumann problem in Lipschitz domains. The first application deals with the invertibility of certain singular operators from potential theory, proved by G. Verchota (1984) for bounded domains and subsequently extended to graph domains by C. Kenig (1985). In the second part of the talk, we will present a new application of the Rellich identity in Harmonic Analysis, involving the Hilbert transform and some ``special'' weights arising from conformal maps. This is joint work with M. J. Carro (Universidad Complutense de Madrid) and V. Naibo (Kansas State University).
Ryan Hynd, University of Pennsylvania
Morrey's inequality measures the Holder continuity of a function whose gradient belongs to an appropriate Lebesgue space. There has been recent interest in understanding the extremals of Morrey's inequality, which are the functions which saturate the inequality. We present a natural variant of Morrey's inequality on a given domain and discuss the question of whether or not an extremal exists.
This is joint work with Simon Larson (Chalmers) and Erik Lindgren (KTH).
Hyunwoo Kwon, Brown University
Abstract: We consider the Dirichlet problems for second-order linear elliptic equations
$$ -\Delta u + \mathrm{div}\, (u\mathbf{b})=f\quad\text{and}\quad -\Delta v -\mathbf{b}\cdot\nabla v =g $$
in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$, $n\geq 2$, where $\mathbf{b}:\Omega\rightarrow\mathbb{R}^n$ is a given vector field. Under the assumption that $\mathbf{b} \in L^n(\Omega)^n$, we establish the existence and uniqueness of solutions in $L^p_\alpha(\Omega)$ for the Dirichlet problem. Here $L_\alpha^p(\Omega)$ denotes the Sobolev space with the pair $(\alpha,p)$ satisfying certain conditions. This result extends the classical work of Jerison-Kenig (1995) for the Poisson equation. We also prove the existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^2(\partial\Omega)$. We also discuss relevant problems for Neumann problems and different regularities on the drift coefficient as well. Part of this presentation is based on the joint work with Prof. Hyunseok Kim (Sogang University, South Korea).
Sarah Strikwerda, University of Pennsylvania
Abstract: In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the surrounding blood circulation. The local and global features can be accounted for through a PDE-ODE coupling. I will discuss the well-posedness results for the PDE and our fixed-point strategy to show well-posedness of the PDE-ODE coupled system.
2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023