Current contact: Gerardo Mendoza
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.
Monday January 25, 2021 at 14:30, Zoom meeting
Analytic and Gevrey regularity for sums of squares in low dimension
Antonio Bove, University of Bologna
We present a couple of results of analytic and Gevrey regularity for sums of squares operators in dimension 2. The reason why we focus on dimension 2 is that we believe it is the only case where Treves conjecture holds. We identify the Poisson strata with some higher multiplicity subvarieties of the characteristic variety.
Monday February 1, 2021 at 14:30, Zoom meeting
Two new local $T1$ theorems on non-homogeneous spaces
Francisco Villarroya, Temple University
We introduce two new $T1$ theorems characterizing all Calder\'on-Zygmund operators $$Tf(x)=\int f(t)K(t,x)d\mu (t)$$ that extend boundedly on $L^{p}(\mathbb R^{n},\mu)$ for $1<p<\infty $ with $\mu$ a non-doubling measure of power growth.
We employ a new proof method that, unlike all currently known works on $T1$ theorems in non-homogeneous spaces, does not use random grids. The new approach allows the use of a countable family of testing functions, and also testing functions supported on cubes of different dimensions.
Monday February 8, 2021 at 14:30, Zoom meeting
Global Solutions of the Nernst-Planck-Euler Equations
Jingyang Shu, Temple University
The transport and the electrodiffusion of ions in homogeneous Newtonian fluids are classically modeled by the Nernst-Planck-Navier-Stokes (NPNS) equations. When the kinematic viscosity term in the Navier-Stokes equation is neglected, the NPNS system becomes the Nernst-Planck-Euler (NPE) system. In this talk, we consider the initial value problem for the NPE equations with two ionic species in two-dimensional tori. We prove the global existence of weak solutions and the global existence and uniqueness of smooth solutions. We also show that in the vanishing viscosity limits, smooth solutions of the NPNS equations converge to the solutions of the NPE equations. This is joint work with Mihaela Ignatova.
Monday March 1, 2021 at 14:30, Zoom meeting
Smoothing effect and Strichartz estimates for some time-degenerate Schroedinger operators
Serena Federico, Ghent University
In this talk we will analyze the smoothing effect and the validity of Strichartz estimates for some classes of time-degenerate Schroedinger operators. In the first part of the talk we will investigate the local smoothing effect (both homogeneous and inhomogeneous) for time-degenerate Schr\"odinger operators of the form $$ \mathcal{L}_{\alpha,c}=i\partial_t+t^\alpha\Delta_x+c(t,x)\cdot \nabla_x,\quad \alpha>0,$$where $c(t,x)$ satisfies suitable conditions. Additionally, we will employ the smoothing effect to prove local well-posedness results for the associated nonlinear Cauchy problem. In the second part of the talk we will analyze Strichartz estimates for a class of operators similar to the previous one, that is of the form $$\mathcal{L}_{b}:=i\partial_t+ b'(t)\Delta_x,$$with $b'$ satisfying suitable conditions. An application of these estimates will give a (different) local well-posedness result for a semilinear Cauchy problem associated with $\mathcal{L}_b$.
Monday March 22, 2021 at 14:30, Zoom meeting
The Neumann problem for symmetric higher order elliptic differential equations
Ariel Barton, University of Arkansas
The second order differential equation $\nabla\cdot A\nabla u=0$ has been studied extensively. It is well known that, if the coefficients $A$ are real-valued, symmetric, and constant along the vertical coordinate (and merely bounded measurable in the horizontal coordinates), then the Dirichlet problem with boundary data in $L^q$ or $\dot W^{1,p}$, and the Neumann problem with boundary data in $L^p$, are well-posed in the half-space, provided $2-\varepsilon<q<\infty$ and $1<p<2+\varepsilon$.
It is also known that the Neumann problem for the biharmonic operator $\Delta^2$ in a Lipschitz domain in $\mathbb{R}^d$ is well posed for boundary data in $L^p$, $\max(1,p_d-\varepsilon)<p<2+\varepsilon$, where $p_d=\frac{2(d-1)}{d+1}$ depends on the ambient dimension~$d$.
In this talk we will discuss recent well posedness results for the Neumann problem, in the half-space, for higher-order equations of the form $\nabla^m\cdot A\nabla^m u=0$, where the coefficients $A$ are real symmetric (or complex self-adjoint) and vertically constant.
Monday March 29, 2021 at 14:30, Zoom meeting
Gustavo Hoepfner, Federal University of Sao Carlos
We introduce the notion of global $L^q$ Gevrey vectors and investigate the regularity of such vectors in global and microglobal settings. We characterize the vectors in terms of the FBI transform and prove global and microglobal versions of the Kotake-Narasimhan Theorem. As a consequence we provided a refinement of an earlier result by Hoepfner and Raich relatingthe microglobal wavefront sets of the ultradistributions $u$ and $Pu$ when $P$ is a constant coefficient differential operator. This is a joint work with A. Raich and P. Rampazzo.
Monday April 5, 2021 at 14:30, Zoom meeting
On Optimal Control Problem related to the Infinity Laplacian
Henok Mawi, Howard University
The infinity Laplacian equation is given by$$\Delta_{\infty} u := u_{x_i}u_{x_j}u_{x_ix_j} = 0 \quad \text{in } \Omega$$where $\Omega$ is an open bounded subset of $\mathbb R^n.$ This equation is a kind of an Euler-Lagrange equation of the variational problem of minimizing the functional $$I[v] := \textrm{ess sup} \, |Dv|,$$among all Lipschitz continuous functions $v,$ satisfying a prescribed boundary value on $\partial\Omega.$ The infinity obstacle problem is the minimization problem $$\min \{ I[v]: v \in W^{1,\infty},\ v\geq \psi \}$$for a given function $\psi \in W^{1, \infty}$ which we refer to as the obstacle.In this talk I will discuss an optimal control problem related to the infinity obstacle problem. This is joint work with Cheikh Ndiaye.
Monday April 12, 2021 at 14:30, Zoom meeting
Global hypoellipticity of sums of squares on compact manifolds
Gabriel Araujo, University of Sao Paulo, Sao Carlos
In a recent work with Igor A. Ferra (Federal Univ. of ABC - Brazil) and Luis F. Ragognette (Federal Univ. of Sao Carlos - Brazil) we investigate necessary and sufficient conditions for global hypoellipticity of certain sums of squares of vector fields. Our model is inspired by a rather general one introduced by (Barostichi-Ferra-Petronilho, 2017) when the ambient manifold is a torus; here we extend their results to more general closed manifolds, which requires a new interpretation of their Diophantine conditions (as these are not available in our framework).
Monday April 19, 2021 at 14:30, Zoom meeting
On the multiparameter distance set problem
Yumeng Ou, University of Pennsylvania
In this talk, we will describe some recent progress on the Falconer distance set problem in the multiparameter setting. The original Falconer conjecture (open in all dimensions) says that a compact set $E$ in $\mathbb{R}^d$ must have a distance set $\{|x-y|: x,y\in E\}$ with positive Lebesgue measure provided that the Hausdorff dimension of $E$ is greater than $d/2$. What if the distance set is replaced by a multiparameter distance set? We will discuss some recent results on this question, which is also related to multiparameter projections of fractal measures. This is joint work with Xiumin Du and Ruixiang Zhang.
Monday April 26, 2021 at 14:30, Zoom meeting
The second boundary value problem for a discrete Monge-Ampere equation
Gerard Awanou, University of Illinois, Chicago
In this work we propose a natural discretization of the second boundary condition for the Monge-Ampere equation of geometric optics and optimal transport. It is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.
Monday September 6, 2021 ,
Labor day (no meeting)
Monday September 20, 2021 at 14:30, Wachman 617
On some electroconvection models
Elie Abdo, Temple University
We consider an electroconvection model describing the evolution of a surface charge density interacting with a 2D fluid. We investigate the model on the two-dimensional torus: we study the existence, uniqueness and regularity of solutions, and we show the existence of a global attractor.
Monday September 27, 2021 at 14:30, Wachman 617
Relaxation of functionals with bulk and surface energy terms
Giovanni Gravina, Temple University
The minimization of energy functionals has a wide range of applications both in pure and applied disciplines, where the existence of minimizes is routinely proved by means of the so-called "direct method in the Calculus of Variations". This, in turn, relies on showing that the energy under consideration is lower semicontinuous. If this property fails, valuable insight may still be gained by characterizing the lower semicontinuous envelop of the energy, referred to as the relaxed energy.
Motivated by problems in the van der Waals-Cahn-Hilliard theory of liquid-liquid phase transitions, and by some classical examples due to Modica, in this talk, we will study the lower semicontinuity of energy functionals with bulk and surface terms. Since the presence of corners in the domain can affect the lower semicontinuity of the energies under consideration, we will focus on uncovering how the roughness of the domain enters the relaxation procedure.
Monday October 4, 2021 at 14:30, Wachman 617
Jeongsu Kyeong, Temple University
Among other things, integral identities of Rellich type allow one to deduce the $L^{2}(\partial \Omega)$ equivalence of the tangential derivative and the normal derivative of a harmonic function with a square integrable non-tangential maximal function of its gradient in a given Lipschitz domain $\Omega \subset \mathbb{R}^{n}$. In this survey talk, I will establish the integral identities in $\mathbb{R}^{n}$ and I will illustrate the role that the aforementioned equivalence plays in establishing invertibility properties of singular integral operators of layer potential type associated with the Laplacian in Lipschitz domains in $\mathbb{R}^{2}$, through an interplay between PDE, Harmonic Analysis, and Complex Analysis methods.
Monday October 11, 2021 at 14:30, Zoom meeting
Heat content and geometric analysis
Patrick McDonald, New College of Florida
The heat content associated with a bounded domain in a Riemannian manifold is a function obtained by solving an initial value problem for the heat operator on the domain. Heat content gives rise to a collection of geometric invariants closely related to the Dirichlet spectrum. In this talk I will survey results that compare and contrast the role of heat content invariants to the role of spectral data in geometric analysis. In particular, I will discuss results involving planar polygons and provide explicit examples of where heat content invariants and Dirichlet spectrum behave similarly, and also where they behave differently.
Monday October 18, 2021 at 14:30, Zoom meeting
A first-order approach to solvability for singular Schrödinger equations
Andrew Morris, University of Birmingham, UK
We will first give a brief overview of the first-order approach to boundary value problems, which factorises second-order divergence-form equations into Cauchy-Riemann systems. The advantage is that the holomorphic functional calculus for such systems can provide semigroup solution operators in tremendous generality, extending classical harmonic measure and layer potential representations. We will then show how recent developments now allow for the incorporation of singular perturbations in the associated quadratic estimates. This allows us to solve Dirichlet and Neumann problems for Schr\"odinger equations with potentials in Sobolev-critical Lebesgue spaces and reverse H\"older spaces. This is joint work with Andrew Turner.
Monday October 25, 2021 , Wachman 617
Irina Mitrea, Temple University
.The $L^p$ Dirichlet Problem for constant coefficient second-order systems satisfying the Legendre-Hadamard strong ellipticity condition is well posed in the upper half-space. Surprisingly, this result may fail if only weak ellipticity is assumed, and the failure manifests itself at a fundamental level through lack of Fredholm solvability. In this talk I will discuss a couple of pathological cases, sought in the class of weakly elliptic systems that fail to possess a distinguished coefficient tensor. This is joint work with Dorina Mitrea and Marius Mitrea.
Monday November 1, 2021 at 14:30, Zoom meeting
Wave decay for domains star-shaped with respect to infinity
Tanya Christiansen, University of Missouri
We wish to understand how the geometry of a domain $X\subset {\mathbb R}^d$ affects the decay of solutions to the wave equation on $X$ with Dirichlet boundary conditions.
The case in which $ \mathcal{O}={\mathbb R}^d\setminus X$ is bounded is a classical obstacle scattering problem. In the special case when $\mathcal{O}$ is star-shaped, decay of solutions of the wave equation is a classical result of Morawetz. We study certain sets $X$ which have ${\mathbb R}^d\setminus X$ unbounded. These sets $X$ are unbounded in some directions, and bounded in others. We introduce a notion of "star-shaped with respect to infinity" and show that this condition has implications for the behavior of the resolvent of the Laplacian. For waveguides which are star-shaped with respect to infinity, this implies some wave decay.
This talk is based on joint work with K. Datchev.
Monday November 8, 2021 at 14:30, Zoom meeting
Uniqueness of the $C^*$ norm in strict deformation quantization
Severino Toscano de Rego Melo, University of São Paulo, São Paulo
Abstract: Rieffel's algebra of pseudodifferential operators, introduced in the context of deformation quantization, will be described from the point of view of somebody who is familiar with pseudodifferential operators. Old results about the characterization of pseudodifferential operators as bounded operators with a smooth orbit under the action of the Heisenberg group will also be explained. Finally I will report on a recent joint paper with Cabral and Forger, in which we prove the uniqueness of the $C^*$ norm on Rieffel's algebra.
Monday November 15, 2021 at 14:30, Virtual presentation
Traveling waves close to the Couette flow
Ángel Castro, Instituto de Ciencias Matemáticas, Madrid
In this talk we shall study the existence of smooth traveling waves close to the Couette flow for the 2D incompressible Euler equation for an ideal fluid. It is well known that this kind of solutions do not exist arbitrarily close to the Couette flow if the distance is measured in $H^{3/2+}$. In this presentation we will deal with the case $H^{3/2-}$.
Monday November 22, 2021 ,
No meeting
Monday November 29, 2021 at 14:30, Zoom meeting
The Cauchy–Szegő projection and its commutator for domains in $\mathbb C^n$ with minimal smoothness
Loredana Lanzani, Syracuse University
Let $D\subset \mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani \& Stein states that the Cauchy–Szego projection $\mathscr S_\omega$ defined with respect to any Leray Levi-like measure $\omega$ is bounded in $L^p(bD, \omega)$ for any $1 < p < \infty$. (We point out that for this class of domains, induced Lebesgue measure is Leray Levi-like.) Here we show that $\mathscr S_\omega$ is in fact bounded in $L^p(bD,\Omega_p)$ for any $1 < p < \infty$ and for any $\Omega_p$ in the optimal class of $A_p$ measures, that is $\Omega_p = \psi_p\sigma$ where $\psi_p$ is a Muckenhoupt $A_p$-weight and $\sigma$ is induced Lebesgue measure. As an application, we characterize boundedness and compactness in $L^p(bD,\Omega_p)$ for any $1 < p < \infty$ and for any $A_p$ measure $\Omega_p$, of the commutator $[b, \mathscr S_\omega]$ for any Leray Levi-like measure $\omega$. We next introduce the notion of holomorphic Hardy spaces for $A_p$ measures, $1 < p < \infty$, and we characterize boundedness and compactness in $L^2(bD,\Omega_2)$ of the commutator $[b, \mathscr S_{\Omega_2}]$ of the Cauchy–Szego projection defined with respect to any $A_2$ measure $\Omega_2$. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates of the Cauchy–Szego kernel, but these are unavailable in the settings of minimal regularity of $bD$; at the same time, newer techniques introduced by Lanzani \& Stein to deal with the setting of minimal regularity are not applicable to $A_p$ measures that are not Leray Levi-like. It turns out that the method of extrapolation is an appropriate replacement for the missing tools.
This is joint work with Xuan Thinh Duong (Macquarie University), Ji Li (Macquarie University) and Brett Wick (Washington University in St. Louis).
Monday December 6, 2021 at 14:30, Wachman 617
Dyadic models for fluid equations
Mimi Dai, University of Illinois-Chicago
Inspired by the study of dyadic models for the Navier-Stokes equation, we propose some simplified models for the magnetohydrodynamics in order to have a better understanding on various topics. Pathological solutions are constructed, for instance, solution that blows up at finite time and non-unique Leray-Hopf solutions. Challenging questions will be discussed too. Most of the work is joint with Susan Friedlander.