# Analysis Seminar

Current contact: Gerardo Mendoza

The seminar takes place Mondays 2:30 - 3:30 pm virtually via Zoom. Please write if you wish to join for a specific meeting. Click on title for abstract.

• Monday January 25, 2021 at 14:30, Zoom meeting

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

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

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

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

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 relating

the 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

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

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

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

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.