# Analysis Seminar

Current contact: Gerardo Mendoza

The seminar takes place Mondays 2:40 - 3:30 pm in Wachman 617. Click on title for abstract.

• Monday November 13, 2017 at 14:40, Wachman 617
TBA

• Monday November 6, 2017 at 14:40, Wachman 617
TBA

Mariana Smit Vega Garcia, University of Washington

• Monday October 30, 2017 at 14:40, Wachman 617

Pierre Albin, University of Illinois, Urbana-Champaign

Stratified spaces arise naturally even when studying smooth objects, e.g., as algebraic varieties, orbit spaces of smooth group actions, and many moduli spaces. There has recently been a lot of activity developing analysis on these spaces and studying topological invariants such as the signature. I will report on joint work with Jesse Gell-Redman in which we study families of Dirac-type operators on stratified spaces and establish a formula for the Chern character of their index bundle.

• Monday October 16, 2017 at 14:40, Wachman 527

Farhan Abedin, Temple University

I will present some new results concerning regularity of solutions to certain degenerate elliptic and parabolic equations in non-divergence form. In the first half of the talk, I will discuss recent work on Harnack’s inequality for operators structured on Heisenberg vector fields, with coefficients that are uniformly positive definite, continuous, and symplectic. This is joint work with Cristian Gutierrez (Temple) and Giulio Tralli (University of Rome). In the second half, I will outline work in progress with Giulio Tralli on parabolic equations of Kolmogorov type.

• Monday October 9, 2017 at 14:40, Wachman 617

Gustavo Hoepfner, Federal University of São Carlos, Brazil

This is a continuation of last week's talk. The goal of this talk is twofold. Firstly, we will recall the notion of quasi $\ell$ Mizohata vector fields first introduced by L. Nunes and R. dos Santos Filho (UFSCar) and show that it can be written normal form which is very special when we are in $\mathbb R^2$.

Secondly, we will establish a connection of these quasi $\ell$ Mizohata vector fields with recent results by Z. Adwan and S. Berhanu on the solutions of first order nonlinear PDE's and extend them to the classes of ultradifferentiable functions.

This is a joint work with R. Medrado from Universidade Federal do Cear\'a.

• Monday October 2, 2017 at 14:40, Wachman 617

Gustavo Hoepfner, Federal University of São Carlos, Brazil

The goal of this talk is twofold. Firstly, we will recall the notion of quasi $\ell$ Mizohata vector fields first introduced by L. Nunes and R. dos Santos Filho (UFSCar) and show that it can be written normal form which is very special when we are in $\mathbb R^2$.

Secondly, we will establish a connection of these quasi $\ell$ Mizohata vector fields with recent results by Z. Adwan and S. Berhanu on the solutions of first order nonlinear PDE's and extend them to the classes of ultradifferentiable functions.

This is a joint work with R. Medrado from Universidade Federal do Cear\'a.

• Monday September 25, 2017 at 14:40, Wachman 617

Luis Fernando Ragognette, Federal University of São Carlos, Brazil

The goal of this talk is to present results on infinite order differential operators and its applications to local solvability of a differential complex associated to a locally integrable structure in a Gevrey environment.

One of the reasons why infinite order differential operators are important in this setting is a structural theorem that says that every ultradistribution of order $s$ can be locally represented by an infinite order differential operator applied to a Gevrey function of order $s$, this new kind of representation is crucial in several applications that we are going to discuss.

• Monday September 18, 2017 at 14:40, Wachman 617

Blair Davey, City College of New York

In the late 1960s, E.M. Landis made the following conjecture: If $u$ and $V$ are bounded functions, and $u$ is a solution to $\Delta u = V u$ in $\mathbb{R}^n$ that decays like $|u(x)| \le c \exp(- C |x|^{1+})$, then $u$ must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded functions $u, V: \mathbb{R}^2 \to \mathbb{C}$ that solve $\Delta u = V u$ in $\mathbb{R}^2$ and satisfy $|u(x)| \le c \exp(- C |x|^{4/3})$. The result of Meshkov was accompanied by qualitative unique continuation estimates for solutions in $\mathbb{R}^n$. In 2005, J. Bourgain and C. Kenig quantified Meshkov's unique continuation estimates. These results, and the generalizations that followed, have led to a fairly complete understanding of the complex-valued setting. However, there are reasons to believe that Landis' conjecture may be true in the real-valued setting. We will discuss recent progress towards resolving the real-valued version of Landis' conjecture in the plane.

• Monday September 11, 2017 at 14:40, Wachman 617

Irina Mitrea, Temple University.

This talk is focused on Singular Integral Operator methods for boundary value problems for the Bi-Laplacian in irregular domains in the Euclidean Space and is largely motivated by the study of the classical free-plate problem arising in the Kirchhoff-Love theory of thin plates. This is based on joint work with Gustavo Hoepfner, Paulo Liboni and Marius Mitrea.

• Monday May 1, 2017 at 14:40, Wachman 617

Tadele Mengesha, University of Tennessee, Knoxville

Global Calderon-Zygmund type estimates are obtained for solutions to fractional elliptic problems over a smooth domain. Our approach is based on the 'extension problem' where the fractional elliptic operator is realized as a Dirichlet-to-Neumann map to a degenerate elliptic PDE in one more dimension. This approach allows the possibility of deriving estimates for solutions to the fractional elliptic equation from that of a corresponding degenerate elliptic equation. We will confirm this first by obtaining weighted estimates for the gradient of solutions to a class of linear degenerate/singular elliptic problems. The class consists of those with coefficient matrix that is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a particular weight that belongs to a Muckenhoupt class. The weighted estimates are obtained under a smallness condition on the mean oscillation of the coefficients with a weight. This is a joint work with T. Phan.

• Monday April 17, 2017 at 14:40, Wachman 617

Yannick Sire, John Hopkins

I will report on recent work with V. Millot and K. Wang on the singular limit for a fractional Allen-Cahn equation leading to stationary nonlocal minimal surfaces. I will introduce these latter concepts and will prove the convergence result, based on a deep Geometric Theory argument from Marstrand.

• Monday March 20, 2017 at 14:40, Wachman 617

Charles Epstein, University of Pennsylvania

• Monday February 27, 2017 at 14:40, Wachman 617

Federico Tournier, University of La Plata and IAM, Argentina

We look at the local problem in free space and in half space of an elliptic operator with Hölder coefficients.

• Monday February 20, 2017 at 14:40, Wachman 617

Guy David, Courant Institute, New York University

Since the work of Cheeger, many non-smooth metric measure spaces are now known to support a differentiable structure for Lipschitz functions. The talk will discuss this structure on metric measure spaces with quantitative topological control: specifically, spaces whose blowups are topological planes. We show that any differentiable structure on such a space is at most 2-dimensional, and furthermore that if it is 2-dimensional the space is 2-rectifiable. This is partial progress on a question of Kleiner and Schioppa, and is joint work with Bruce Kleiner.

• Monday January 30, 2017 at 14:40, Wachman 617

Irina Mitrea

In this talk I will discuss well-posdness results for the Dirichlet problem for second-order, homogeneous, elliptic systems, with constant complex coefficients, in the upper half space, with boundary data from Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. A key tool in this analysis is establishing boundedness of the Hardy-Littlewood maximal operator on appropriate Köthe function spaces. This is joint work with Dorina Mitrea, Marius Mitrea and Jose Maria Martell.

• Monday January 23, 2017 at 14:40, Wachman 617

Scott Armstrong, NYU

I will describe recent developments in the quantitative homogenization of linear elliptic equations in divergence form, emphasizing some ideas arising from the calculus of variations and the role played by a new elliptic regularity theory for equations with random coefficients.