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Current contact: Gerardo Mendoza
The seminar takes place Mondays 2:40 - 3:30 pm in Wachman 617. Click on title for abstract.
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.
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.
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.
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.
Charles Epstein, University of Pennsylvania
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.
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.
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.
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.
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.
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.
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.
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.
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.
Mariana Smit Vega Garcia, University of Washington
In this talk I will overview the Signorini problem for a divergence form elliptic operator with Lipschitz coefficients, and I will describe a few methods used to tackle two fundamental questions: what is the optimal regularity of the solution, and what can be said about the singular free boundary in the case of zero thin obstacle. The proofs are based on Weiss and Monneau type monotonicity formulas. This is joint work with Nicola Garofalo and Arshak Petrosyan.
Ahmad Sabra, University of Warsaw
In this talk we study the following variational problem: \begin{equation} \inf\Big\{\int_\Omega |Du|:u\in \mathrm{BV}(\Omega),\ u|_{\partial\Omega}=f\Big\}, \end{equation} with $\Omega$ a Lipschitz bounded domain and $f\in L^1(\partial\Omega)$. Solutions to (1) are called least gradient functions and do not always exist for every boundary data $f$. It is well-known that level sets of $\textrm{LG}$ functions are minimal surfaces. Using this geometrical property, we construct solutions to (1) for convex domains $\Omega$ and $f\in \mathrm{BV}(\partial\Omega)$ satisfying some monotonicity properties. We also establish a connection between solutions to (1) and variational problems that appear in Free Material Design applications.
Shif Berhanu, Temple University
We will discuss results on local unique continuation at the boundary for holomorphic functions of one variable and for the solutions of the Helmholtz equation $L_cu=\Delta u+cu=0,\, c\in \mathbb R$ in an open set of the half space $\mathbb R^n_+$ generalizing the theorems proved by Baouendi and Rothschild for harmonic functions. The results involve a local boundary sign condition on the product of the solution and a monomial. Applications to unique continuation for CR mappings will also be discussed.
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