# Poster Session and Reception

Friday, May 29, 2015 - 3:30pm - 5:00pm

Lind 400

**Recent Developments in Multi-parameter Singular Integrals**

Yumeng Ou (Brown University)

In this poster, we survey several recent results on the Calder\'on-Zygmund theory of multi-parameter singular integrals, with one of the most classical examples being the tensor product of Hilbert transforms. We develop a mixed type characterization of the multi-parameter Calder\'on-Zygmund operator class that has been studied by Journ\'e, and prove for this class of operators a dyadic representation theorem. As an application, new results on iterated commutators of singular integrals with BMO symbols are obtained. And a T(b) theorem on product spaces is proved in a similar fashion as well.**Convergence of a Semidiscrete Scheme for a Forward-backward Parabolic Equation**

Carina Geldhauser (Rheinische Friedrich-Wilhelms-Universität Bonn)

We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation $u_t = (W'(u_x))_x$

with periodic boundary conditions in one space dimension, where $W$ is a standard double-well potential. We characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. We also characterize the long-time behavior of the limit solutions.

Using an approximation argument, we show that it is possible to flow initial data with derivative in the concave region ${ W**Gradient Flow of Chevron Structures in Liquid Crystal Cells**

Lidia Mrad (Purdue University)

Liquid crystal phases are understood to be intermediate states between a liquid and a crystalline. Possessing both flow-like properties of liquids and molecular packing of solids, liquid crystals are widely used in optical devices. In addition to having a specific orientation, molecules in the Smectic-C phase tend to organize in layered structures. In a surface-stabilized cell however, the uniform layers deform into V-shaped layers, referred to as a chevron structure, causing distortions. Moreover, the helical structure of the chiral Smectic-C is suppressed in the thin cell which allows for two stable states.

In this poster, we present molecular reorientation dynamics between these stable states under the influence of an applied electric field. Our model is based on the Chen-Lubensky energy involving the molecular director and a general complex-valued layer order parameter. We construct a discrete-in-time gradient flow through an iterative minimization procedure. To establish the existence of a solution to this system of parabolic equations, we use elliptic estimates and prove discrete-to-continuous convergence of the gradient flow. This implicit time-discretization method, called Rothe method, has a strong numerical aspect and gives an insight into the structure of the solution unlike some abstract methods for the analysis of existence problems.**Domain Decomposition Methods for Problems Describing Materials with Complex Geometry and Physics**

Daria Kurzanova (University of Houston)

The point of interest are so-called high contrast composite materials. High contrast means that the ratio between highest and lowest values of the parameter, that describes materials property, is very high, even infinite. Another key aspect is a complex geometry of materials of interest. Mathematical modeling of such materials yields in a PDE that has roughly oscillating coefficients with large variability in the domain. Solving such problems by traditional numerical methods is expensive due to a fine mesh needed in thin regions of the computational domain. The main goal is a numerical treatment of problems associated with described materials. The novelty idea is to take advantage of properties of structured materials to build new numerically efficient schemes. In particular, one of the focuses of proposed research is on the domain decomposition methods for the problems that describe media whose parts might have high contrast constituents. The key step here is to split a large domain into subdomains in a natural way to deal with homogeneous and high contrast parts. With that, we obtain a coupled problem where subdomains are bridged though the interface. Then, one can build an iterative method based on the resulted partition.**Vector-Valued Extensions for Bilinear Operators**

Cristina Benea (Cornell University)

The Bilinear Hilbert Transform(BHT) exhibits a one-dimensional modulation invariance, which places it half way between singular (multi-)linear operators and Carleson's operator. The study of BHT in the 90s was the beginning of time-frequency analysis.

A question regarding a specific system of ODEs led to an operator resembling Rubio de Francia's square function: that is, an operator given by Fourier projections of BHT associated with arbitrary intervals. This further led to vector-valued extensions for BHT, for which some partial results were already known. Similarly, one can get vector-valued extensions for paraproducts, and as a consequence, estimates for tensor products of paraproducts on L^p spaces with mixed norms.**On Monotonicity for Strain-Limiting Theories of Elasticity**

Tina Mai (Texas A & M University)

This presentation addresses certain notions of convexity for strain-limiting theories of elasticity in which the Green-St.Venant strain tensor is written as a nonlinear response function of the second Piola-Kirchhoff stress tensor. Previous results on strong ellipticity for special strain-limiting theories of elasticity required invertibility of the Frechet derivative of the response function as a fourth-order tensor. The present contribution generalizes the theory to cases in which the Frechet derivative of the response function is not invertible, by studying a weaker rank-1 convexity notion, monotonicity, applied to a general class of nonlinear strain-limiting models. It is shown that the generalized monotonicity holds for Green St. Venant strains with sufficiently small norms, and fails (through demonstration by counterexample) when the small strain constraint is relaxed.