May 28 - 31, 2015
The theory of boundary-value problems for the Laplacian in Lipschitz domains is by now very well developed. Furthermore, many of the existing tools and known results for the Laplacian have been extended to the case of second-order linear equations of the form div (A grad u), where A is a matrix of variable coefficients.
However, at present there are many open questions in the theory of higher-order elliptic differential equations. I will describe a generalization of layer potentials to the case of higher-order operators in divergence form; layer potentials are a common and very useful tool in the theory of second-order equations. I will then describe some applications of layer potentials to the theory of boundary-value problems.
This is joint work with Steve Hofmann and Svitlana Mayboroda.
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.
Keywords of the presentation: decay estimate, water wave equation
In this talk, we discuss recent work concerning a decay estimate for solutions to the linear dispersive equation iu_t-(-Delta)^{1/4}u=0, for (t,x)in R times R, which corresponds to a factorization of the linearized two-dimensional water wave equation. Our arguments are based on a Littlewood-Paley decomposition and stationary phase estimates, and we obtain optimal decay of order |t|^{-1/2} for solutions, assuming only bounds in L^2-based spaces (with weights).
Keywords of the presentation: Variational inequalities, free boundary problems, Signorini problem, semi-permeable walls
In this talk we will present an overview of regularity results (both for the
solution and for the free boundary) in a class of problems which arises in
permeability theory. We will mostly focus on the parabolic Signorini (or thin obstacle) problem, and discuss the modern approach to this classical problem, based on several families of monotonicity formulas. In particular, we will present the optimal regularity of the solution, the classi?cation of free boundary points, the regularity of the regular set, and the structure of the singular set. These results have been obtained in joint work with N. Garofalo, A. Petrosyan, and T. To.
We will also discuss the regularity of solutions in a related model arising in problems of semi-permeable walls and of temperature control. This is joint work with T. Backing.
Keywords of the presentation: elliptic theory, parabolic theory, Carleman estimates
Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we demonstrate a method for producing parabolic theorems from their elliptic analogues. Specifically, we show that an L^2 Carleman estimate for the heat operator may be obtained by taking a high-dimensional limit of L^2 Carleman estimates for the Laplacian. Other applications of this technique will be indicated.
Keywords of the presentation: Free boundary problems, regularity theory.
In this talk we will provide an overview of the regularity theory for a one-phase (Bernoulli) free boundary problem. We will describe several results which parallel the regularity theory of minimal surfaces, including flatness theorems, monotonicity formulae, regularity in low dimensions, a priori gradient bounds. We will then talk about the thin one-phase problem, that can be viewed as a non-local version of the classical one-phase problem. We will highlight the differences in the strategy developed to investigate regularity issues with respect to the classical case.
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 ${ W''< 0}$ of $W$, where the backward character of the equation manifests.
Keywords of the presentation: T1 theorem, singular integral, non-homogeneous, haar functions
We prove the $T1$ theorem for singular integrals with general Calderon-Zygmund kernels in the non-homogeneous setting by modifying the dyadic representation theorem in a convenient manner. This presentation is based on a joint work with T. Hytonen.
Keywords of the presentation: NLS, probability, statistical mechanics
Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body dynamics, a step towards large deviations for Bose-Einstein condensation.
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.
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.
Keywords of the presentation: elliptic equations, polyhedral domain, finite element method
I discuss well-posedness and regularity for strongly elliptic, linear systems, like the system of elasticity, on polyhedral domains using weighted Sobolev spaces. The domain need not be convex and can have curvilinear sides. If time permits, I will discuss how the regularity estimates for the solution can be used to study convergence of the Finite Element Method, a widely used numerical method to solve PDEs.
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.
Keywords of the presentation: Periodic NLS, supercritical regime, random data
In this talk we first review recent progress in the study of certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem) which stems from incorporating to the deterministic toolbox, powerful but still classical tools from probability as well. We will explain some of these ideas and describe in more detail joint work with Gigliola Staffilani on the almost sure well-posedness for the periodic 3D quintic nonlinear Schrodinger equation in the supercritical regime; that is, below the critical space $H^1(mathbb T^3)$.
Keywords of the presentation: Homogenization, Bernoulli Problem, Plateau Problem
In this talk, we present some recent results on the regularity and the asymptotic behavior of some geometric variational problems. First analysis is about a free boundary value problem in random media that is motivated by the one-phase Bernoulli problem. Second analysis is about the classical problem of Plateau with a highly oscillatory boundary condition. This part of the talk will present mostly open problems that are emerged from some interesting behaviors of solutions in different settings.
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.
Keywords of the presentation: quantum many body systems, nonlinear dispersive PDE
The derivation of nonlinear dispersive PDE, such as the nonlinear
Schr"{o}dinger (NLS) or nonlinear Hartree equations, from many body quantum
dynamics is a central topic in mathematical physics, which has been approached
by many authors in a variety of ways. In particular, one way to derive NLS is
via the Gross-Pitaevskii (GP) hierarchy, which is a coupled system of linear
non-homogeneous PDE that describes the dynamics of a gas of infinitely many
interacting bosons, while at the same time retains some of the features of a
dispersive PDE.
In this talk we will discuss the process of going from a quantum many body
system of bosons to the NLS via the GP. Also we will look into what a nonlinear
PDE such as the NLS can teach us about the GP hierarchy and quantum many body
systems.
The talk is based on joint works with T. Chen, C. Hainzl, R. Seiringer and N.
Tzirakis.
There has been recent interest in the use of PDEs to gain understanding of various complex systems in the social sciences. In this talk I will introduce a system modeling social segregation. We analyze this system to understand the effect of social preference, economic disparity, and the environment on segregation. We discuss the existence of steady-state solutions as well as the local and global well-posedness of the system. These results are of broader interest as similar systems arise in the modeling opinion dynamics and rioting activity among other phenomena.
Keywords of the presentation: Signorini problem, thin obstacle problem, free boundary, regularity
We will describe the Signorini, or lower-dimensional obstacle problem, for a uniformly elliptic, divergence form operator L = div(A(x)nabla) with Lipschitz continuous coefficients. We will give an overview of this problem and discuss some recent developments, including the optimal regularity of the solution and the $C^{1,alpha}$ regularity of the regular part of the free boundary. These are obtained by proving a new monotonicity formula for a generalization of the celebrated Almgren’s frequency functional, a new Weiss type monotonicity formula and an appropriate version of the epiperimetric inequality.
This is joint work with Nicola Garofalo and Arshak Petrosyan.
In the first part of these lectures I will present certain, by now classic, results on local and global well-posedness for the NLS via Strichartz estimates.
In the second part I will show how one can use randomization of the initial data to prove well-posedness almost surely even when the problem lack enough regularity for a more deterministic approach. In this context I will also introduce certain Gibbs type measures and how their invariance can be used in certain cases to extend local
solutions to a global ones.
In the 1920's Besicovitch studied linearly measurable sets in the plane, that is sets with locally finite "length". The basic question he addressed was whether the infinitesimal properties of the "length" of a set E in the plane yield geometric information on E itself. This simple question marks the beginning of the study of the geometry of measures and the associated field known as Geometric Measure Theory (GMT).
In this series of lectures we will present some of the main results in the area concerning the regularity of the support of a measure in terms of the behavior of its density or in terms of its tangent structure. We will discuss applications to PDEs, free boundary regularity problem and harmonic analysis. The aim is that the mini-course will be self contained.