May 28 - 31, 2015
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.
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: 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.