Skip to main content

Analysis and PDE

Analysis and PDE seminar

Gradient Corrections in Atomic Physics

 

During the last few years there has been a systematic pursuit for sharp estimates 

of the energy components of atomic systems in terms of their single particle density. 

The common feature of these estimates is that they include corrections that 

depend on the gradient of the density. In this talk I will review these results. 

The most recent result is the sharp estimate of P.T. Nam on the kinetic energy. 

I will also present some recent results concerning geometric estimates 

for generalized Poincaré inequalities obtained in collaboration with C. Vallejos and H. Van Den Bosch. 

These geometric estimates are a useful tool to estimate the numerical value of the constant of 

Nam's gradient correction term.

Date:
-
Location:
POT 745
Tags/Keywords:

Analysis and PDE seminar

Title: Steady State of Rotating Stars and Galaxies

Abstract: The equilibrium shape and density distribution of rotating fluid under self-gravitation is a classical problem in mathematical physics. Early efforts in and before the nineteenth century discovered ellipsoidal shapes with constant density. In the twentieth century, major progress was made by studying steady rotating solutions to the compressible Euler-Poisson equations. Two methods of constructing solutions have been used. Assuming a polytropic equation of state $p=\rho^\gamma$, a variational method, pioneered by the work of Auchmuty and Beals, proves existence of solutions if $\gamma>\frac43$. On the other hand, we present in this talk a perturbative method that establishes specially structured solutions for $\gamma>\frac65$. The method is built upon an old work of Lichtenstein. We also examine analogous results for the Vlasov-Poisson equations and for magnetic stars, and other extensions.

 

Date:
-
Location:
POT 745
Tags/Keywords:

Analysis and PDE seminar

Title: Localization and delocalization for two interacting 1D quasiperiodic particles.

 

Abstract: The talk is about several tentative results, joint with J. Bourgain and S. Jitomirskaya. We consider a model of two 1D almost Mathieu particles with a finite range interaction. The presence of interaction makes it difficult to separate the variables, and hence the only known approach is to treat it as a 2D model, restricted to a range of parameters (both frequencies and phases of the particles need to be equal). In the usual 2D approach, a positive measure set of frequency vectors is usually removed, and extra care needs to be taken in order to keep the diagonal frequencies (which is a zero measure set) from being removed. We show that the localization holds at large disorder for energies separated from zero and from certain values associated to the interaction.

 

We also study the model in the regime of strong interaction, in which case an additional band of spectrum (“droplet band”) is created. We show that this droplet band is localized in the regime of large interaction and fixed difference between phases (in particular, it covers the ``physical’’ regime of equal phases). However, there is another regime where the difference between phases is close to pi/2, in which case the droplet band has some ac spectrum.

Date:
-
Location:
POT 745
Tags/Keywords:

Analysis and PDE seminar

Title: ​At Last: Astala & Paivarinta's Solution to Calderon's Problem 

Abstract: The Inverse Conductivity Problem was first posed by A.P. Calder\'on in 1980. Let $\Omega \subset \mathbb{C}$ be a simply connected domain.  Consider the following Dirichlet problem

\begin{equation*}

\begin{cases}

\nabla \cdot (\sigma \nabla u) = 0 & \text{ in } \Omega \\

u = f & \text{ on } \partial \Omega

\end{cases}

\end{equation*}

and the Dirichlet-to-Neumann operator $\Lambda_\sigma$,

\begin{equation*}

\Lambda_{\sigma} f = \left. \sigma \frac{\partial u}{\partial \nu}\right|_{\partial \Omega}

\end{equation*}

The question Calder\'on posed was: is a conductivity $\sigma \in L^\infty(\mathbb{C})$ uniquely determined by its Dirichlet-to-Neumann operator $\Lambda_\sigma$?  The focus of this talk is the solution found by Kari Astala and Lassi P\"aiv\"arinta in 2006, where they study a Beltrami equation related to the conductivity problem through a change of variable.  Using the properties of quasiconformal maps and a D-bar method, they define a nonlinear Fourier transform and use its properties to obtain uniqueness.  This is also a shameless advertisement for Francis Chung's upcoming topics course on inverse problems.

Date:
-
Location:
POT 745
Tags/Keywords:

Analysis and PDE

Title: Logarithmic decay of waves on spacetimes bounded by event horizons.

 

Abstract: I will describe the decay of linear waves on a class of stationary spacetimes bounded by non-degenerate Killing horizons. Without any assumptions on the trapped set solutions of the wave equation exhibit logarithmic energy decay. This is analogous to well known results in other geometric settings. The proof follows from new high frequency bounds on the resolvent.

Date:
-
Location:
POT 745
Tags/Keywords:

Analysis and PDE seminar

Global Well-Posedness of the defocussing Davey-Stewartson II Equation: The Work of Nachman-Regev-Tataru

The defocussing Davey-Stewartson II equation is a dispersive nonlinear equation which describes weakly nonlinear surface waves on a two-dimensional surface. In a remarkable paper (see https://arxiv.org/abs/1708.04759), Nachman, Regev, and Tataru use ideas of inverse scattering and harmonic analysis to prove global well-posedness and scattering in $L^2$. We'll describe key elements of their result including (i) new estimates on fractional integrals and pseudo differential operators, (ii) resolvent bounds via concentration compactness, and (iii) Lipschitz estimates on the scattering transform. A key role is played in the estimates by the Hardy-Littlewood maximal function. 

Date:
-
Location:
POT 745
Tags/Keywords:
Subscribe to Analysis and PDE