Analysis & PDE

Abstract: The catenoid, which is a minimal surface, can be viewed as a stationary solution of the hyperbolic vanishing mean curvature equation in Minkowski spacetime. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in…

I will present a result obtain in collaboration with Hitoshi Ishii and Ariela Briani, that studies the asymptotic behaviors of solutions of fully nonlinear equations in thin domains with oblique boundary conditions with a test function approach à la  Evans. The limit equation…

Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock…

I will present several results on linear and nonlinear models, relating to the problem of stability/instability of the exterior of extremal Reissner-Nordstrom black hole spacetimes.

Abstract: The existence of multi black hole solutions in asymptotically flat spacetimes is one of the expectations from the final state conjecture. In this talk, I will present preliminary works towards showing the existence of multi black hole solutions via a semilinear toy model in dimension 3…

Abstract:

Quantum harmonic analysis on phase space uses representations of the Heisenberg group to define analogs of the Fourier transform and of convolutions for bounded operators, and where the Schatten classes of compact operators play the role of the Lebesgue spaces. We will briefly…

Abstract:

We study the spectral inequalities of Schödinger operators for polynomial type growth potentials in the whole space. The spectral inequalities quantitatively depend on the density of the sensor sets, growth rate of the potentials and spectrum (or eigenvalues). We are able to…

Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent…

Abstract: I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range. Central to the framework is a new formulation of nonlinear gravitational perturbations of Kerr in a geometric gauge tailored to the…

Abstract: Extremal black holes are special solutions of Einstein’s equations which have absolute zero temperature in the thermodynamic analogy of black hole mechanics. In this talk, I will present a proof that extremal black holes arise on the critical threshold between gravitational collapse…