Analysis Seminar

Higher order rectifiability and Reifenberg parametrizations

Silvia Ghinassi
Member, School of Mathematics
March 9, 2020
We provide geometric sufficient conditions for Reifenberg flat sets of any integer dimension in Euclidean space to be parametrized by a Lipschitz map with Hölder derivatives. The conditions use a Jones type square function and all statements are quantitative in that the Hölder and Lipschitz constants of the parametrizations depend on such a function. We use these results to prove sufficient conditions for higher order rectifiability of sets and measures.

The singular set in the fully nonlinear obstacle problem

Ovidiu Savin
Columbia University
November 18, 2019

For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1,\log^\eps}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.

The Surface Quasigeostrophic equation on the sphere

Angel Martinez Martinez
Member, School of Mathematics
October 28, 2019

In this talk I will describe joint work with D. Alonso-Orán and A. Córdoba where we extend a result, proved independently by Kiselev-Nazarov-Volberg and Caffarelli-Vasseur, for the critical dissipative SQG equation on a two dimensional sphere. The proof relies on De Giorgi technique following Caffarelli-Vasseur intermingled with a nonlinear maximum principle that appeared later in the approach of Constantin-Vicol. The final result can be paraphrased as follows: if the data is sufficiently smooth initially then it is smooth for all times.

Strong ill-posedness of the logarithmically regularized 2D Euler equations in the borderline Sobolev space

Hyunju Kwon
Member, School of Mathematics
October 21, 2019

The well-posedness of the incompressible Euler equations in borderline spaces has attracted much attention in recent years. To understand the behavior of solutions in these spaces, the logarithmically regularized Euler equations were introduced. In borderline Sobolev spaces, local well-posedness was proved Chae-Wu when the regularization is sufficiently strong, while strong ill-posedness of the unregularized case was established by Bourgain-Li. In this talk, I will discuss the strong ill-posedness of the remaining intermediate regime of regularization.