Analysis Seminar

Given a compact set $E$ in $\mathbb{R}^d$ and a point $y \in E$, define the pinned distance set at $y$ as $\Delta^y(E)=\{|x-y|\colon x\in E\}.$ Peres and Schlag proved that if the Hausdorff dimension of $E$ is larger than $(d+2)/2$, then there exists $y \in E$ such that $\Delta^y(E)$ has nonempty interior. However, such a threshold is only useful for dimensions at least 3 and does not give information when $d=2$. In this talk, we will discuss how local smoothing estimates for the wave equation can be used to get a nontrivial threshold for such a problem in the plane and to improve the known threshold in $d=3$. We proved that if $E$ is a compact subset of the plane with Hausdorff dimension at least 7/4, then there exists $x \in E$ such that the pinned distance set at $x$ contains an interval. This talk is based on joint work with B. Foster, Y. Ou, and E. Palsson.

Abstract: We review recent developments in the theory of weak convergence of pde-constrained sequences. We consider the weak lower semicontinuity problem along weakly convergent $\mathcal{A}$-free sequences, where $\mathcal{A}$ is a linear pde system of constant rank and provide improvements of the $\mathcal{A}$-quasiconvexity theory of Fonseca-Müller and compensated compactness theory of Murat-Tartar. Special emphasis will be placed on concentration effects of weak convergence, in particular by presenting the resolution of a question due to Coifman-PL Lions-Meyer-Semmes, which leads to a recent connection between quasiconvexity and higher integrability.  Joint work with André Guerra, Jan Kristensen, Matthew Schrecker.