### Arbeitsgruppe Analysis und Partielle Differentialgleichungen

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

The seminar may occasionally take place online on the Zoom platform. In case, the Zoom details will be comunicated separately.

Please join the pdg-l mailing list or contact Gael Diebou Yomgne for further information.

## Next weeks

### January 14 - Gianmarco Brocchi (University of Birmingham)

### Title:

Two weight estimates for the Bergman projection

### Abstract:

The space of holomorphic functions which are $L^2$-integrable on a domain are at the cross-road of Operator Theory, Harmonic Analysis and Complex analysis in several variables.An active direction of research aims to better understand the projection from $L^2$ onto this space in terms of weighted estimates. Several weighted estimates can be obtained by using a powerful technique in Harmonic Analysis called sparse domination. Sparse domination consists in controlling a non-local, non-positive operator by a sum of local, positive averages. As a consequence, plenty of quantitative weighted estimates are easily derived, and they are often optimal. In this talk we introduce the concept of sparse domination and show how this tool can be used to derive sufficient conditions for the

*two weight*inequality for the Bergman projection on the unit ball.

### January 21 - Armin Schikorra (University of Pittsburgh)

### Title:

A Harmonic Analysis perspective on $W^{s,p}$ as $s \to 1^-$

### Abstract:

We revisit the Bourgain-Brezis-Mironescu result that the Gagliardo-Norm of the fractional Sobolev space $W^{s,p}$, up to rescaling, converges to $W^{1,p}$ as $s\to 1$. We do so from the perspective of Triebel-Lizorkin spaces, by finding sharp $s$-dependencies for several embeddings between $W^{s,p}$ and $F^{s,p}_{q}$ where $q$ is either 2 or $p$. We recover known results, find a few new estimates, and discuss some open questions.Joint work with Denis Brazke, Po-Lam Yung.

### January 28 - Xian Liao, KIT (To be confirmed!)

### Title:

TBA

### Abstract:

TBA

### February 04 - Anna Balci (Universität Bielefeld)

### Title:

Lavrentiev gap and regularity of PDE: a differential forms perspective, borderline cases and numerical analysis.

### Abstract:

We present new density results and examples on Lavrentiev gap for general classes of differential forms using fractal contact sets. We design the finite element scheme to study numerically so called W-minimizers for such kind of problems.This talk is based on several joint works with Lars Diening (Bielefeld), Mikhail Surnachev (Keldysh Institute of Applied Mathematics, Moscow), Johannes Storn(Bielefeld) and Christoph Ortner(UBC, Vancouver).

## Previous weeks

### December 17 - Rachel Greenfeld, UCLA

### Title:

The periodic tiling conjecture

### Abstract:

A finite subset $F$ of a finitely generated abelian group $G$ is a translational tile of $G$ if it is possible to partition $G$ into translates of $F$. We will motivate and discuss the study of the periodic tiling conjecture, which asserts that any translational tile of $G$ admits a periodic tiling. We will present a new structure theorem for translational tilings, which implies new results regarding the periodic tiling conjecture. This is joint work with Terence Tao.

### December 10 - Christian Brennecke, Bonn Junior Fellow (Universität Bonn)

### Title:

Low-Energy Spectrum of Bose-Einstein Condensates in the Gross-Pitaevskii Regime

### Abstract:

In this talk, I will consider systems of $N$ trapped bosons in $\mathbb{R}^3$ that interact through a two-body potential whose scattering length is of order $N^{-1}$, characterizing the so-called Gross-Pitaevskii regime. At low energy, such systems form a Bose-Einstein condensate. I will review dynamical and spectral properties of such Bose-Einstein condensates, describing in particular the low-energy excitation spectrum of the system, up to errors that vanish in the limit $N\to\infty$.The talk is based on joint work with B. Schlein and S. Schraven.

### December 3 - Micheal Winkler (Universität Paderdorn)

### Title:

Gagliardo-Nirenberg-type inequalities for rapidly decreasing functions and applications in the large time analysis of nonlinear heat flows.

### Abstract:

The objective of the presentation is to discuss options of Gagliardo-Nirenberg type interpolation in situations when the considered functions are known to decay faster than algebraic in space. A result in this regard, obtained in joint work with Marek Fila, in fact establishes a quantitative connection between Lebesgue and first-order Sobolev norms of such functions, where in contrast to the classical case of algebraic decay this connection involves a non-algebraic correction. This new interpolation inequality is thereafter applied to derive temporal decay rates for solutions rapidly decreasing in space to some nonlinear diffusion equations of fast diffusion type. As these rates can be seen to be essentially optimal, this a posteriori shows optimality of the obtained interpolation results.

### November 26 - Jan Bohr (University of Cambridge)

### Title:

The transport Oka-Grauert principle for simple surfaces.

### Abstract:

In the talk we will consider matrix valued transport equations on simple Riemannian surfaces. The main result I want to explain establishes the existence of solutions with controlled Fourier modes in their velocities, so called matrix holomorphic integrating factors. This is applied to characterise the range of an associated nonlinear X-ray transform. The result can further be interpreted as non-existence result for holomorphic vector bundles on a complex surface, reminiscent of the classical Oka-Grauert principle.The talk is based on joint work with Gabriel Paternain.

### November 19 - Marco Fraccaroli (Universität Bonn)

### Title:

Outer $L^p$ spaces: K\"{o}the duality, Minkowski inequality and more.

### Abstract:

The theory of $L^p$ spaces for outer measures, or outer $L^p$ spaces, was developed by Do and Thiele to encode the proof of boundedness of certain multilinear operators in a streamlined argument. Accordingly to this purpose, the theory was developed in the direction of the real interpolation features of these spaces, such as versions of H\"{o}lder's inequality and Marcinkiewicz interpolation, while other questions remained untouched.For example, the outer $L^p$ spaces are defined by quasi-norms generalizing the classical mixed $L^p$ norms on sets with a Cartesian product structure; it is then natural to ask whether in arbitrary settings the outer $L^p$ quasi-norms are equivalent to norms and what other reasonable properties they satisfy, e.g. K\"{o}the duality and Minkowski inequality. In this talk, we will answer these questions, with a particular focus on two specific settings on the collection of dyadic intervals in $\mathbb{R}$ and the collection of dyadic Heisenberg boxes in $\mathbb{R}^2$. This will allow us to clarify the relation between outer $L^p$ spaces and tent spaces, and get a glimpse at the use of this language in the proof of boundedness of prototypical multilinear operators with invariances.

### November 12 - Phuc Cong Nguyen (Louisiana State University)

### Title:

Choquet Integrals, Capacitary Inequalities, and the Hardy-Littlewood Maximal Function

### Abstract:

We obtain the boundedness of the Hardy-Littlewood maximal function on $L^q$ type spaces defined via Choquet integrals associate to Sobolev capacities. The bounds are obtained in full range of exponents including a weak type end-point bound. We also obtain a capacitary inequality of Maz’ya type which resolves a problem proposed by D. Adams.This talk is based on joint work with Keng Hao Ooi.

### November 05 - Giuseppe Negro (University of Birmingham)

### Title:

Stability of sharp Fourier restriction to spheres.

### Abstract:

In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$ (that is, the classical Stein--Tomas estimate).Joint work with E.Carneiro and D. Oliveira e Silva.

### October 29 - Olli Saari (Universität Bonn)

### Title:

A right inverse of the divergence in time-dependent non-cylindrical domains

### Abstract:

Consider an open and connected set in the (1+n)-dimensional space time. We discuss a construction of a right inverse of divergence that respects vanishing boundary values at the lateral boundary. The domain is assumed to be Hölder regular both in time and space variables, and depending on the actual regularity, a priori estimates in Sobolev spaces of positive and negative order can be proved. In particular, we show that the construction exhibits a non-trivial amount of regularity in time variable, which is of interest from the point of view of fluid mechanics. This is based on joint work with S. Schwarzacher.

### October 22 - Dongxiao Yu (Universität Bonn)

### Title:

Asymptotic completeness for a scalar quasilinear wave equation satisfying the weak null condition

### Abstract:

I will discuss an asymptotic completeness result for a scalar quasilinear wave equation in three space dimensions. This equation satisfies the weak null condition introduced by Lindblad and Rodnianski, and it admits small data global existence which was proved by Lindblad. In this talk, I will first introduce a new notion of asymptotic profile by deriving a new reduced system for the model equation. I will then present a proof of the asymptotic completeness. In other words, given a global solution to the model equation, I will show how to find the corresponding asymptotic profile.

### October 15 - Organisational meeting

## Aktuelles

Corona: Maßnahmen im Mathematik-Zentrum

Hausdorff Edition „Felix Hausdorff - Gesammelte Werke“

Prof. Catharina Stroppel ist *plenary speaker* beim ICM 2022 in Sankt Petersburg

Bonner Mathematik belegt bei Shanghai Ranking den 1. Platz in Deutschland und weltweit den 13. Platz

Prof. Georg Oberdieck erhält Heinz Maier-Leibnitz-Preise 2020

Prof. Daniel Huybrechts erhält gemeinsam mit Debarre, Macri und Voisin ERC Synergy Grant