Arbeitsgruppe Analysis und Partielle Differentialgleichungen

S5B1: Graduate Seminar on Advanced Topics in PDE

Winter term 2016/2017

We discuss various topics in Analysis and PDE. The seminar may lead to research topics in the area that can lead to a Master or PhD thesis. Students interested in participating should contact one of the organizers.



  • Friday, 14:15 - 16:00, SR 0.006 (Endenicher Allee 60)

Schedule and abstracts

This seminar in the Bonn Mathematical Events calendar
No talk
Athanasios Chatzikaleas
Blow-up of co-rotational wave maps in odd dimensions
We study wave maps from the (1+d)-dimensional Minkowski space to the d-sphere and in particular we prove the asymptotic non-linear stability of the ground-state solution.
No talk
Olli Saari
Regularity of the maximal function and Poincaré inequalities
We study regularity properties of the Hardy-Littlewood maximal function by means of Poincaré type inequalities. The main result gives a unified approach to proving boundedness in BMO, Lipschitz, and Sobolev spaces. In addition, we discuss the open problem about the Sobolev endpoint p=1.
Yang Lan
On asymptotic dynamics for L2 critical generalized KdV equations with a saturated perturbation
We consider the L2 critical gKdV equation with a saturated perturbation. For any initial data in H1, the corresponding solution is always global and bounded in H1. This equation has a family of solitons, and our goal is to study the behavior of solutions with initial data near the soliton. Together with a suitable decay assumption, there are only 3 possibilities:
  1. the solution converges asymptotically to a solitary wave;
  2. the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at +∞;
  3. the solution leaves any small neighborhood of the modulated family of the solitary waves.
This result can be viewed as a perturbation of the rigidity dynamics near ground state for L2 critical gKdV equations proved by Martel, Merle and Raphaël.
Joris Roos
Maximal operators related to curves in the plane
The talk will consist of two parts. In the first part, the discussion will focus on an analogue of Carleson's operator associated with integration along a monomial curve. In that context it is natural to ask whether the methods of time-frequency analysis carry over to an anisotropic setting. We answer that question and also provide certain partial bounds for the Carleson operator along monomial curves using entirely different methods. In the second part, I will present some results for maximal operators and Hilbert transforms along variable curves. Apart from the intrinsic interest in these operators, another motivation stems from Zygmund's conjecture on differentiation along Lipschitz vector fields. In particular, we can prove a curved variant of the conjecture.
Yi Zhang
Planar Sobolev extension domains
A full geometric characterization for a bounded simply connected planar domain to be a W^{1,p} - extension domain, for p = 2, was established around 30 years ago. The case of p > 2 was completed in 2010. We describe the solution for the remaining case 1 < p < 2, and discuss some follow-up work.
Polona Durcik
Entangled multilinear forms and applications
I will discuss boundedness of some multilinear singular integral forms with applications to sharp quantitative norm convergence of ergodic averages with respect to two commuting transformations, quantitative cancellation estimates for the simplex Hilbert transform, and a problem on side lengths of corners in dense subsets of the Euclidean space.
Festkolloquium für Prof. Dr. Werner Ballmann, Lipschitz-Saal, 16:30 (tea at 16:00).
Winfried Sickel
Pointwise multipliers for Besov spaces
Colloquium in commemoration of Felix Hausdorff, Lipschitz-Saal, 15:00 (tea at 14:45).
Blazej Wrobel
Dimension-free Lp estimates for vectors of Riesz transforms associated with orthogonal expansions
An explicit Bellman function is used to prove a bilinear embedding theorem for operators associated with general multi-dimensional orthogonal expansions on product spaces. This is then applied to obtain Lp, 1 < p < ∞, boundedness of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the Lp norms of these Riesz transforms are both dimension-free and linear in max(p,p/(p-1)). The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.
Yuan Zhou
Regularity of viscosity solutions to Aronsson equations
In this talk, I will recall the recent progress on the regularity of infinity harmonic functions by Savin, Evans and Smart, and then present our recent result on viscosity solutions to a class of Aronsson equations (joint with J. Siljander and C. Wang). Several related open questions will also be discussed here.
Marco Fraccaroli
On distributions with GL2(ℝ) dilation symmetry
In the paper On a trilinear integral form with determinental kernel by Gressman, He, Kova\v c, Street, Thiele, and Yung, the authors studied a tempered distribution satisfying some invariance conditions. The question about its uniqueness gives rise to the study of GLn(ℝ)-homogeneity for tempered distributions in ℝ. In dimension n=1, the degree of homogeneity is known to identify the tempered distributions. In this talk I will present a complete classification theorem in dimension n=2 we managed to prove.