Mathematical Institute of the University of Bonn

RG Analysis and Partial Differential Equations

S5B1 - Graduate Seminar on Advanced Topics in PDE

Winter term 2013/14

Instructors

Prof. Dr. Herbert Koch, Prof. Dr. Christoph Thiele

Lectures

Friday, 14 (c.t.) - 16, Seminar room 0.011

Talks

• October 18, 2013: Regularity of the free boundaries for the elliptic thin obstacle problem [Wenhui Shi] (postponed to Oberseminar on October, 24)
  Abstract: I will describe an ongoing project on the higher regularity of the free boundaries in the elliptic thin obstacle problem. The main method we use is the appropriate generalization of the partial hodograph-Legendre transformation, which was used by Kinderlehrer and Nirenberg in the classical obstacle problem.

• October 25, 2013: postponed
• November 8, 2013 at 14:45 pm: Global maximizers for the sphere adjoint Fourier restriction inequality [Diogo Oliveira e Silva]
  Abstract: M. Christ and S. Shao were the first ones to study extremizers for the endpoint restriction problem on a compact manifold. Their work generated a lot of interest in the area, but basic questions remained unanswered. In particular, how do global maximizers for the Tomas-Stein inequality on the 2-sphere look like? Just this past week, D. Foschi uploaded a paper to the arXiv with a beautiful solution of this problem. I plan to cover his proof in detail and discuss some possible further applications of his methods.

• November 15, 2013: Multi-parameter singular integrals on homogeneous nilpotent groups [Genadi Uraltsev]
• November 22, 2013: Weak type (1, 1) inequalities for discrete rough maximal functions [Mariusz Mirek]
• November 27, 2013 -- Extra Seminar in Room 1.007, 12:15-14:00: Pointwise estimates outside exceptional sets for the Carleson operator and the directional Hilbert transform [Francesco DiPlinio]
  Abstract: One of the possible approaches to L^p bounds for the Carleson operator, somewhat reminiscent of the original proof by Carleson, and originating in the work of Demeter-Lacey-Tao-Thiele on improved range in Bourgain's return times theorem, relies on pointwise estimates outside exceptional sets. In the same spirit, but in the combinatorially more challenging two-parameter phase plane setting, we will prove sharp L^p bounds for a finitary model of the related Hilbert transform along vector fields in the plane. We also describe an adaptation of the argument proving bounds near L^1 for the lacunary Carleson operator.

• November 29, 2013: L^p estimates for entangled multilinear forms [Polona Durcik]
• December 6, 2013: Partial Regularity with Angular Integrability for the Navier-Stokes Equation [Renato Luca]
  Abstract: We focus on suitable weak solutions of the Navier-Stokes equation with small initial data in weighted Lebesgue spaces with angular integrability. We show how the regularity of the solutions improves under higher angular integrabilty assumption.

• December 20, 2013: We will have two seminar talks:
  14:15: Polynomial Carleson operators on the paraboloid [Po Lam Yung]
  15:25: Inner-outer factorization of analytic matrix-valued functions [Joris Roos]

• January 10, 2014: Hilbert transform along measurable vector fields constant on Lipschitz curves [Shaoming Guo]
• January 24, 2014: Linear inviscid damping in low Sobolev regularity [Christian Zillinger]