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Analysis Seminar

The seminars are held on Thursdays at 16:00 in Room B3.02 - Mathematics Institute unless noted otherwise

Term 2 2019/20
Organiser: Lucas Ambrozio, Filip Rindler

09 Jan Alessandro Pigati (ETH-Zürich) Codimension two min-max minimal submanifolds from PDEs

16 Jan No seminar

23 Jan No seminar

30 Jan Spencer Becker-Kahn (Cambridge) Stationary varifolds near multiplicity two planes

06 Feb Lucas Ambrozio (Warwick) Min-max width and volume of Riemannian three dimensional spheres

13 Feb Giacomo del Nin (Warwick) Pattern formation in planar partitions

20 Feb Pieralberto Sicbaldi (Granada) Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold

27 Feb Jonas Azzam (Edinburgh) Poincaré inequalities, uniform rectifiability, and Dorronsoro's Theorem

05 Mar Angkana Rüland (Max-Planck Leipzig) Uniqueness, stability and single measurement recovery for the fractional Calderon problem

11 Mar 2pm Bobby Wilson (Washington) Marstrand's Theorem for uniformly convex norms

12 Mar Pierre Raphaël (Cambridge) On energy super critical blow up for waves and fluids

Term 1 2019/20
Organiser: Filip Rindler, Lucas Ambrozio
3/10/19 Ovidiu Munteanu (Connecticut, visiting UCL)
Green's function estimates and the Poisson equation
Abstract: The Green's function of the Laplace operator has been widely studied in geometric analysis. Manifolds admitting a positive Green's function are called non-parabolic. By Li and Yau, sharp pointwise decay estimates are known for the Green's function on nonparabolic manifolds that have nonnegative Ricci curvature. The situation is more delicate when curvature is not nonnegative everywhere. While pointwise decay estimates are generally not possible in this case, we have obtained sharp integral decay estimates for the Green's function on manifolds admitting a Poincare inequality and an appropriate (negative) lower bound on Ricci curvature. This has applications to solving the Poisson equation, and to the study of the structure at infinity of such manifolds.
10/10/19 Franz Gmeineder (Bonn)

Regularity for quasiconvex functionals with linear growth from below

Abstract: I shall report on some recently established and forthcoming results on the regularity of minima of quasiconvex functionals. As a main feature, we focus on linear growth from below but, opposed to standard settings, allow for superlinear growth from above. Such results also apply to functionals depending on more general differential operators than the gradients. This talk comprises joint work with J. Kristensen (Oxford).

17/10/19 Diogo Oliveira e Silva (Birmingham)

Global maximisers for spherical restriction

Abstract: This talk is based on recent results obtained with René Quilodrán.
We prove that constant functions are the unique real-valued maximisers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximisers coincide with nonnegative maximisers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents.

24/10/19 Adolfo Arroyo-Rabasa (Warwick)

A variational characterization of oscillations and concentrations ocurring along a PDE-constrained sequence

There are two main phenomena preventing a sequence (u_j) from converging strongly in L^1. One, are high oscillations; and second, concentration of mass. An interesting problem is to impose a (generic) PDE-constraint on the elements of sequence, and ask whether the PDE brings enough rigidity to prevent oscillations or concentrations? The answer to this question lies in a somewhat gray area where the L^1-compactness interacts in a non-trivial way with the PDE constraint. I will motivate and provide an answer to this problem from a variational viewpoint. More precisely, I will give a Hahn-Banach (separation) characterization of the set of Young measures which are generated by such sequences (x-parametrized families of probability distributions generated by sequences of points {u_j(x)}_x) in terms of Jensen inequalities with quasiconvex integrands.

31/10/19 NO TALK  
7/11/19 David Bourne (Heriot-Watt)

Optimal Lattice Quantizers and Best Approximation in the Wasserstein Metric

Abstract: In this talk I will discuss the problem of the best approximation of the three-dimensional Lebesgue measure by a discrete measure supported on a Bravais lattice. Here 'best approximation' means best approximation with respect to the Wasserstein metric W_p, p \in [1,\infty). This problem is known as the quantization problem and it arises in numerical integration, electrical engineering, discrete geometry, and statistics.

14/11/19 NO TALK  
21/11/19 Konstantinos Koumatos (Sussex)
On the vectorial Weierstrass problem and some applications
Abstract: In the calculus of variations, the Weierstrass problem consists in finding sufficient conditions for a given map to be a local minimiser in an appropriate topology. The talk presents some recent developments in the vectorial setting, since its resolution by Grabovsky and Mengesha in 2009, including an alternative proof and an extension to non-smooth domains. Surprisingly, the proof has consequences in the theory of conservation laws with involutions which are discussed.
28/11/19 Stefano Borghini (Uppsala)

Static vacuum spacetimes with positive cosmological constant

Abstract: Static vacuum spacetimes are solutions to the Einstein Field Equations with vanishing stress-energy tensor and featuring a very special metric structure (warped product). Such a structure induces a natural foliation of the spacetime into space-like slices which are all isometric to each other, so that the corresponding physical universe is static. We discuss the problem of the classification of such solutions in the case of positive cosmological constant. To this end, we develop some new or improved tools to study extremal points of real analytic functions. Building on this, we deduce a new characterization of the Schwarzschild-de Sitter solution based on the geometry of the maximum set of the lapse. This is a joint work with P. T. Chru´sciel and L. Mazzieri.

5/12/19 Jonathan Ben-Artzi (Cardiff)
Convergence rates in dynamical systems lacking a spectral gap
Abstract: Our world is neither compact nor periodic. It is therefore natural to consider dynamical systems on unbounded domains, where typically there is no spectral gap. I will present a (simple) method for studying the generators of such systems where a spectral gap assumption is replaced with an estimate of the Density of States (DoS) near zero. There are two main applications:
1) Dissipative systems: when the generator is non-negative, an estimate of the DoS leads to a so-called "weak Poincaré inequality" (WPI). This in turn leads (in some cases) to an algebraic decay rate for the $L^2$ norm of the solution. For instance, in the case of the Laplacian (generator of the heat equation) the WPI is simply the Nash inequality which leads to the optimal decay rate of $t^{-d/4}$.

2) Conservative systems: when the generator is skew-adjoint, an estimate of the DoS leads to a uniform ergodic theorem on an appropriate subspace. Examples include the linear Schrödinger equation and incompressible flows in Euclidean space.

Based on joint works with Amit Einav (Graz) and Baptiste Morisse (Cardiff).