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).

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.

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

Abstract: 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.

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.

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.