Nicolas Dirr

Stochastic Homogenization for Hamilton-Jacobi Equations in Carnot Groups

Abstract We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable, but associated to a Carnot group, i.e. it is coercive in the intrinsic derivative of the Carnot group, which omits directional derivatives in some “forbidden” directions. The rescaling considered is consistent with this underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the epsilon-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable homogenized deterministic Hamilton-Jacobi problem. The methods are variational by applying ideas from Gamma-convergence to the Lagrangians.

Nikos Katzourakis

On isosupremic vectorial minimisation problems in \(L^\infty\) with general nonlinear constraints

Abstract In this talk I will discuss minimisation problems in \(L^\infty\) for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints.

Jan Kristensen

Morrey's problem in classes of homogeneous integrands

Abstract The question of whether rank-one convexity implies quasiconvex is often called Morrey's problem. Sverak has shown that the answer is no in general and so a number of modifications have been proposed over the years. In this talk I will discuss some of these where positive answers have been found recently. The talk is based on joint work with Kari Astala (Helsinki), Daniel Faraco (Madrid), Andre Guerra (IAS) and Aleksis Koski (Helsinki).

Carlo Mercuri

The effect of a local charge on a nonlinear and non local PDE on \(\mathbb{R}^{N}\)

Abstract I will make some reflections highlighting the role of \(\rho\) in the problem of seeking variational solutions to the nonlinear and nonlocal PDE \begin{equation*} -\Delta u+u+\lambda^{2}\bigg(\frac{1}{\omega|x|^{N-2}} * \rho u^2\bigg) \rho(x)u=|u|^{q-1} u \ \ \ x\in \mathbb{R}^{N}, \end{equation*} where \(\omega= (N-2)|S^{N-1}|\), \(\lambda >0\), \(q\in(1,2^{*}-1)\), \(\rho:\mathbb{R}^{N} \rightarrow \mathbb{R}\) is nonnegative, locally bounded, and possibly non-radial, \(N=3,4,5\) and \(2^{*}= 2N/(N-2)\) is the critical Sobolev exponent. This talk is also an homage to a great mathematician and friend, Antonio Ambrosetti, from whom I have learnt a lot.

Monica Musso

Leapfrogging for Euler equations, and the vortex filament conjecture

Abstract We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. In this talk I will discuss the construction of helical filaments, associated to a translating-rotating helix, and of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).

Enrico Valdinoci

(Non)local phase coexistence models

Abstract We will discuss classical and recent results concerning the Allen-Cahn equation and its long-range counterpart, especially in relation to its limit interfaces provided by (possibly nonlocal) minimal surfaces.