Abstract
This talk is about integral functionals of the form
$$
I_n(\varphi)=\int_Q |\nabla \varphi|^n+ f(x)\det \nabla \varphi \, dx,
$$
which can appear in the Calculus of Variations as excess functionals
.
This makes it important to know for which $f \in L^{\infty}(Q)$ it holds that $I_n(\varphi) \geq 0$
for all $\varphi \in W_0^{1,n}(Q;\mathbb{R}^n)$.
We prove that there are piecewise constant $f$ such that $I_n\geq 0$ holds and, moreover, that this is
strictly stronger than any inequality obtained by using only the pointwise Hadamard inequality
$n^{\frac{n}{2}}|\det A|\leq |A|^n$ for $n \times n$ matrices $A$.
When $f$ takes just two values we find that $I_n\geq 0$ holds if and only if the variation of $f$ in $Q$
is at most $2n^{\frac{n}{2}}$, a fact that is connected to work of A. Mielke and P. Sprenger
(in J. Elasticity, 1998) on quasiconvexity at the boundary.
For more general (but still piecewise constant) $f$, we show that $I_n \geq 0$ is decided by both
the geometry of the jump sets and the values taken by $f$.
Abstract (I) Our first result concerns the classical Darboux theorem, i.e. given two skew-symmetric matrices $h,g=g\left( x\right) \in\mathbb{R}^{n\times n},$ with $n=2m$ even, we want to find a map $u:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ such that% \[ \left( \nabla u\right) ^{t}h\,\nabla u=g\quad\Leftrightarrow\quad u^{\ast }\left( h\right) =g. \] We discuss the existence, regularity and uniqueness of solutions; the "best" choice will be made by adding an appropriate condition making the system elliptic.
(II) We then apply the above result to the so-called symplectic factorization. We show that any map $u,$ satisfying appropriate assumptions, can be written as \[ u=\psi\circ\chi \] where ($\omega_{m}$ denoting the constant symplectic matrix of rank $n=2m$)% \[ \left( \nabla\psi\right) ^{t}\omega_{m}\nabla\psi=\omega_{m}\quad \text{and}\quad\left\langle \nabla\chi;\omega_{m}\right\rangle =\sum_{i=1}% ^{m}\left( \frac{\partial\chi^{2i}}{\partial x_{2i-1}}-\frac{\partial \chi^{2i-1}}{\partial x_{2i}}\right) =0. \]
(III) The analogy with mass transportation and the Monge-Amp\`{e}re equation, as well as with the polar factorization, will be emphasized.
This is a joint work with Wifrid Gangbo and Olivier Kneuss.
Abstract In this talk I will give a brief overview on the evolution by horizontal mean curvature flow (i.e. the evolution in sub-Riemannian geometries) and compare that with the corresponding Euclidean and Riemannian evolutions. I will briefly introduce two methods for proving existence: the stochastic representation formula and the Riemannian approximation, showing then some connections between the two methods. The results are from a series of works, in collaboration with Nicolas Dirr, Raffaele Grande and Max von Renesse.
Abstract We consider nonlinear viscoelastic materials of Kelvin-Voigt type with stored energies satisfying an Andrews-Ball condition, allowing for non-convexity in a compact set, and a linear viscous stress. We show the existence of weak solutions with deformation gradients in $H^1$ for energies of any superquadratic growth. In two space dimensions, and in a striking analogy to the incompressible Euler equations with bounded vorticity, weak solutions turn out to be unique in this class.
Conservation of energy for these solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are also established under additional restrictions on the growth of the stored energy. This is joint work with C. Lattanzio, S. Spirito, and A. E. Tzavaras.
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 (ETH) and Aleksis Koski (Helsinki).
Abstract
The goal of the seminar is to report on recent joint work with Daniele Semola, motivated by a question of
Gromov to establish a synthetic regularity theory
for minimal surfaces in non-smooth ambient spaces.
In the setting of non-smooth spaces with Ricci curvature lower bounds:
Optimal transport plays the role of underlying technical tool for addressing various points.
Abstract We propose a density functional theory of Thomas-Fermi-(von Weizsacker) type to describe the response of a single layer of graphene to a charge some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers. We further provide conditions under which those minimizers are unique. The associated Euler-Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. For a class of special potentials we also establish a precise universal asymptotic decay rate, as well as an exact charge cancellation by the graphene sheet. In addition, we discuss the existence of nodal minimizers which leads to multiple local minimizers in the TFW model.
Abstract In this talk I will present gradient estimates of elliptic types (specifically, of Hamilton and Souplet-Zhang types) for a class of nonlinear parabolic equations involving the Witten Laplacian. The context is that of a smooth metric measure space where the metric and potential evolve with time (a geometric flow). The estimates are established under different curvature conditions and lower bounds on the Bakry-Emery Ricci tensor and are then used to prove a number of important results such as Harnack inequalities, spectral bounds, sharp Logarithmic Sobolev inequalities (LSI) and general Liouville and global constancy results. If time allows, I will present applications of the above to the (super) Perelman-Ricci flow, heat entropy formula and ancient/eternal solutions.
Abstract I will present our latest developments on the generalization of the Aw-Rascle model of traffic. These include existence and non-uniqueness results and also singular limit analysis leading to certain hard-congestion model. The latter model has been studied in the context of lubrication forces for the motion of rigid bodies in a viscous fluid. The proof of the singular limit will be presented in one-space dimensional case with periodic boundary conditions.
Abstract Partial differential equations often arise as a (rigorous or formal) scaling limit of particle models. We sketch some limit passages to derive PDEs describing the collective motion of interacting particles. In particular, we discuss the Hamiltonian picture. For the Carleman model of stochastic interacting particles, we show how this leads to a class of Hamilton–Jacobi partial differential equations in the space of probability measures. For these equations, a comparison principle can be proved and some structure of the Hamiltonian, which looks mysterious at first sight, can be unravelled as the Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models.
Joint work with Jin Feng and Toshio Mikami.
Abstract In this lecture we shall present some recent results on the interplay between Control and Machine Learning, and more precisely, Supervised Learning, Universal Approximation and Normalizing Flows.
We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets). Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNets, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control.
We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies.
This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfil.
These results are then re-interpreted in the context of transport equations and normalising flows and their control through neural vector fields. We also discuss the interplay between the time and space-like complexity of these vector fields.