Talks

• Scott Armstrong: Homogenization on percolation clusters
I will give a review of recent results in quantitative stochastic homogenization for uniformly elliptic equations and show how they can be adapted to handle problems in random porous media and/or degenerate equations. A focus is to understand harmonic functions on a (near-critical) supercritical percolation cluster.
• Lisa Beck: On the Neumann problem related to convex, variational integrals
of linear growth
We study the minimization of functionals of the form
$$w \mapsto \int_\Omega [ f(|\nabla w|) – T_0 \cdot \nabla w ] \, dx$$
with a strictly convex integrand $f$ of linear growth and a regular vector-field $T_0$, among all functions in the Sobolev space $W^{1,1}$. Equivalently, we may study weak solutions to the associated Euler–Lagrange system subject to a Neumann-type boundary constraint. Due to the lack of weak compactness properties of the space $W^{1,1}$, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations. While for the Dirichlet problem, where prescribed boundary values are imposed, solutions exist only in a suitably generalized sense via relaxation to the space of functions of bounded variations, the Neumann problem turns out to be always solvable in $W^{1,1}$, under a sharp and natural boundedness condition on $T_0$. The results presented in this talk are based on a joined project with Miroslav Bulíček (Prag) and Franz Gmeineder (Oxford).
• Barbora Benešová: The concept of gradient polyconvexity and applications in elasticity
It is an open problem to date whether weak lower semicontinuity of quasiconvex functions still holds if these functions take infinity values on a non-convex set as required in elasticity. In modelling, one possible remedy to assure the existence of minimizers is to include for such energies terms depending on higher gradients which are often linked to some type of interfacial energies. Within this talk, we will highlight the observation that actually including the whole second gradient is not necessary; instead one can work only with gradients of the determinant and cofactor that have a clear physical interpretation. We show how in this situation existence of minimizers is obtained and that it can be assured that they are globally injective. We also point out connections to higher-order polyconvexity and relaxation techniques under so-called locking constraints.
• Maria Colombo: Stability of optimal paths in branched transport
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given source onto a given target measure along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow.
The talk introduces the model and focuses on the stability for optimal traffic paths, with respect to variations of the source and target measures. The stability of optimal traffic paths was known when $\alpha$ is bigger than a critical threshold, but can be generalized to other exponents (for instance, to any  $\alpha \in (0,1)$ in dimension 2) for a fairly large class of measures.
(joint work with Antonio De Rosa and Andrea Marchese)
• Daniel Faraco: Mixing solutions for the Muskat problem
The Muskat Problem describes the evolution of the interphase between two fluids evolving through a porous media. The theory is very different depending on whether the heavier or lighter fluid are in top of each other. The case of the heavier fluid on top is  ill posed in Sobolev Spaces.  In spite of that there is a number of results  and numerical experiments showing the presence of  a so called mixing zone, related to the phenomena of fingering. In this talk I will describe how a combination of convex integration and contour dynamics yields the existence of weak solutions for an arbitrary initial interphase. This is a joint work with Angel Castro and Diego Cordoba.
• Alessio Figalli: Description of almost-flat d-cones
A developable-cone (also “d-cone” in short) is the shape one obtains when pushing an elastic sheet at its center into a hollow cylinder by an amount \epsilon>0. Starting from a nonlinear model depending on the thickness h>0 of the sheet, we can first prove a rigorous \Gamma-convergence result as h\to 0, and then study the exact shape of minimizers for \epsilon small enough. In particular, we can rigorously justify previous results in the physics literature. This is a joint work with Connor Mooney.
• Nicola Fusco: Evolution of material voids by surface diffusion
We consider the evolution by surface diffusion of material voids in a linearly elastic solid. We prove short time existence and asymptotic stability when the initial configuration is close to a stable critical point for the energy. Similar results are also obtained for the evolution by surface diffusion of epitaxially thin films.
• Jan Kristensen: Partial regularity for BV minimizers
I discuss a partial regularity result for BV minimizers of quasiconvex integrals of linear growth.
• Bernd Kirchheim: Convexity notions in the Calculus of Variations
There is an entire zoo of convexity concepts, which reflect lower semicontinuity or regularity properties of vectorial variational problems. We are going to present less and more recent results, stating that such notions do agree of we restrict the class of functionals used or the domain of them.
• Konstantinos Koumatos: Quasiconvex elastodynamics: weak-strong uniqueness for measure-valued solutions
A weak-strong uniqueness result is presented for measure-valued solutions to the system of conservation laws arising in elasticity with a stored-energy function which is strongly quasiconvex. The proof combines the relative entropy method and tools borrowed from the calculus of variations. This is joint work with Stefano Spirito (University of L’Aquila).
• Annalisa Massaccesi: On the geometric structure of normal and integral currents
In this joint work with G. Alberti and E. Stepanov we consider normal currents of the form $T=\xi\mu$ (where $\xi$ is a $C^1$ vectorfield and $\mu$ is a measure) and we investigate the interplay between the integrability of $\xi$ (in the sense of Frobenius Theorem) and the absolute continuity of $\mu$ with respect to the Lebesgue measure. This result is connected with the geometric structure of the boundary $\partial T$.
• Christoph Melcher: Variational framework for chiral skyrmions
Chiral skyrmions are topological solitons occurring in magnets without inversion symmetry. In this talk I shall explain the analytical structure and variational consequences of chiral interactions responsible for the occurrence of new magnetic phases in the spirit of Ginzburg-Landau, and for the stabilization magnetic skyrmions.
• Giuseppe Mingione: A tour in the dark forest of Lipschitz bounds
Lipschitz regularity is the threshold between lower and higher regularity in several elliptic and parabolic nonlinear problems. Once the gradient of solutions is bounded, you can then start getting higher regularity. I will present a short survey of recent Lipschitz regularity results including: borderline cases for uniformly elliptic and parabolic problems, nonlinear potential estimates, regularity for non-uniformly elliptic problems.
• Clément Mouhot: Some applications of De Giorgi-Nash-Moser regularity theory in kinetic theory
We discuss a recent extension of the celebrated De Giorgi-Nash regularity theory to kinetic equations of Kolmogorov type, and applications of it to the regularity Landau-Coulomb equation as well as to the global well-posedness for a nonlinear toy model.
• Felix Otto: Regularity for Monge-Ampere: A variational proof.
We give a completely variational proof of an $\epsilon$-regularity result for the Monge-Ampere equation. Such a result is already known from the work of Figalli & Kim; however our proof of it by-passes the regularity theory of Caffarelli. The one-step improvement in the Campanato-iteration is based on constructing suitable competitors within the Benamou-Brenier formulation of optimal transportation. This is joint work with Michael Goldman.
If time permits, we present a (new) application of this variational approach
to the matching problem between the Poisson point process and the Lebesgue measure. This is joint work in progress with Michael Goldman and Martin Huesmann.
• Mircea Petrache: Sharp asymptotics for N-marginal optimal transport with singular costs
Motivated by the study of optimal point configurations and of questions from quantum mechanics, we study an optimal transport problem with N marginals and with singular cost such as |x-y|^{-s} for 0
• Tristan Rivière: Minmax Hierarchies in the Viscosity Method : a PDE approach to the construction of minimal surfaces of increasing indices
We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in Banach manifolds equipped with Finsler structures. We call the resulting tree  a “minmax hierarchy”. We shall combine this algorithm with a viscosity approach for producing smooth minimal surfaces of strictly increasing area in arbitrary codimension. We implement this scheme to the case of the 3−dimensional sphere. In particular we are giving a min-max characterization of the Clifford Torus and conjecture what are the next minimal surfaces to come in the S3 hierarchy.
• Sylvia Serfaty: Microscopic description of Coulomb-type systems
We are interested in systems of points with Coulomb, logarithmic
or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in various settings: an instance is the classical Coulomb gas which in some cases happens
to be a random matrix ensemble, another is vortices in the Ginzburg-Landau
model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named
Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. After reviewing the motivations, we will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes are given.
This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions.
The main results are joint with Thomas Leblé and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache.
• Joaquim Serra: The De Giorgi conjecture for the half-Laplacian in dimension 4
A celebrated conjecture of Giorgi  for the Allen-Cahn equation states that global monotone solutions are 1D up to dimension 8. This conjecture is motivated by the connection of Allen-Cahn and minimizers of the perimeter, and by classical classification results for entire minimal graphs. The well-konwn version of this conjecture for boundary reactions in half-spaces can be reduced to study the problem in the whole space for the Allen-Cahn equation with the half-Laplacian. In the talk I will present a recent result with Alessio Figalli that establishes the validity of the De Giorgi conjecture for stable solutions in dimension 3.
As a corollary, we obtain that every monotone solution (without further assumptions) is 1D in dimension 4.
• Barbara Zwicknagl: Martensitic inclusions in low-hysteresis shape memory alloys
Shape memory alloys undergo a martensitic phase transition, that is, a diffusionless, first order solid-solid phase transformation. It has been found that the width of the associated thermal hysteresis is often closely related to the minimal energy that is necessary to build a martensitic nucleus in an austenitic matrix. Mathematically, this energy barrier is typically modeled by (singularly perturbed) nonconvex elasticity functionals. In this talk, I will discuss recent results on the resulting variational problems, in particular on stress-free martensitic inclusions. This talk is partly based on joint works with S. Conti, M. Klar, A. Rüland and C. Zillinger.