List of Titles and Abstracts

Luigi De Rosa: Turbulent flows at critical regularity
Since the celebrated Kolmogorov Theory of Turbulence from 1941 it has been clear that, at first approximation, a good understanding of incompressible Turbulence is subject to the study of weak solutions to the Euler equations having critical regularity. After presenting the problem, and propose a definition of “critical” regularity, I will show how to prove a formula for the anomalous dissipation measure which improves the previously known ones. This is then used to prove interior local energy conservation for bounded solutions with bounded variation, building on measure theoretic ideas introduced by Luigi Ambrosio in the context of the DiPerna-Lions theory for linear transport equations. This is the first result proving absence of dissipative anomaly heavily relying on the incompressibility of the flow, thus distinguishing incompressible Turbulence from the compressible one. This also connects to Intermittency.

Gilles Francfort: Uniqueness and characteristic flow: the case of a non-homogeneous functional with linear growth
This is joint work with J.F. Babadjian. We investigate a functional of the gradient arising out of the theory of elasto-plasticity. It exhibits linear growth at infinity while not being a norm (so it is different from a least gradient type problem). The relaxed functional has BV minimizers. In 2d, their uniqueness may be tackled through hyperbolic methods. A mix of those with geometric measure theoretic arguments eventually leads to a uniqueness result for pure Dirichlet boundary conditions, while uniqueness is false if other types of boundary conditions are considered.

Stanislav Hencl: Ball-Evans approximation problem
It is well-known that Sobolev mapping may be approximated by smooth ones in the Sobolev norm. In the problems connected with Nonlinear Elasticity one needs to approximate Sobolev homeomorphisms (from open subset of R^n to R^n) by a sequence of diffeomorphisms or piecewise affine homeomorphisms which is much more difficult. In our talk we review some recent positive planar results and show some negative results in higher dimensions. The main result we show is that we can approximate planar W^{1,1} homeomorphism with a sequence of diffeomorphisms. This is a joint result with A. Pratelli.

Jonas Hirsch: On the Lawson-Osserman conjecture on the minimal surfaces system
(joint work with Connor Mooney and Riccardo Tione)

In the renowned paper by Lawson and Osserman, non-existence, non-uniqueness, and irregularity of solutions to the minimal surface system, Conjecture 2.1 stands out and states roughly:
”The outer variations for the minimal surfaces system are sufficient for a graph that is locally Lipschitz continuous.”
put differently ”Does the outer variations for a locally Lipschitz continuous graph imply that the inner variation holds as well?”
We affirmatively resolve the conjecture in dimension two. Our main result can be succinctly stated as follows: A two-dimensional graph that is locally Lipschitz continuous and is a critical point of the area with respect to outer variations is smooth.
Having presented the conjecture and our result, the remainder of the talk will be devoted to outlining the ideas behind the proof and elucidating the role of working in two dimensions.

Dominik Inauen: Flexibility of very weak solutions to the 2-dimensional Monge-Ampère equation
The Monge-Ampère equation, given by det D^2v = f, is a fundamental equation in analysis and geometry and arises in problems such as the isometric embedding problem, the prescribed Gaussian curvature problem, or optimal transportation. The notion of 'very weak solutions' naturally emerges in the context of von Kármán theories in nonlinear elasticity.
After reviewing some results about the flexibility and rigidity of isometric embeddings, I want to indicate how the connection between isometric embeddings and the Monge-Ampère equation can be used to prove a similar flexibility in the case of very weak solutions to the Monge-Ampère equation of regularity C^{1,\alpha} for \alpha < 1/3.

Jan Kristensen: The Burkholder functional on classes of planar quasiregular maps
The area formula of Gronwall and Bieberbach can be viewed as a precise way to express that the Jacobian of a planar Sobolev map is a null Lagrangian. In this talk I discuss a quasiconvexity inequality for the Burkholder functional in the context of planar quasiconformal maps. This inequality can be viewed as an extension of the area formula to an Lp context. If combined with Stoilow factorization and blow-up arguments it also allows a proof of semicontinuity, and hence to prove existence of minimizers for the Burkholder energy on classes of planar quasiregular maps. The talk is based on joint work with Kari Astala (Helsinki), Daniel Faraco (Madrid), Andre Guerra (ETH), and Aleksis Koski (Aalto).

Hyunju Kwon: Strong Onsager conjecture
Smooth solutions to the incompressible 3D Euler equations conserve kinetic energy in every local region of a periodic spatial domain. In particular, the total kinetic energy remains conserved. When the regularity of an Euler flow falls below a certain threshold, a violation of total kinetic energy conservation has been predicted due to anomalous dissipation in turbulence, leading to Onsager's theorem. Subsequently, the L^3-based strong Onsager conjecture has been proposed to reflect the intermittent nature of turbulence and the local evolution of kinetic energy. This conjecture states the existence of Euler flows with regularity below the threshold of $B^{1/3}_{3,\infty}$ which not only dissipate total kinetic energy but also exhibit intermittency and satisfy the local energy inequality. In this talk, I will discuss the resolution of this conjecture based on recent collaboration with Matthew Novack and Vikram Giri.

Xavier Lamy: On regularity and rigidity of 2×2 differential inclusions into non-elliptic curves
I will describe regularity and rigidity results obtained with A.Lorent and G.Peng on differential inclusions $Du\in \Gamma$ for a vector field $u$ in a 2D domain. Here $\Gamma$ is a connected curve of 2x2 matrices without rank-one connection, but non-elliptic: tangent lines might have rank-one connections, so that classical regularity and rigidity results do not apply. For a large class of such curves, we show that $Du$ must be Lipschitz outside a discrete set, and is rigidly characterized around each singularity. We also identify conditions on the curve $\Gamma$ under which there are no singularities. The proof generalizes methods and ideas from the theory of the Aviles-Giga functional.

Filip Rindler: Geometric Rademacher-type theorems and dislocation motion
Dislocation lines in crystalline solids, whose motion constitutes the microscopic mechanism for plastic deformation, can be seen as fundamentally governed by the geometric (Lie) transport equation for a time-indexed family of integral or normal 1-currents. This talk will report on recent progress on the analysis of this equation, covering in particular existence and uniqueness of solutions as well as Rademacher-type differentiability theorems with respect to the "geometric derivative". The latter are related to an interplay between differential and pointwise constraints, reminiscent of Tartar's framework for differential inclusions. This is joint work with Paolo Bonicatto and Giacomo Del Nin.

Kostas Zemas: Stability aspects of the Möbius group of the sphere
In this talk I would like to discuss quantitative stability aspects of the class of Möbius transformations of the sphere among maps in the critical Sobolev space (with respect to the dimension). The case of sphere- and R^n-valued maps will be addressed. In the latter, more flexible setting, unlike similar in flavour results for maps defined on domains, not only a conformal deficit is necessary, but also a deficit measuring the distortion of the sphere under the maps in consideration, which is introduced as an associated isoperimetric deficit. The talk will be based on previous works in collaboration with Stephan Luckhaus and Jonas Hirsch, and more recent ones with Xavier Lamy and André Guerra.

Martina Zizza: Non-admissibility of Spiral-like strategies in Bressan's Fire Conjecture
In this talk we will introduce Bressan's Fire Conjecture: it is concerned with the model of wild fire spreading in a region of the plane and the possibility to block it using barriers constructed in real time. The fire starts spreading at time $t=0$ from the unit ball $B_1(0)$ in every direction with speed $1$, while the length of the barrier constructed within the time $t$ has to be lower than $\sigma t$, where $\sigma$ is a positive constant (construction speed). If $\sigma\leq 1$ Bressan proved that no barrier can block the spreading of the fire, while if $\sigma>2$ there exists always a strategy that confines the fire. In 2007 Bressan conjectured that if $\sigma\leq 2$ then no barrier can block the fire. In this talk we will prove Bressan's Fire Conjecture in the case barriers are spirals. Spirals are thought to be the best strategies a firefighter can do in order to confine the fire for $\sigma\leq 2$. We will introduce the new concept of family of generalized barriers and we will prove that, if there exists such a family satisfying a diverging condition, then no spiral can confine the fire. This is a joint work with Stefano Bianchini.