Titles and Abstracts:

Giovanni Alberti (Univ. Pisa): Loss of regularity for the transport equation with a non-smooth velocity field.

We consider the transport equation ρt+u·∇ρ=0, where u=u(t,x) is a given, divergence-free velocity field on the plane. If u is sufficiently regular, e.g., Lipschitz in space uniformly in time, then the solution of the corresponding initial-value problem is given by the composition of the initial datum ρ0and the flow associated to u, and this shows that (some) regularity of the initial datum is preserved for all times by the solution ρ. In this talk I will show that this result may fail completely if u is less regular, that is, if we only require that u belongs to some Sobolev space that does not embed in the space of Lipschitz functions. In particular, I will describe an example of a velocity field u and an initial datum ρ0 such that

This is joint work with Gianluca Crippa (University of Basel) e Anna Mazzuccato (Penn State University).

Luigi Ambrosio (SNS Pisa): Semigroups and Geometric Measure Theory

I will illustrate with a few examples how semigroup tools can have a crucial role in the proof of extensions of classical results in Geometric Measure Theory and Real Analysis.

Sören Bartels (Univ. Freiburg): Approximating gradient flow evolutions of self-avoiding inextensible curves and elastic knots

We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of a bending energy and a so-called tangent-point functional. We define evolutions via the gradient flow for the total energy within a class of arclength parametrized curves, i.e., given an initial curve we look for a family of inextensible curves that solves the nonlinear evolution equation. Our numerical approximation scheme for the evolution problem is specified via a semi-implicit discretization of the equation with an explicit treatment of the tangent-point functional and a linearization of the arclength condition. The scheme leads to sparse systems of linear equations in the time steps for cubic splines and a nodal treatment of the constraints. The explicit treatment of the nonlocal and nonlinear tangent-point functional avoids working with fully populated matrices and furthermore allows for a straightforward parallelization of its computation. Based on estimates for the second derivative of the tangent-point functional and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization. We present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in certain knot classes, so-called elastic knots. This is joint work with Philipp Reiter (University of Georgia)

Giovanni Bellettini (Univ. Siena): New results on the relaxation of the area functional of graphs of nonsmooth maps from the plane to the plane.

We shall discuss some new results concerning the expression of the relaxation of the area functional, for graphs of singular maps from the plane (source space) to the plane (target space). We shall focus attention to the case where the map takes only three values (the vertices of a triangle), without any symmetry assumptions. The value of the relaxed functional turns out to be related to the solution of three Plateau-type problems, for three surfaces sitting in the four-dimensional space, and entangled together on the vertical plane corresponding to the target space.

Elisabetta Chiodaroli (Univ. Pisa): On the energy conservation for the 3D Navier-Stokes equations.

In this talk we consider weak solutions to the 3D Navier-Stokes equations in a smooth domain with Dirichlet conditions and we discuss the validity of the energy equality in this class. We prove some new conditions for energy conservation and we compare them with classical and more recent results of the existing literature, in particular in view of the famous Onsager conjecture. Finally, we analyze the problem of energy conservation for very weak solutions. This is a joint work with Luigi C. Berselli (Pisa).

Sergio Conti (Bonn): Stress-space relaxation

The theory of relaxation, based on the concept of quasiconvexity, has been very successful in the study of microstructure in nonlinear elasticity. There are situations, however, in which a formulation with the elastic deformation as the only independent variable is not appropriate, even after minimizing out some internal variables. We consider here a setting in which the natural independent variable is the stress and not the strain field, such as critical-state theory of plasticity. We give a general relaxation framework building upon the general tools of $\mathcal A$-quasiconvexity and discuss its application to the relaxation of isotropic models in which the yield surface depends on the first two invariants only. Our results can be used to interpret numerical results on fused silica glass. The talk is based on joint work with Stefan Müller and Michael Ortiz.

Lucia De Luca (Univ. Pisa): Γ-convergence of the Heitmann-Radin sticky disc energy to the crystalline perimeter.

We investigate low energy configurations of atomic systems in two dimensions interacting via the Heitmann-Radin sticky disc potential. We consider the energy functional which is obtained by subtracting from the Heitmann-Radin potential the minimal energy per particle, i.e., the so-called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function. Whenever the limit configuration is a single crystal, namely, it has constant orientation, we show that the Γ-limit is the anisotropic perimeter, corresponding to the Finsler metric determined by the orientation of the single crystal. Finally, we exhibit two classes of examples showing that, depending on the macroscopic shape of the limit configuration, single crystals and polycrystals can be both energetically favorable. Joint work with M. Ponsiglione (Rome) and M. Novaga (Pisa).

Alessio Figalli (ETH Zurich): Free boundaries in obstacle problems

The aim of this talk is to discuss recent progresses on the regularity of free boundaries in the elliptic/parabolic obstacle problems. I'll then mention some applications to the Stefan and Hele-Shaw problem.

Franco Flandoli (SNS, Pisa): Remarks on 2D inverse cascade turbulence

Recently the interest in certain invariant measures of 2D Euler equations was renewed, motivated for instance by questions of existence for almost every initial condition, similarly to the case of dispersive equations where probability on initial conditions allowed very succesful progresses. Obviusly invariant measures of 2D Euler equations are primarily of interest for turbulence but those known are not realistic from several viewpoints, beside some element of great interest. We discuss this issue and show modifications, based also on variational problems of statistical mechanics type, that could give much better results for turbulence.

Nadine Große (Univ. Freiburg): Poisson-type boundary value problems on manifolds of bounded geometry

Abstract: We consider Poisson problems on manifolds with boundary and bounded geometry and assume that they have finite width (that is, that the distance from any point to the boundary is bounded uniformly). We include Robin boundary conditions. As an application, we establish the connection to the Poisson problem on certain domains in the plane and higher dimensional stratified spaces. In particular we get the well-posedness of strongly elliptic equations on domains with oscillating conical singularities, a class of domains that generalizes the class of bounded domains with conical points. This is joint work with Bernd Ammann (Regensburg) and Victor Nistor (Metz).

Andrea Malchiodi (SNS, Pisa): Prescribing scalar curvature in high dimension

We consider the classical problem of prescribing the scalar curvature of a manifold via conformal deformation of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions are proven to be "isolated simple". We find natural conditions to guarantee this also in arbitrary dimensions, when the prescribed curvatures are Morse functions. As a consequence, we improve some pinching conditions in the literature and derive existence results of new type. This is joint work with M. Mayer.

Simon Masnou (Université Lyon 1): Weak and approximate curvatures of a measure: a varifold perspective

I will present a notion of weak curvature tensor which makes sense for a suitable class of varifold measures and can be extended to ALL varifolds of any dimension and codimension through a regularization procedure. The resulting approximate second fundamental forms are defined not only for piecewise smooth surfaces but also for datasets of very general type, e.g. point clouds. These weak and approximate curvature tensors are explicitly computable. Some compactness, consistency and convergence results will be presented. I will illustrate the effectiveness of the approach with various numerical tests on point clouds (evaluation of curvatures and geometric flows, also in presence of noise and singularities). It is a joint work with Blanche Buet (Université Paris-Sud) and Gian Paolo Leonardi (Università di Trento).

Marcello Ponsiglione (La Sapienza, Roma): Low energy configurations of topological singularities in two dimensions

We analyze two continuous models for the study of topological singularities in 2D: the core-radius approach and the Ginzburg-Landau theory, discussing applications to screw dislocations in crystals. It is well known that, as the length scale parameter goes to zero, these models are characterized by energy and topological concentration around a finite number of points, the so-called vortices (or dislocations). We focus on low energy regimes that prevent the formation of isolated vortices in the limit, but that are compatible with configurations of short (in terms of the length scale parameter) dipoles, and more in general with short clusters of vortices having zero average. By using a Gamma-convergence approach, we provide a quantitative analysis of the energy induced by such configurations on a continuous range of length scales. (Joint work with L. De Luca.)

Aldo Pratelli (Univ. Pisa): Stability results for the Coulomb energy

Joint work with Nicola Fusco (Napoli)

Guofang Wang (Univ. Freiburg): Optimal geometric inequalities in the hyperbolic space

In this talk, we are interested in establishing Alexandrov-Fenchel type inequalities for hypersurfaces in the hyperbolic space, which are optimal geometric inequalities involving the integral of the mean curvature or the higher order mean mean curvature and the area. As an application, we obtain Penrose type inequalities for graphical manifolds. The talk bases on the joint work with Yuxin Ge, Jie Wu and Chao Xia.


Last modified: Wed Dec 26 19:46:05 CET 2018