Anisotropy 2017

23.02.2017 - 25.02.2017 | Warsaw



Thursday   Friday
coffee 9:30 coffee
*Moll 10:00 Giacomelli
*Arroyo-Rabasa 11:00 Górny
lunch break 12:00 lunch break
Karch 15:00 Mercier
coffee 16:00 coffee
Schmidt 16:30 Maringova
Titi 17:30  
  19:00 conference dinner


***NOTE*** Talks marked with "*" take place in room 403. All the other talks take place in room 321. 

Coffe breaks are held at club room, 4th floor. 


Adolfo Arroyo-Rabasa -- Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints. 

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities in measure solutions to linear PDE’s and of the corresponding generalized convexity classes. This is a joint work with G. De Philippis and F. Rindler

Lorenzo Giacomelli -- Waiting-time phenomena and shock formation in flux-saturated parabolic equations. 

Flux saturated parabolic equations are characterized by a cross-over between a nonlinear parabolic scaling in the "small-gradient" regime and a (possibly nonlinear) hyperbolic scaling in the "large-gradient" one. For this reason, they share many features with nonlinear conservation laws, two of which will be discussed in this talk: the occurrence of waiting time phenomena, which is also closely related to shock formation at the boundary of the support, and shock formation in the bulk of the support, which seems to be strongly influenced by the form of the nonlinearity. Concentrating on suitable prototypical equations, we will present results, examples, and ongoing research of a joint project with Salvador Moll and Francesco Petitta, as well as open questions. 

Wojciech Górny -- Non-uniqueness of solutions for a nonsmooth anisotropy in least gradient problem. 

In the two-dimensional least gradient problem it is well known that for a smooth anisotropy and continuous boundary data solutions exist and are unique. If we consider p-norms, then we may prove non-uniqueness of solutions even for smooth boundary data in the nonsmooth cases: p = 1 or infinity. We can also determine the structure of solutions for p between 1 and infinity and their relation to the isotropic case. 

Grzegorz Karch -- Instabilities due to diffusion in reaction-diffusion-ODE systems. 

Mathematical models of a pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations will be discussed. In such models, a certain natural property of the system leads to the instability of all inhomogeneous stationary solutions. We have proved, moreover,  that space inhomogeneous solutions of these models become unbounded in either finite or infinite time, even if space homogeneous solutions are bounded uniformly in time. These are recent results obtained jointly with Anna Marciniak-Czochra and Kanako Suzuki.

Erika Maringová -- Globally Lipschitz minimizers for variational problems with linear growth. 

The classical example of a variational problem with linear growth is the minimal surface problem, which for smooth data possesses a regular (up to the boundary) solution if the domain is convex. On the other hand, for non-convex domains we know that there always exist data for which the solution does exist only in the space BV (the trace is not attained). In the work we sharply identify the class of functionals (the minimal surface problem is prototypic example of this class) for which we always have regular (up to the boundary) solution in any dimension for arbitrary regular domain. This is a joint work with L. Beck and M. Bulicek. 

Gwenael Mercier -- Continuity results for anisotropic total variation minimizers. 

In this talk, we investigate the preservation of continuity in minimizing anisotropic total variation + data term, where the data term can either be a Dirichlet condition u=g on the boundary or a L2 norm of u-g on the domain. We will in particular focus on the differences between the Euclidean and the anisotropic frameworks. 

Salvador Moll -- On the Kobayashi-Warren-Carter system for grain boundary motion in a polycrystal. 

In this talk, I will make a review on the model of grain boundary motion in a polycrystal proposed by Kobayshi-Warren-Carter. First, the model will be introduced as a gradient-like system derived from a free energy functional which includes a generalized total variation term. Then, I will present recent results joint with K. Shirakawa (Chiba University) and H. Watanabe (Oita University) about existence of a suitable class of solutions (energy dissipating) both for the homogeneous Neumann problem and for the non-homogeneous Dirichlet one. Asymptotic behaviour will also be shown.

Thomas Schmidt -- Dirichlet problems with rough data for the 1-Laplace equation. 

The talk is concerned with BV solutions and BV supersolutions to the 1-Laplace equation (or to the minimal surface equation), and the focus is on notions of up-to-the-boundary BV supersolutions which are based on Anzellotti type pairings and precise representatives. On the basis of recent joint works with Lisa Beck (Augsburg) and Christoph Scheven (Duisburg-Essen), it will be shown that these notions relate to convex duality and make sense even on rough domains, for generalized Dirichlet data, and/or in the presence of additional constraints. 

Edriss Titi -- Global regularity for certain incompressible fluid models with anisotropic dissipation. 

In this talk  I will present some recent results concerning the global regularity of certain incompressible fluid dynamics models with anisotropic dissipation. Specifically,  I will consider the two-dimensional incompressible  Boussinesq equations (Rayleigh-Bénard problem),  and the three-dimensional primitive equations of oceanic and atmospheric dynamics. In addition, I will show that for certain class of initial data the solutions to the inviscid primitive equations develop finite-time singularity. Notably, such a result is still out of reach for the two-dimensional Boussinesq equations without any dissipation. The latter is considered to be major open problem in applied analysis due to its similarity to the question of global regularity of the three-dimensional axi-symmetric incompressible  Euler equations with swirl.

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