Topics in variational problems arising from models in physics- postponed from 06-11.06.2021

24.07.2022 - 29.07.2022 | Będlewo


Problems in physics can often be given a variational formulation, or more precisely, can be formulated in terms of differential equations whose solutions are critical points of a suitable Euler-Lagrange functional. Below we give some particular examples but the scope of the conference will not be restricted to them.

1. Segregation, interaction of species and related free boundary problems.
Partial differential equations are often used to describe physical laws governing a complicated system in a confined space with relatively simpler behaviour on the boundary. Thus, reaction-diffusion systems with strong interaction terms appear in many multi-species physical problems as well as in population dynamics, chemistry and material science. The qualitative properties of the solutions and their limiting profiles in different regimes have been at the center of the community's attention in recent years. A prototypical example appears when looking for solitary wave solutions for Bose-Einstein condensates of two (or more) different hyperfine states which overlap in space. Typically the forces between particles in the same state are attractive while those between particles in different states can be either attractive or repulsive. If the condensates repel, they eventually separate spatially giving rise to a free boundary. This phenomenon is called phase separation and has been described in recent literature, both physical and mathematical.
One of the most interesting problems researchers investigate is when different phases of matter, populations, or clusters exist in a single space (i.e. in adjacent cells). Their interest focuses not only in how these different phases/populations/clusters interact with one another, but also on the properties of the boundaries separating them. The recent literature shows that the walls separating the different phases are geometrically tractable surfaces, as well as multiple junctions among them. This involves developing novel variational methods and geometric measure theory and free boundary tools. Relevant connections have been established with optimal partition problems involving spectral functionals. The classification of entire solutions and the geometric aspects of the nodal sets of solutions are of fundamental importance as well. We intend to focus on the most recent development of the theory in connection with problems featuring anomalous diffusions, long-range and non symmetric interactions.

2. Curl-curl equation.
The system of nonlinear equations involving the curl-curl operator can be derived from Maxwell's equations and, depending on the ansatz, describes time-harmonic or travelling electromagnetic waves. Although Maxwell’s equations are linear, the polarization contains in general a non-negligible nonlinear part unless the amplitude is small (a typical example is a Kerr-like nonlinearity). There is some earlier work on this subject where by either imposing suitable symmetry assumptions or by making a so-called scalar approximation the system is reduced to a single semilinear equation involving the Laplacian. Only recently it became possible to treat the full system. There are many major open problems concerning this system. E.g., existence and symmetries of instantons; almost nothing is known if permittivity is positive. We intend to discuss the latest development and hopefully find some new methods for approaching the problems.

We mention a few more examples, without elaborating on the details: nonlocal problems, e.g. the Choquard equation; nonlinear perturbation of Maxwell equations, e.g. the Born-Infeld equations; the Allen-Cahn equation; N-vortex Hamiltonian systems.



8:00-9:00 Breakfast Breakfast Breakfast Breakfast Breakfast
9:00-9:50 Bartsch Pacella Jeanjean Pistoia Kowalczyk
9:50-10:00 Break Break Break Break Break
10:00-10:50 Siemianowski Bieganowski De Filippis Baldelli Karch
10:50-11:20 Coffee break Coffee break Coffee break Coffee break Coffee break
11:20-12:10 Pomponio d'Avenia Reichel Mandel  
12:10-13:00 Saldaña Schino Lunch De Regibus Lunch
13:00-14:30 Lunch Lunch 13:30 Excursion Lunch  
14:30-15:20 Grossi Ruiz   Maia  
15:20-16:10 Vita Gallo   Bernini  
16:10-16:20 Break Break   Break  
16:20-17:10 Kryszewski Pellacci   Jendrej  
17:10-18:00       Terracini  
18:00-- Barbecue Dinner Reception 18:30 Dinner  


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