Journée FNRS/Séminaire Analyse-EDP (joint avec ULB).

Tuesday 3 Dec 2019, 10:00 → 17:00 Europe/Brussels

Bâtiment NO, 9ème étage (ULB)

10.00h-11.00h

 

Frédéric Robert (Nancy), Impact of localization of the Hardy potential on the stability of Pohozaev obstructions

Abstract: The Pohozaev obstruction yields a sufficient condition on the potential for the nonexistence of solutions to some nonlinear elliptic PDE on a star-shaped domain. This condition (C) is not stable under perturbation of the potential. Unlike large dimensions, Druet-Laurain have proved that the Pohozaev obstruction is stable in small dimension. In this talk, I will discuss this issue on PDEs involving Hardy-type potential. The stability of the obstruction depends both on the dimension and on the localization of the Hardy weight.

11.00h-11.30h: Coffee break

11.30h-12.30h

 

Henrik Schumacher (RWTH Aachen), Gradient Flows for the Möbius Energy

Abstract: Aiming at optimizing the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to certain fractional-order Sobolev scalar products that are adapted to the Möbius energy. In contrast to L^2-gradient flows, the resulting flows are ordinary differential equations on an infinite-dimensional manifold of embedded curves. In the fully discrete setting, this allows us to completely decouple the time step size from the spatial discretization, resulting in a very robust optimization algorithm that is orders of magnitude faster than following the discrete L^2-gradient flow.

12.30h-14.30h: Lunch break

14.30h-15.30h

 

Angela Pistoia (Rome), Elliptic systems with critical growth

Abstract: I will present some results concerning the existence of nodal solutions to the Yamabe equation on the sphere and their connections with the existence of positive solutions to competitive elliptic systems with critical growth in the whole space.

15.30h-16.00h: Coffee break

16.00h-17.00h

 

Flaviana Iurlano (Paris), Concentration versus oscillation effects in brittle damage

Abstract: This talk is concerned with an asymptotic analysis, in the sense of Γ-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order ε, the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at ε fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as ε→0, concentration and saturation of damage are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at ε fixed) to being of linear-growth type (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. We explicitly identify the Γ-limit in two and three dimensions for isotropic Hooke tensors. The expression of the limit effective energy turns out to be of Hencky plasticity type.

The abstracts with a proper display of LaTeX code can be found at https://perso.uclouvain.be/heiner.olbermann/seminar.html