Mathematical models for elastic materials undergoing growth will be considered. The characteristic feature is a multiplicative decomposition of the deformation gradient into an elastic part a growth-related part. Approaches towards the existence of solutions will be discussed in
various settings, including models with and without codimension. This is joint work with Kira Bangert and Julian Blawid.

In this talk, we focus on the existence of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations, thereby providing a rigorous realization of a phenomenon first conjectured by Helmholtz (1858). In the leapfrogging motion, two coaxial vortex rings periodically exchange positions, a striking behavior repeatedly observed in experiments and numerical simulations, yet lacking complete mathematical justification. Our construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, which yields a Hamiltonian description of nearly concentric vortex rings. The main difficulty stems from a singular small-divisor problem arising in the linearized transport dynamics, where the effective time scale degenerates with the ring thickness parameter. To overcome this obstruction, we develop a degenerate KAM-type analysis combined with pseudo-differential operator techniques to control the linearized dynamics around symmetric configurations. Combining these tools with a Nash-Moser iteration scheme, we construct families of nontrivial time-periodic solutions in an almost uniformly translating frame. This establishes the first rigorous construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence. 

This is a joint work with Zineb Hassainia and Taoufik Hmidi.

In this talk I will discuss recent work, with R. Tione, on the regularity of stationary points for a class of planar polyconvex integrands which are conformally-invariant, a natural assumption in view of geometric applications. We prove that, in two dimensions, stationary points are smooth away from a discrete set. We also show full C^1-regularity for orientation-preserving solutions, which appear naturally in minimization problems of Teichmüller type.

In this talk we describe several aspects related to the theory of some anisotropic parabolic equations. The anisotropy shown in such equations will appear in the form of porous medium, in the fast and porous medium diffusion regime. In particular, we show the existence of selfsimilar fundamental solutions, which is uniquely determined by its mass, and the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. 

The investigation of both models are objects of joint works with F. Feo and J. L. V´azquez.