In the talk I will discuss some results concerning geometric flows. In particular, we will focus on flows in the periodic setting with a volume constraint and whose motion is depending on their mean curvature, which (formally) arise as gradient flows for the perimeter functional. After a general introduction on the topic, we will discuss a novel geometric inequality, which takes the form of a quantitative Alexandrov theorem. We will then show how to use this inequality to prove global existence and to characterize the asymptotic behaviour for some instances of volume-preserving geometric flows. Our results apply to the volume-preserving mean curvature flow, the surface diffusion flow and the Mullins-Sekerka flow. This work is based on a collaboration with Anna Kubin (TU Wien), Andrea Kubin (University of Jyväskylä) and Antonia Diana (Sapienza University).

Geometric gradient flows for elastic energies play an important role in mathematics and in many applications. In this talk, we consider a curve with boundary points free to move on a line in $\mathbb{R}^2$, which evolves by the $L^2$-gradient flow of the elastic energy, that is a linear combination of the Willmore and the length functionals. For this planar evolution problem, we first study the short and long-time existence, then we analyze the stability using a Lojasiewicz-Simon type inequality.

We study a diffuse interface model that describes the flow of a fluid through a deformable porous medium composed of two phases. The system non-linearly couples Biot’s equations for poroelasticity, including phase-field dependent material properties, with the Cahn--Hilliard equation to model the evolution of the solid, where we further distinguish between the absence and presence of a visco-elastic term of Kelvin--Voigt type. Among many exciting applications, tumour growth modelling is particularly promising since associating the phase-field variable with malignant and healthy tissue allows for the inclusion of varying properties such as (visco-)elastic responsiveness or porosity. The coupling to a deformation field, which is also influenced by interstitial fluid pressure, further permits the introduction of growth-inhibiting effects due to mechanical stress. In this talk, we will derive the system as the gradient flow of a generalised Ginzburg--Landau energy and discuss recent results on the existence of weak and strong solutions.

In this talk, we investigate a variant of a two-phase Canham-Helfrich energy, proposed as a model for lipid raft formation in cell membranes. The model features a coupling between an order parameter and the local curvature of the membrane, which energetically favors the emergence of patterns in some parameter regimes. As a first step toward understanding the mechanisms of pattern formation, we derive a scaling law for the infimum of the energy in terms of the model parameters. For the proof of an Ansatz-free lower bound, we develop novel nonlinear and nonlocal interpolation inequalities which bound fractional Sobolev seminorm in terms of Modica-Mortola energy. In addition, we study the associated gradient flow equation and establish the existence of weak solutions, both in the case of regular and singular potentials. Finally, we present numerical simulations that exhibit pattern formation in various parameter regimes. The talk is based on joint work with Janusz Ginster and Barbara Zwicknagl, and on a collaboration with Patrik Knopf and Dennis Trautwein.

"Multiphase elastoplastic materials" is the name given to a class of media which present both elastic and plastic properties. In particular, we work in the single-slip, finite crystal plasticity regime, namely we assume that dislocations can move along one active direction only and with a finite amount of slip. Moreover, we focus on materials that are characterized by two different elastic phases which can interact with each other under suitable rank-one connectedness assumptions. After stating the specifics of the model, we focus on the understanding of the effective behaviour of the composite, namely the quasiconvex hull of the class of deformation gradients and the quasiconvex envelope of the energy density.

In this talk, we will establish a relaxation result in the function space $BV^{\mathcal{B}}$. The operators $\mathcal{B}$ are linear, homogeneous partial differential operators of arbitrary order satisfying the constant rank property. Such operators can be viewed as the potential of the classical differential operators $\mathcal{A}$, introduced by Fonseca and Mueller. We will give an integral representation of the relaxed energy of functionals with linear growth. Moreover, we will present an application of such a relaxation result. This talk is based on joint work with E. Zappale.

In this talk we will present a Gamma-convergence result for a non-local Modica-Mortola type energy $\lambda_\varepsilon \int W(u_\varepsilon) + \varepsilon |u_\varepsilon|_{H^{1/2}}^2$. We first review some basic facts about Modica-Mortola type energy functionals and known results and then focus on explaining some of the ideas which allow to show Gamma-convergence in an elementary way.

In the talk, we discuss scaling laws for singular perturbation problems of "staircase type" within the geometrically linearized theory of elasticity. More precisely, we focus on a three-well problem and show that the scaling depends both on the lamination order of the prescribed Dirichlet boundary data and on the number of (non-)degenerate symmetrized rank-one directions in the symmetrized lamination convex hull. Our analysis is based on localization techniques in Fourier space. This is joint work with Angkana Rüland.