Conference Agenda

We discuss interpolation inequalities that bound variants of fractional Sobolev seminorms in terms of Modica-Morta-type functionals. As an application, we consider a model for pattern formation in biological membranes. The latter are thin structures that are composed of various components, which are often arranged in complex patterns. We consider an energy functional from the literature that consists of a Canham-Helfrich type energy for the thin membrane and a Modica-Mortola type energy for the order parameter describing the local chemical composition. Pattern formation is driven by a coupling between the local chemical composition and the local curvature of the membrane. We will in particular discuss the scaling law for the infimal energy, and the limiting behaviour in some parameter regimes.
This talk is based on joint work with Janusz Ginster and Anastasija Pesic.

In this talk, we discuss the asymptotic behaviour of a system of nonlinear reaction--diffusion--advection equations in a domain consisting of two bulk regions connected via microscopic channels distributed within a thin membrane. Both the width of the channels and the thickness of the membrane are of order $\varepsilon \ll 1$, and the geometry evolves in time in an a priori known way.
We consider nonlinear flux boundary conditions at the lateral boundaries of the channels and critical scaling of the diffusion inside the layer. Employing two-scale convergence and unfolding techniques in thin layers we derive an effective model in the limit $\varepsilon \to 0$, in which the membrane is reduced to a lower-dimensional interface. We obtain jump conditions for the solution and the total fluxes, which involve the solutions of local, space--time-dependent cell problems in the reference channel.

In this talk I will discuss the optimal regularity of solutions to the variable coefficient parabolic Signorini problem (or parabolic thin obstacle problem), where the coefficients are in the Sobolev space $W^{1,1}_p$ with $p>n+2$ (here $n$ is the space dimension). I will talk about two main ingredients used in the proof: a parabolic Carleman estimate, and an energy/energy-dissipation inequality. This is based on joint work with Vedansh Arya.