# PDE CDT Lunchtime Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

For nematic elastomers in a membrane limit, one expects in the elastic theory an interplay of material and structural non-linearities. For instance, nematic elastomer material has an associated anisotropy which allows for the formation of microstructure via nematic reorientation under deformation. Furthermore, polymeric membrane type structures (of which nematic elastomer membranes are a type) often wrinkle under applied deformations or tractions to avoid compressive stresses. An interesting question which motivates this study is whether the formation of microstructure can suppress wrinkling in nematic elastomer membranes for certain classes of deformation. This idea has captured the interest of NASA as they seek lightweight and easily deployable space structures, and since the use of lightweight deployable membranes is often limited by wrinkling.

In order to understand the interplay of these non-linearities, we derive an elastic theory for nematic elastomers of small thickness. Our starting point is three-dimensional elasticity, and for this we incorporate the widely used model Bladon, Terentjev and Warner for the energy density of a nematic elastomer along with a Frank elastic penalty on nematic reorientation. We derive membrane and bending limits taking the thickness to zero by exploiting the mathematical framework of Gamma-convergence. This follows closely the seminal works of LeDret and Raoult on the membrane theory and Friesecke, James and Mueller on the bending theory.

ill-posed PDE, thus any limit dynamics might feature measure-valued solutions, phases interfaces, and hysteretic interface motion.

interfaces with non-trivial dynamics and study the rigorous passage to the limit for a piecewise affine nonlinearity.

In this talk, I will describe how to use the partial hodograph-Legendre transformation to show the analyticity of the free boundary in the elliptic thin obstacle problem. In particular, I will discuss the invertibility of this transformation and show that the resulting fully nonlinear PDE has a subelliptic structure. This is based on a joint work with Herbert Koch and Arshak Petrosyan.

Those mentioned in the title are integral functionals of the Calculus of Variations characterized by the fact of having an integrand switching between two different kinds of degeneracies, dictated by a modulating coefficient. They have introduced by Zhikov in the context of Homogenization and to give new examples of the related Lavrentiev phenomenon. In this talk I will present some recent results aimed at drawing a complete regularity theory for minima.

We shall discuss the problem of the 'trend to equilibrium' for a degenerate kinetic linear Fokker-Planck equation. The linear equation is assumed to be degenerate on a subregion of non-zero Lebesgue measure in the physical space (i.e., the equation is just a transport equation with a Hamiltonian structure in the subregion). We shall give necessary and sufficient geometric condition on the region of degeneracy which guarantees the exponential decay of the semigroup generated by the degenerate kinetic equation towards a global Maxwellian equilibrium in a weighted Hilbert space. The approach is strongly influenced by C. Villani's strategy of 'Hypocoercivity' from Kinetic theory and the 'Bardos-Lebeau-Rauch' geometric condition from Control theory. This is a joint work with Frederic Herau and Clement Mouhot.

We consider the layer potentials associated with operators $L=-\mathrm{div}A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent. A "Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi-Nash-Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.