Meeting 9

Location: University of Reading, Reading, UK.

Room: Agriculture Building, 1U02

Date: 6th of February 2018


11.30-12.00 Coffee-Tea impetus

12.00-12.45 Manuel Del Pino (Bath, UK)

Title: Singularity formation and bubbling in nonlinear diffusions
Abstract: A fundamental question in nonlinear evolution equations is the analysis of solutions which develop singularities (blow-up) in finite time or as time goes to infinity. We review recent results on the construction of solutions to certain notable nonlinear parabolic PDE which exhibit this kind of behavior in the form of “bubbling”. This means solutions that at main order look like asymptotically singular time-dependent scalings of a fixed finite energy entire steady state. We carry out this analysis for the classical two-dimensional harmonic map flow into the sphere and the energy-critical semilinear heat equation.

12.55-13.40 Jonathan Bevan (Surrey, UK)

Title: Estimating critical cavitation loads in nonlinear elasticity
Abstract: A well-known approach to modelling elastic materials postulates that the deformed configuration is an energy minimiser. It is possible for such a minimiser to cavitate, i.e. open a `hole’ inside the material, which corresponds to a discontinuity in the deformation. It is of particular interest to know when the minimiser is an affine map since this guarantees that phenomena like cavitation do not lower the energy. This amounts to a study of quasiconvexity, a fundamental condition discovered by C. B. Morrey in the 1950s, in the setting of nonlinear elasticity. The talk will show how quasiconvexity, rigidity estimates from the calculus of variations and singular values of gradient matrices combine to give sufficient conditions for global minimisers to be affine maps of suitable stored-energy functionals.

13.45-15.00 Lunch break

15.00-15.45 Monica Musso (Bath, UK)

Title: Existence, Compactness and Non Compactness for fractional Yamabe Problem
Abstract: Let (X^{n+1}, g^+) be an (n+1)-dimensional asymptotically hyperbolic manifold with a conformal infinity (M^n, [h]). The fractional Yamabe problem consists in finding a metric in the conformal class [h] whose fractional scalar curvature is constant. In this talk, I will present some recent results concerning existence of solutions to the fractional Yamabe problem, and also properties of compactness and non compactness of its solution set, in comparison with what is known in the classical case. These results are in collaboration with Seunghyeok Kim and Juncheng Wei.

15.55-16.40 Igor Velcic (Zagreb, Croatia)

Title: Sharp operator-norm asymptotics for linearised elastic plates with rapidly oscillating periodic properties
Abstract: In this talk we analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions. We assume that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response. Following the application of an appropriate version of the Floquet transform, we analyse the operator-norm resolvent behaviour of the operators in each fibre of the resulting direct integral, as the period and the plate thickness go to zero simultaneously. The convergence estimates we obtain are uniform with respect to both the Floquet parameter and the plate thickness, which yields order-sharp resolvent estimates for the convergence of the original plate problems as the plate thickness goes to zero. This is a joint work with Kirill Cherednichenko (university of Bath).

16.45-17.00 Coffee-Tea Break

Student session:

17.00-17.20 Ahmed Jama (Cardiff, UK)

Title: Generalised translations and periodic sets with applications to the Grushin plane
Abstract: In this talk we will introduce a new notion of translations, namely generalized translations along vector fields, and an associated notion of periodicity for sets, which apply to very general geometrical structures. We are in particular interested in applying these notions to the case of Grushin spaces. We will also prove a Poincare inequality for an unbounded periodic set in this setting.

17.20-17.40 Birzhan Ayanbayev (Reading, UK)

Title: Separable infinity-harmonic functions
Abstract: In this talk I will discuss certain newly discovered explicit solutions to the infinity-laplace equation, the fundamental PDE arising in Calculus of Variations in the space L-Infinity. These solutions obey certain symmetry requirements, and contain as particular sub-cases the already known classes of infinity-harmonic functions.

18.00 Supper