Forthcoming Seminars › Seminar Series

OxPDE Lunchtime Seminar

Thu, 20/10/2011
12:30 		Dmitry Beliaev (Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Systems of PDE in percolation and other lattice models
Thu, 17/11/2011
12:30 		Parth Soneji (Oxford Centre for Nonlinear PDE) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Lower Semicontinuity in BV, Quasiconvexity, and Super-linear Growth
An overview is given of some key issues and definitions in the Calculus of Variations, with a focus on lower semicontinuity and quasiconvexity. Some well known results and instructive counterexamples are also discussed. We then move to consider variational problems in the BV setting, and present a new lower semicontinuity result for quasiconvex integrals of subquadratic growth. The proof of this requires some interesting techniques, such as obtaining boundedness properties for an extension operator, and exploiting fine properties of Sobolev maps.
Thu, 24/11/2011
12:30 		Federica Dragoni (Cardiff University) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Properties of $ \mathcal{X} $-convex functions and $ \mathcal{X} $-subdifferential
In the first part of the talk I will introduce a notion of convexity ($ \mathcal{X} $-convexity) which applies to any given family of vector fields: the main model which we have in mind is the case of vector fields satisfying the Hörmander condition. Then I will give a PDE-characterization for $ \mathcal{X} $-convex functions using a viscosity inequality for the intrinsic Hessian and I will derive bounds for the intrinsic gradient and intrinsic local Lipschitz-continuity for this class of functions.
In the second part of the talk I will introduce a notion of subdifferential for any given family of vector fields (namely $ \mathcal{X} $-subdifferential) and show that a non empty $ \mathcal{X} $-subdifferential at any point characterizes the class of $ \mathcal{X} $-convex functions. As application I will prove a Jensen-type inequality for $ \mathcal{X} $-convex functions in the case of Carnot-type vector fields. (Joint work with Martino Bardi).
Thu, 01/12/2011
12:30 		Guido De Philippis (Scuola Normale Superiore di Pisa) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Sobolev regularity for solutions of the Monge-Ampère equation and application to the Semi-Geostrophic system
I will talk about $ W^{2,1} $ regularity for strictly convex Aleksandrov solutions to the Monge Ampère equation
\[ \det D^2 u =f \]
where $ f $ satisfies $ \log f\in L^{\infty} $. Under the previous assumptions in the 90's Caffarelli was able to prove that $ u \in C^{1,\alpha} $ and that $ u\in W^{2,p} $ if $ |f-1|\leq \varepsilon(p) $. His results however left open the question of Sobolev regularity of $ u $ in the general case in which $ f $ is just bounded away from $ 0 $ and infinity. In a joint work with Alessio Figalli we finally show that actually $ |D^2u| \log^k |D^2 u| \in L^1 $ for every positive $ k $.
If time will permit I will also discuss some question related to the $ W^{2,1} $ stability of solutions of Monge-Ampère equation and optimal transport maps and some applications of the regularity to the study of the semi-geostrophic system, a simple model of large scale atmosphere/ocean flows (joint works with Luigi Ambrosio, Maria Colombo and Alessio Figalli).

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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