12/11/18 New preprint: Dimensional estimates and rectifiability for measures satisfying linear PDE constraints by Adolfo Arroyo-Rabasa, Guido De Philippis, Jonas Hirsch, Filip Rindler
We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators (including all first-order and all second-order operators). Our general theorem provides a new proof of the rectifiability results for functions of bounded variations and functions of bounded deformation. For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.
https://arxiv.org/abs/1811.01847
15/10/18 Preprint: Critical L^p-differentiability of BV^𝔸-maps and canceling operators (revised) by Bogdan Raita
We give a generalization of Dorronsoro’s Theorem on critical L^p-Taylor expansions for BV^k-maps on R^n, i.e., we characterize homogeneous linear differential operators 𝔸 of k-th order such that D^{k−j}u has j-th order L^{n/(n−j)}-Taylor expansion a.e. for all u∈BV^𝔸_loc (here j=1,…,k, with an appropriate convention if j≥n). The space BV^𝔸_loc consists of those locally integrable maps u such that 𝔸u is a Radon measure on ℝ^n. A new L^∞-Sobolev inequality is established to cover higher order expansions. Lorentz refinements are also considered. The main results can be seen as pointwise regularity statements for linear elliptic systems with measure-data.
https://arxiv.org/abs/1712.01251v2
28/06/18 Book Calculus of Variations by Filip Rindler has appeared with Springer
See www.calculusofvariations.com for details.
05/06/18 New preprint: Relaxation for partially coercive integral functionals with linear growth by Filip Rindler, Giles Shaw
We prove an integral representation theorem for the L1(Ω;ℝm)-relaxation of the functional
F : u↦∫ f(x,u(x),∇u(x)) dx, u∈ W^{1,1}(Ω;ℝ^m), Ω⊂ℝ^d open,
to the space BV(Ω;ℝ^m) under very general assumptions, requiring principally that f be Caratheodory, partially coercive, and quasiconvex in the final variable. Our result is the first of its kind which applies to integrands which are unbounded in the u-variable and thus allows to treat many problems from applications. Such functionals are out of reach of the classical blow-up approach introduced by Fonseca & Muller [Arch. Ration. Mech. Anal. 123 (1993), 1-49]. Our proof relies on an intricate truncation construction (in the x and u arguments simultaneously) made possible by the theory of liftings as introduced in the companion paper arXiv:1708.04165, and features techniques which could be of use for other problems featuring u-dependent integrands.
https://arxiv.org/abs/1806.00343
06/04/18 New preprint: On the two-state problem for general differential operators by Guido De Philippis, Luca Palmieri, Filip Rindler
In this note we generalize the Ball-James rigidity theorem for gradient differential inclusions to the setting of a general linear differential constraint. In particular, we prove the rigidity for approximate solutions to the two-state inclusion with incompatible states for merely L1-bounded sequences. In this way, our theorem can be seen as a result of compensated compactness in the linear-growth setting.
https://arxiv.org/abs/1803.09302
01/04/18 New team members
Adolfo Arroyo-Rabasa (Leipzig) and Bogdan Raita (Oxford) will join the project as postdocs in autumn. Anna Skorobogatova will be a PhD student in the project from autumn as well.
01/04/18 Project now officially underway
‘The funding has officially started!
24/12/17 Survey article online
You can read a more technical description of some recent results here: AfreeSurvey.
03/12/17 Postdoc job available!
We are looking for a Postdoctoral Research Fellow to join the team at the University of Warwick! The position is for 24 months with the possibility of an extension for another 12 months. Please apply here.
