We prove that weak solutions to a quasilinear hypoelliptic equations with bounded measurable coefficients are H\”older continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and regularity transfers developed in kinetic theory. The latter tool is used repeatedly: first in the proof of the local gain of integrability of sub-solutions, second in proving that the gradient with respect to the speed variable is , third, in the proof of an “hypoelliptic isoperimetric De Giorgi lemma”. To get such a lemma, we develop a new method which combines the classical isoperimetric inequality on the diffusive variable with the structure of the integral curves of the first-order part of the operator. It also uses that the gradient of solutions w.r.t. v is .
Cédric Villani was on BBC 2 on the occasion of the publication of the book “Birth of a Theorem” (translation of his book “Théorème vivant” describing the making of our theorem).
Here is the link (Villani starts at 37:45):
Also, BBC RADIO 4 has a series of programs with Cedric Villani this week:
1. The Mathematical Mind with Cedric Villani: http://www.bbc.co.uk/programmes/b054pj6y
2. Book of the week (Birth of a Theorem by Cedric Villani):
Episode 1: http://www.bbc.co.uk/programmes/b054pmgr
Episode 2: http://www.bbc.co.uk/programmes/b054qc1m
Episode 3: http://www.bbc.co.uk/programmes/b054qd81
Episode 4: http://www.bbc.co.uk/programmes/b054t3s4
Episode 5: http://www.bbc.co.uk/programmes/b054tl6z
A 10mn movie about the scientific research of Sara Merino-Aceituno, PhD student co-supervised by myself and James Norris, and featuring several other members of the “geometric analysis and partial differential equations” group in Cambridge:
It will held this year in Cambridge on the 20-21 March, 2014 in the CMS.
All the informations can be found on: http://www.maths.cam.ac.uk/postgrad/cca/oxbridgepde2014.html
Some postdoctoral and lectureship positions are open to applications in the DPMMS (follow the link). Please note the closing date 4 january 2014 for the postdoctoral positions and 17 january 2014 for the lectureship positions.
Contact us if you wish to participate.
All informations available on the website: