204 research outputs found
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation
Starting from the quantitative stability result of Bianchi and Egnell for the
2-Sobolev inequality, we deduce several different stability results for a
Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the
connection between this inequality and a fast diffusion equation, we get a
quantitative stability for the Log-HLS inequality. Finally, using all these
estimates, we prove a quantitative convergence result for the critical mass
Keller-Segel system
Stability for Borell-Brascamp-Lieb inequalities
We study stability issues for the so-called Borell-Brascamp-Lieb
inequalities, proving that when near equality is realized, the involved
functions must be -close to be -concave and to coincide up to
homotheties of their graphs.Comment: to appear in GAFA Seminar Note
Second order stability for the Monge-Ampere equation and strong Sobolev convergence of Optimal Transport Maps
The aim of this note is to show that Alexandrov solutions of the Monge-Ampere equation, with right hand side bounded away from zero and infinity, converge strongly in W2,1loc if their right hand side converge strongly in L1loc. As a corollary we deduce strong W1,1loc stability of optimal transport maps
Optimal regularity and structure of the free boundary for minimizers in cohesive zone models
We study optimal regularity and free boundary for minimizers of an energy
functional arising in cohesive zone models for fracture mechanics. Under
smoothness assumptions on the boundary conditions and on the fracture energy
density, we show that minimizers are , and that near non-degenerate
points the fracture set is , for some .Comment: 39 page
On the Regularity of Optimal Transportation Potentials on Round Spheres
In this paper the regularity of optimal transportation potentials defined on
round spheres is investigated. Specifically, this research generalises the
calculations done by Loeper, where he showed that the strong (A3) condition of
Trudinger and Wang is satisfied on the round sphere, when the cost-function is
the geodesic distance squared. In order to generalise Loeper's calculation to a
broader class of cost-functions, the (A3) condition is reformulated via a
stereographic projection that maps charts of the sphere into Euclidean space.
This reformulation subsequently allows one to verify the (A3) condition for any
case where the cost-fuction of the associated optimal transportation problem
can be expressed as a function of the geodesic distance between points on a
round sphere. With this, several examples of such cost-functions are then
analysed to see whether or not they satisfy this (A3) condition.Comment: 24 pages, 4 figure
Mass Transportation on Sub-Riemannian Manifolds
We study the optimal transport problem in sub-Riemannian manifolds where the
cost function is given by the square of the sub-Riemannian distance. Under
appropriate assumptions, we generalize Brenier-McCann's Theorem proving
existence and uniqueness of the optimal transport map. We show the absolute
continuity property of Wassertein geodesics, and we address the regularity
issue of the optimal map. In particular, we are able to show its approximate
differentiability a.e. in the Heisenberg group (and under some weak assumptions
on the measures the differentiability a.e.), which allows to write a weak form
of the Monge-Amp\`ere equation
Some new well-posedness results for continuity and transport equations, and applications to the chromatography system
We obtain various new well-posedness results for continuity and transport
equations, among them an existence and uniqueness theorem (in the class of
strongly continuous solutions) in the case of nearly incompressible vector
fields, possibly having a blow-up of the BV norm at the initial time. We apply
these results (valid in any space dimension) to the k x k chromatography system
of conservation laws and to the k x k Keyfitz and Kranzer system, both in one
space dimension.Comment: 33 pages, minor change
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