Tuesday Seminar of Analysis

Seminar information archive ~04/01Next seminarFuture seminars 04/02~

Date, time & place Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.)
Organizer(s) ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Xavier Cabre (ICREA and UPC, Barcelona)
New isoperimetric inequalities with densities arising in reaction-diffusion problems (English)
[ Abstract ]
In joint works with X. Ros-Oton and J. Serra, the study of the
regularity of stable solutions to reaction-diffusion problems
has led us to certain Sobolev and isoperimetric inequalities
with weights.  We will present our results in these new
isoperimetric inequalities with the best constant, that we
establish via the ABP method.  More precisely, we obtain
a new family of sharp isoperimetric inequalities with weights
(or densities) in open convex cones of R^n. Our results apply
to all nonnegative homogeneous weights satisfying a concavity
condition in the cone. Surprisingly, even that our weights are
not radially symmetric, Euclidean balls centered at the origin
(intersected with the cone) minimize the weighted isoperimetric
quotient. As a particular case of our results, we provide with
new proofs of classical results such as the Wulff inequality and
the isoperimetric inequality in convex cones of Lions and Pacella.
Furthermore, we also study the anisotropic isoperimetric problem
for the same class of weights and we prove that the Wulff shape
always minimizes the anisotropic weighted perimeter under the
weighted volume constraint.