[CvGmt News] Seminario Matteo Bonforte 12/7/2016

Alfonso Sorrentino sorrentino at mat.uniroma2.it
Mon Jul 4 12:14:25 CEST 2016


SEMINARIO DI ANALISI ED EQUAZIONI DIFFERENZIALI
Dipartimento di Matematica
Università degli Studi di Roma "Tor Vergata"

Martedi' 12 Luglio 2016, ore 14:30 Aula Dal Passo

Matteo Bonforte
Universidad Autonoma de Madrid

Title: Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains

Abstract. We investigate  quantitative properties of nonnegative solutions
$u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t
u + \mathcal{L} F(u)=0$ posed in a bounded domain, $x\in\Omega\subset
\mathbb{R}^N$\,, with appropriate homogeneous Dirichlet boundary
conditions. As $\mathcal{L}$ we can use a quite general class of linear
operators that includes the three most common versions of the fractional
Laplacian $(-\Delta)^s$, $0<s<1$, in a bounded domain with zero Dirichlet
boundary conditions, but it  also includes many other examples, since our
theory only needs some basic properties that are typical of  ``linear heat
semigroups''.  The nonlinearity $F$ is assumed to be increasing and is
allowed to be degenerate, the prototype being $F(u)=|u|^{m-1}u$, with
$m>1$\,.
  We will present some recent results about existence, uniqueness and a
priori estimates for a quite large class of very weak solutions, that we
call weak dual solutions. We then show sharper lower and upper estimates
up to the boundary, which fairly combine into various forms of Harnack
inequalities. The standard Laplacian case $s=1$ is included and the
linear case $m=1$ can be recovered in the limit in most of the results.
  When the nonlinearity is of the form $F(u)=|u|^{m-1}u$, with $m>1$,
global Harnack estimates are the key tool to understand the sharp
asymptotic behaviour of the solutions.
   We finally show that solutions are classical,  and even C^{\infty} in
space in the interior of the domain, when the operator $\mathcal{L}$ is
the (restricted) fractional Laplacian.
The above results are contained on a series of recent papers with J. L.
Vazquez, and also with A. Figalli, Y. Sire and X. Ros-Oton.


Website: http://www.mat.uniroma2.it/~castorin/Seminario-PDE.html

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Email: sorrentino at mat.uniroma2.it



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