March 19 2015, Prof. M. Bonforte (Universidad Autónoma de Madrid) Prof. A. Lunardi (University of Parma)

Giovedì 19 Marzo 2015, ore 14,30 precise, presso la sala conferenze dell'IMATI-CNR di Pavia,
il Prof. Matteo Bonforte (Universidad Autónoma de Madrid)
terrà un seminario dal titolo:
"A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains",

e alle ore 15,30 la

Prof. Alessandra Lunardi (Università degli studi di Parma)
terrà un seminario dal titolo:
"Integrali superficiali e tracce su superfici regolari di funzioni di Sobolev in dimensione infinita"

nell'ambito del Seminario di Matematica Applicata (IMATI-CNR e Dipartimento di Matematica, Pavia).
Nell'intervallo tra le due conferenze sarà organizzato un piccolo rinfresco.

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Abstract seminario Bonforte.

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L} (u^m)=0$, posed in a bounded domain, $x\in\Omega\subset \RR^N$ for $t>0$ and $m>1$. As $\mathcal{L}$ we can take the most common definitions of the fractional Laplacian $(-\Delta)^s$, $0<s<1$, in="" a="" bounded="" domain="" with="" zero<br="">Dirichlet boundary conditions, as well as more general classes of operators. We consider a class of very weak solutions for the equation at hand, that
we call weak dual solutions, and we obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. The standard Laplacian case $s=1$ or the linear case $m=1$ are recovered as limits. The method is quite general, suitable to be applied to a number of similar
problems that will be briefly discussed as examples. As a consequence, we can prove existence and uniqueness of minimal weak dual solutions with data in $L^1_{\Phi_1}$\,, where $\Phi_1$ is the first eigenfunction of $\mathcal{L}$. We also briefly show existence and uniqueness of $H^{-s}$
solutions with a different approach. As a byproduct, we derive similar estimates for the elliptic semilinear equation $\mathcal{L}S^m=S$ and we prove
existence and uniqueness of $H^{-s}(\Omega)$ solutions via parabolic techniques. Solutions to this elliptic problem represents the asymptotic profiles of the rescaled solutions, namely the stationary states of the rescaled equation $\partial_t v = -\mathcal{L} (v^m)+ v$.
Finally, we will study the asymptotic behaviour. We will prove sharp rates of decay of the rescaled solution to the unique stationary profile S and also for the
relative error $v/S-1$. The sharp rates of convergence can be obtained with two different methods: one is based on the above estimates, that guarantee
existence of the "friendly giant". Another approach is given by a new entropy method, based on the so-called Caffarelli-Silvestre extension. This is a joint work with J. L. Vazquez (UAM, Madrid, Spain) and Y. Sire (Univ. Marseille, France).

References:
[BV1] M. B., J. L. Vazquez, A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains. To appear in
Arch. Rat. Mech. Anal (2015) http://arxiv.org/abs/1311.6997
[BSV] M. B., Y. Sire, J. L. Vazquez, Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains.
To appear in Discr. Cont. Dyn. Sys. (2015) http://arxiv.org/abs/1404.6195

[BV2] M. B., J. L. Vazquez, Nonlinear Degenerate Diffusion Equations on bounded domains with Restricted Fractional Laplacian. In Preparation (2015).

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Abstract seminario Lunardi.

Si considerano spazi di Sobolev di funzioni con valori reali, definite in un sottoinsieme O={x: G(x)<0} di uno spazio di Banach dotato di una misura gaussiana
m, e si studiano le tracce di tali funzioni sulla superficie S= {x: G(x)=0}. Sotto ipotesi abbastanza generali su G, l'operatore traccia e' ben definito e continuo da W^{1,p}(O,m) a L^1(S,n) dove n e' una misura superficiale naturalmente associata a m. Se O e' un semispazio si caratterizza
l'immagine dell'operatore traccia come una sorta di spazio di Sobolev frazionario, in analogia al caso finito dimensionale; ma in generale, appena consideriamo insiemi diversi da semispazi, sul comportamento delle tracce si sa dire poco. Un risultato in positivo importante e' una formula di
integrazione per parti per funzioni di Sobolev che coinvolge le loro tracce su S. 

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The Applied Mathematics Seminar usually meets on Tuesdays at 3 pm in the Conference Room of IMATI-CNR, Pavia. Refreshments are offered after the talk. For further information, please contact the organizers Carlo Lovadina, Stefano Lisini and Laura Spinolo.

http://www-dimat.unipv.it/~seminari/matematica-applicata.html