Descrizione

Pasquale Ambrosio (University of Naples “Federico II”)

“Regularity results for a class of strongly degenerateparabolic equations”

Motivated by applications to gas filtration problems, we study the

regularity of weak solutions to the strongly degenerateparabolic PDE

ut – div [((|Du| – 𝜈)+)^{p-1}Du/|Du|] = f in ΩT = Ω × (0, T),

where Ω is a bounded domain in Rn for n ≥ 2, p ≥ 2, 𝜈 is a positive

constant and (⋅)+ stands for the positive part. Assumingthat the datum

f belongs to a suitable Lebesgue-Sobolev parabolic space, weestablish

the Sobolev spatial regularity of a nonlinear function ofthe spatial

gradient Du of the weak solutions, which in turn implies theexistence

of the weak time derivative ut. Next, we weaken theassumption on the

right-hand side, by assuming that f belongs to a suitablelocal

Lebesgue-Besov space. This leads us to obtain fractionalSobolev

regularity results for a function of Du, which in turn yieldthe higher

summability of Du with respect to the spatial variable. Themain novelty

here is that the structure function of the above equationsatisfies

standard growth and ellipticity conditions only outside theball with

radius 𝜈 centered at the origin.

This is a joint research with Antonia Passarelli di Napoli(University

of Naples “Federico II”).

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Emanuele Cristoforoni (SSM)

Titolo: A thermal insulation problem witha prescribed heat source

Abstract: We study the thermal insulationof a bounded body $Omegasubsetmathbb{R}^n$, under a prescribed heat source$f > 0$, via a bulk layer of insulating material. We consider a model ofheat transfer between the insulated body and the environment determined byconvection; this corresponds to Robin boundary conditions on the free boundaryof the layer. We show that a minimal con guration exists and that it satisfiesuniform density estimates.