# Seminario

#### Otros Seminarios

SEMINARIO DOBLE DE ECUACIONES EN DERIVADAS PARCIALES - Nonlocal Fast Diffusion Equation on Bounded Domains // A Steklov Version of the Torsional Rigidity

Fecha:  viernes 18 de marzo de 2022 - 12:30
Lugar:  Aula 520, Departamento de Matemáticas, UAM
Online:  us06web.zoom.us/j/84512065488

Resumen:

"Nonlocal Fast Diffusion Equation on Bounded Domains", Peio Ibarrondo (Universidad Autónoma de Madrid)

Abstract: We study the Cauchy-Dirichlet Problem for a nonlinear and  nonlocal diffusion equation of singular type on bounded domains. Namely, the equations is of the form $$\partial_t u=-\mathcal{L}u^m$$, where $$\mathcal{L}$$ is a nonlocal diffusion operator and $$m$$ belongs to $$(0,1)$$. Our results provide a complete basic theory including existence and uniqueness in the biggest class of data known so far, sharp smoothing estimates with weighted and unweighted $$L^p$$-norms, and extinction of solutions in finite time. We will compare two strategies to prove smoothing effects: Moser iteration VS Green function method. This is a joint work with M. Bonforte and M. Ispizua.

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"A Steklov Version of the Torsional Rigidity", Mikel Ispizua (Universidad Autónoma de Madrid)

Abstract: Torsional Rigidity is the twisting of an object due to an applied torque.  The correct formulation of the torsion problem for a beam with a certain cross section was given by the French mechanician and mathematician A.B. de Saint Venant in the middle of XIXth century. He also stated that  the simply connected cross section with maximal torsional rigidity is a circle.

We study a boundary variant of the torsional rigidity problem. This corresponds to the sharp constant for the continuous trace embedding of W1,2(Ω) in L1(∂Ω). We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.