Prelectura de tesis

Mikel Ispizua Moreno

Miércoles 17 de abril del 2024, aula 520 del departamento

Hora: 12:00

Título: Some topics in Parabolic and Elliptic PDEs

Directores de tesis: Matteo Bonforte y María del Mar González

Abstract: This thesis is  a compilation of three independent works within the common area of parabolic and elliptic PDEs.

In the first one, motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, we try to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of W1,2(Ω) into L1(∂Ω). We 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.

Then, we study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type posed on a bounded domain . The linear diffusion operator is a sub-Markovian operator, allowed to be of nonlocal type, the prototype equation being the Fractional Fast Diffusion Equation. The main results shall provide a complete basic theory for solutions to (CDP): existence and uniqueness in the biggest class of data known so far, sharp smoothing estimates, boundary behaviour, extinction time and regularity of solutions.

We conclude by studying  the singular limit of a chemotaxis model of bacterial collective motion recently introduced in arXiv:2009.11048 . The equation models an aggregation-diffusion phenomena with an advection term that is discontinuous and depends sharply on the gradient of the density itself. The quasi-linearity of the problem poses major challenges in the construction of the solution and complications arise in the proof of regularity. Our method overcomes these obstacle by relying only on entropy inequalities and the theory of monotone operators. We provide existence, uniqueness and smoothing estimates in any dimensional space.