Bruno Pini Mathematical Analysis Seminar

Intrinsic fractional Taylor formula

2022-01-04T17:43:56+01:00

We consider a class of non-local ultraparabolic Kolmogorov operators and a suitable fractional Holder spaces that take into account the intrinsic sub-riemannian geometry induced by the operators. We prove an intrinsic fractional Taylor formula in such spaces with global bounds for the remainder given in terms of the norm naturally associated to the differential operator.

The synergistic interplay of Amyloid beta and tau proteins in Alzheimer's disease: a compartmental mathematical model

2022-01-05T09:26:56+01:00

The purpose of this Note is to present and discuss some mathematical results concerning a compartmental model for the synergistic interplay of Amyloid beta and tau proteins in the onset and progression of Alzheimer's disease. We model the possible mechanisms of interaction between the two proteins by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of misfolded tau and a kinetic-type transport equation for a function taking into accout the degree of malfunctioning of neurons. We provide a well-posedness results for our system of equations. This work extends results obtained in collaboration with M.Bertsch, B.Franchi and A.Tosin.

Towards semi-classical analysis for sub-elliptic operators

2022-01-05T10:54:25+01:00

We discuss the recent developments of semi-classical and micro-local analysis in the context of nilpotent Lie groups and for sub-elliptic operators. In particular, we give an overview of pseudo-differential calculi recently defined on nilpotent Lie groups as well as of the notion of quantum limits in the Euclidean and nilpotent cases.

Higher-order fractional Laplacians: An overview

2022-01-05T11:58:05+01:00

We summarize some of the most recent results regarding the theory of higher-order fractional Laplacians, i.e., the operators obtained by considering (non-integer) powers greater than 1 of the Laplace operator. These can also be viewed as the nonlocal counterparts of polylaplacians. In this context, nonlocality meets polyharmonicity and together they pose new challenges, producing at the same time surprising and complex structures. As our aim is to give a fairly general idea of the state of the art, we try to keep the presentation concise and reader friendly, by carefully avoiding technical complications and by pointing out the relevant references. Hopefully this contribution will serve as a useful introduction to this fascinating topic.

On minmax characterization in non-linear eigenvalue problems

2022-01-05T12:24:41+01:00

This is a note based on the paper [20] written in collaboration with N. Fusco and Y. Zhang. The main goal is to introduce minimax type variational characterization of non-linear eigenvalues of the p-Laplacian and other results related to shape and spectral optimization problems.

Optimization problems with non-local repulsion

2022-01-05T12:42:14+01:00

We review some optimization problems where an aggregating term is competing with a repulsive one, such as the Gamow liquid drop model, the Lord Rayleigh model for charged drops, and the ground state energy for the Hartree equation. As an original contribution, we show that for large values of the mass constraint, the ball is an unstable critical point of a functional made up as the sum of the first eigenvalue of the Dirichlet-Laplacian plus a Riesz-type repulsive energy term, in support to a recent open question raised in [MR21]

Regularity of the free boundary in the one-phase Stefan problem: a recent approach

2022-01-05T14:39:14+01:00

In this note, we discuss about the regularity of the free boundary for the solutions of the one-phase Stefan problem. We start by recalling the classical results achieved by I. Athanasopoulos, L. Caffarelli, and S. Salsa in the more general setting of the two-phase Stefan problem. Next, we focus on some recent achievements on the subject, obtained with Daniela De Silva and Ovidiu Savin starting from the techniques already known for one-phase problems governed by elliptic operators.