Bruno Pini Mathematical Analysis Seminar 2022-01-17T10:45:13+01:00 Annamaria Montanari Open Journal Systems <strong>Bruno Pini Mathematical Analysis Seminar (BPMAS) – ISSN 2240-2829</strong> publishes seminars solicited by the Department of Mathematics, University of Bologna. It features recent developments in mathematical analysis. Intrinsic fractional Taylor formula 2022-01-04T17:43:56+01:00 Maria Manfredini <p>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.</p> 2022-01-17T00:00:00+01:00 Copyright (c) 2021 Maria Manfredini The synergistic interplay of Amyloid beta and tau proteins in Alzheimer's disease: a compartmental mathematical model 2022-01-05T09:26:56+01:00 Maria Carla Tesi <p>The purpose of this Note is to present and discuss some mathematical results concerning&nbsp;a compartmental model for the synergistic interplay of Amyloid beta and tau proteins&nbsp;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.&nbsp;We provide a well-posedness results for our system of equations.&nbsp;This work extends results obtained in collaboration with M.Bertsch, B.Franchi and A.Tosin.</p> 2022-01-17T00:00:00+01:00 Copyright (c) 2021 Maria Carla Tesi Towards semi-classical analysis for sub-elliptic operators 2022-01-05T10:54:25+01:00 Véronique Fischer <p>We discuss the recent developments of semi-classical and micro-local analysis&nbsp;in the context of nilpotent Lie groups and for sub-elliptic operators.&nbsp;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.</p> 2022-01-17T00:00:00+01:00 Copyright (c) 2021 Véronique Fischer Higher-order fractional Laplacians: An overview 2022-01-05T11:58:05+01:00 Nicola Abatangelo <p>We summarize some of the most recent results regarding the theory of higher-order fractional Laplacians,&nbsp;i.e., the operators obtained by considering (non-integer) powers&nbsp;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.&nbsp;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.</p> 2022-01-17T00:00:00+01:00 Copyright (c) 2021 Nicola Abatangelo On minmax characterization in non-linear eigenvalue problems 2022-01-05T12:24:41+01:00 Shirsho Mukherjee <p>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.</p> 2022-01-17T00:00:00+01:00 Copyright (c) 2021 Shirsho Mukherjee Optimization problems with non-local repulsion 2022-01-05T12:42:14+01:00 Berardo Ruffini <p>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]</p> 2022-01-17T00:00:00+01:00 Copyright (c) 2021 Berardo Ruffini Regularity of the free boundary in the one-phase Stefan problem: a recent approach 2022-01-05T14:39:14+01:00 Nicolò Forcillo <p><span id="WEBEXT">In this note, we discuss about the regularity of the free boundary for the&nbsp;</span>solutions of the one-phase Stefan problem. We start by recalling the classical results&nbsp;achieved by I. Athanasopoulos, L. Caffarelli, and S. Salsa in the more general setting of&nbsp;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.</p> 2022-01-17T00:00:00+01:00 Copyright (c) 2021 Nicolò Forcillo