Bruno Pini Mathematical Analysis Seminar

Preface

2023-06-19T11:27:28+02:00

Nonlinear PDEs is one of the traditional topics developed by the Italian school of Analysis since its early days. These have recently met with the theory of nonlocal operators, which has been a trending topic in the international community in the past decade or so. The awareness of this fact has been the driving idea for organizing an event going over some of the most recent results about nonlinear and/or nonlocal equations, in a mixture of the two bridging classical ideas and new challenges, classical problems and new approaches. Such an event has taken place at the Mathematics Department of the University of Bologna on September 8th and 9th 2022 under the title “NonPUB - Nonlocal and Nonlinear Partial Differential Equations at the University of Bologna”.

Recurrence of the random process governed with the fractional Laplacian and the Caputo time derivative

2023-06-19T11:41:51+02:00

We are addressing a parabolic equation with fractional derivatives in time and space that governs the scaling limit of continuous-time random walks with anomalous diffusion. For these equations, the fundamental solution represents the probability density of finding a particle released at the origin at time 0 at a given position and time. Using some estimates of the asymptotic behaviour of the fundamental solution, we evaluate the probability of the process returning infinite times to the origin in a heuristic way. Our calculations suggest that the process is always recurrent.

The Brezis-Nirenberg problem for mixed local and nonlocal operators

2023-06-19T11:53:52+02:00

In this note we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a perturbed critical problem driven by a mixed local and nonlocal linear operator. We develop an existence theory, both in the case of linear and superlinear perturbations; moreover, in the particular case of linear perturbations we also investigate the mixed Sobolev inequality associated with this problem, detecting the optimal constant, which we show that is never achieved.

Some notes on functions of least W^s,1-fractional seminorm

2023-06-19T12:12:27+02:00

In this survey we discuss some existence and asymptotic results, originally obtained in [4,3], for functions of least W^s,1-fractional seminorm. We present the connection between these functions and nonlocal minimal surfaces, leveraging this relation to build a function of least fractional seminorm. We further prove that a function of least fractional seminorm is the limit for p → 1 of the sequence of minimizers of the W^s,p-energy. Additionally, we consider the fractional 1-Laplace operator and study the equivalence between weak solutions and functions of least fractional seminorm.

Nonlocal Neumann boundary conditions

2023-06-19T14:35:47+02:00

We present some properties of a nonlocal version of the Neumann boundary conditions associated to problems involving the fractional p-Laplacian. For this problems, we show some regularity results for the general case and some existence results for particular types of problems. When p=2, we give a generalization of the boundary conditions in which both the nonlocal and the classic Neumann conditions are present, and we consider problems involving both nonlocal and local interactions.

Liouville-type results for the Lane-Emden equation

2023-06-19T12:31:10+02:00

We present some Liouville-type result for the Lane-Emden equation in the subcritical and in the critical regimes. In particular, we focus on the so-called critical p-Laplace equation.

A detour on a class of nonlocal degenerate operators

2023-06-19T12:42:33+02:00

We present some recent results on a class of degenerate operators which are modeled on the fractional Laplacian, converge to the truncated Laplacian, and are extremal among operators with fractional diffusion along subspaces of possibly different dimensions. In particular, we will recall basic properties of these operators, validity of maximum principles, and related phenomena.