TEACHING

TEACHING IN GRADUATE COURSES in Sapienza (Analysis) 

Phd in MATHEMATICAL MODELS FOR ENGINEERING, ELECTROMAGNETICS AND NANOSCIENCES

A.A. 2021/2022

Fractional Calculus and Probability, I

Mirko D'Ovidio, Raffaela Capitanelli

Abstract: In Part 1 (Mirko D'Ovidio), we introduce some basic aspects related to fractional calculus and probability. In particular, we consider time-changed Markov processes driven by Cauchy problems written in terms of non-local (time and/or space) operators. Moreover, we consider the boundary value problem in the framework of the time changes and in general, we show some connection between multiplicative functionals, semigroups and boundary value problems for PDEs and non-local PDEs. We introduce some basic notions about limit theorems, stochastic processes, PDEs connections, non-local operators with special attention for the case of fractional (Caputo) derivative and fractional Laplacian, additive and multiplicative functionals associated with boundary conditions, time changes associated with boundary conditions. We focus on Dirichlet, Neumann, Robin and Wentzell boundary conditions together with the probabilistic reading of killed, reflected, elastic and sticky processes. We also discuss some applications concerned with regular and irregular domains in the macroscopic analysis introduced by fractional equations. In Part 2 (Raffaela Capitanelli), we discuss approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general   subordinators or inverses to general subordinators. Our analysis is  based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes.

A.A. 2020/2021

Introduction to fractals and boundary control problems in irregular domains

Anna Chiara Lai, Maria Rosaria Lancia,  Raffaela Capitanelli

Part 3 (Raffaela Capitanelli-CFU 3)

Nonlinear Analysis on Fractal Structures

This module presents an introduction to some nonlinear problems on fractal structures.

1 We present existence, uniqueness and approximation results for variational solutions  for some quasilinear obstacle problems on domains with a fractal boundary. Our mail tools are suitable extension theorems, sharp quantitative trace results (on polygonal curves) in terms of the increasing numbers of sides and Poincaré type estimates  adapted to the geometry. [CFV]

2 In order to study  mass transport problem on fractal structures, we study the limit of p-Laplace type problems with obstacles as p tends to infinity. [CF]

3 Finally, we present some similar asymptotic  results on fractal domains like the Sierpinski gasket. [CCV]

[CCV] F. Camilli,  R. Capitanelli, M.A. Vivaldi, "Absolutely Minimizing Lipschitz Extensions and infinity harmonic functions on the Sierpinski gasket", Nonlinear Anal. 163 (2017), 71–85. https://doi.org/10.1016/j.na.2017.07.005

[CF] R. Capitanelli, S. Fragapane, "Asymptotics for quasilinear obstacle problems in bad domains", Discrete and Continuous Dynamical Systems - Series S 12 (2019), no.1, 43-56. http://aimsciences.org/article/doi/10.3934/dcdss.2019003

[CFV] R. Capitanelli,  S. Fragapane, M. A. Vivaldi, "Regularity results for p-Laplacian in pre-fractal domains", Advances in Nonlinear Analysis  8 (2019), no. 1, 1043–1056. https://doi.org/10.1515/anona-2017-0248