MSCA IF 892017

Free Boundary (FB) problems arise in many applications such us Biology, Physics, Finance, Fluid Dynamics: they are usually described by some Partial Differential Equations (PDEs) that exhibit in addition some unknown interfaces. A classical example is the Stefan problem, which describes the melting of an ice block into water: the main idea is to model the space-time variation of the temperature through a parabolic problem, distinguishing between the region where it is zero (ice) and where it is positive (water). The most challenging problem is to study the evolution and regularity of both the temperature distribution and its FB, that is the region which separates the ice from the water. The peculiarity of the "double unknown" makes this class of problems significantly complex: the advancements in this field have been possible thanks to some innovative techniques which combine tools from different areas like PDEs, Calculus of Variations and Geometric Measure Theory.

In the meantime, nonlocal diffusion problems have been the subject of considerable research too and, as FB problems, they possess significant applications in Finance, Industry as well as in Probability and Statistics. The interest in nonlocal problems comes from a simple empiric observation: a large class of physical phenomena are characterized by long-range interactions between particles involved in the diffusion process, which cannot be investigated with models with local diffusion (that is, based on  Brownian motion). Models with nonlocal diffusion (for example, the fractional heat equation) seem to be the right tools to fruitfully describe them, since they take into account that the moving particles can jump from a value to another in a time step: this generates probability densities exhibiting fat tails, in deep constrast with standard diffusion.

The MSCA 892017 (LNLFB-Problems) has been mainly developed around this two main areas. Below, you can find the complete list of research products and activities related to the project.

RESEARCH PRODUCTS

• A. Audrito, T. Sanz-Perela. Elliptic regularization of some semilinear parabolic free boundary problems, Interfaces Free Bound. 26 (2024), 135-159.

• A. Audrito, T. Kukuljan. Regularity theory for fully nonlinear parabolic obstacle problems, J. Funct. Anal. 285 (2023), 1-57.

• A. Audrito. On the existence and Hölder regularity of solutions to some nonlinear Cauchy-Neumann problems, J. Evol. Equ. 23 (2023), 1-45.

• A. Audrito, A. Garriz, F. Quiros. Convergence in relative error for the Porous Medium equation in a tube, Ann. Mat. Pura Appl. 203 (2023), 149-171.

• A. Audrito, J. Serra. Interface regularity for semilinear one-phase problems, Adv. Math 403 (2022), 1-51.

RESEARCH VISITS

• 11-23 June 2022: Department of Mathematics, Universidad Autónoma de Madrid.

• 20-24 December 2021: Department of Mathematics, Università degli Studi di Torino.

• 25-28 October 2021: Department of Mathematics, Università di Pisa.

RESEARCHERS HOSTED at ETHZ

• 25 June to 10 July 2022: Zihui Zhao, University of Chicago, USA.

• 26 June to 9 July 2022: Hui Yu, University of Singapore, Singapore.

• 3-9 July 2022: Bozhidar Velichkov, Università di Pisa, Italy.

• 8-12 November 2021: Giorgio Tortone, Università di Pisa, Italy.

• 11-15 October 2021: María Medina, Universidad Autónoma de Madrid, Spain.

TALKS

• Seminar: The parabolic obstacle problem with fully nonlinear diffusion, Universidad Autónoma de Madrid, Madrid, June 2022.

• Online seminar: A rigidity result for a class of elliptic semilinear one-phase problems, University of California San Diego, San Diego, USA, January 2022.

• Seminar: A variational approach to a class of nonlinear Cauchy-Neumann problems, Politecnico di Torino, Italy, December 2021.

• Seminar: A variational approach to a class of nonlinear Cauchy-Neumann problems, Università di Pisa, Italy, October 2021.

• Online seminar: 1D symmetry of solutions to a class of semilinear one-phase elliptic PDE, Analysis Seminar, Universidad Autónoma de Madrid, Madrid, Spain, 13/10/2021.

• Seminar: Interface regularity for semilinear one-phase problems, Levico Terme, Italy, September 2021.

• Online seminar: The Neumann problem for the fractional laplacian, SIAM annual meeting, July 2021.

• Online seminar: Boundary regularity of solutions to some nonlocal elliptic Neumann problems, ``Italia vs España online seminars'', 06/05/2021.

• Online seminar: The Neumann problem for some nonlocal elliptic operators: Regularity up to the boundary, ETHZ, Zürich, Switzerland, 24/11/2020.

WORKSHOPS and CONFERENCES

• "Regularity for nonlinear diffusion equations. Green functions and functional inequalities", Madrid, Spain, 13-17 June 2022.

• "Analytical Methods in Quantum and Continuum Mechanics", Torino, Italy, 29 November - 3 December 2021.

• "Regularity Theory for Free Boundary and Geometric Variational Problems", Levico Terme, Italy, 06-10 September 2021.

TEACHING

• Lecturer for the PhD course "Reaction-diffusion equations: traveling fronts and long-time behaviour", ETHZ, September-October 2021.

• Teaching assistant for the course/seminar "Classical Theory of Elliptic Partial Differential Equations", Undergraduate and Postgraduate Studies at ETHZ, 2021. Professor responsible for the course: Joaquim Serra.

COURSES and TRAINING

• "Facilitating Meetings and Workshops", March 2021.

• "Leadership Essentials", April 2021.

• "Project Management for research for postdocs and senior researchers", May-June 2021.

• "Lateral Leadership", September-November 2021.

• "Time Management", October-December 2021.

• A1.1 German course organized by ETHZ and UZH (October-December 2020).

SCIENTIFIC POPULARIZATION and COMMUNICATION ACTIVITIES

• Seminar: Some rigidity results for a class of semilinear elliptic PDE, Welcome Home 2021, Dipartimento di Matematica, Università degli Studi di Torino, 21-22/12/2021.

• Online seminar: Some Wonderful PDEs, Science is Wonderful 2021, 24/11/2021.

• Organizer of the workshop ``Zome and soap bubbles'', 2021 European Researcher's Night, Torino, Italy, 25/09/2021.

• Speaker in the online workshop ``MSCA - Postdoctoral fellowships in Horizon Europe'', Università degli Studi di Torino, 19/07/2021.

• Seminar: MSCA Proposal Preparation: some hints, in the online workshop ``Online Applicants Training for Marie Skłodowska-Curie Individual Fellowships'', ETHZ, 25/06/2020.

ORGANIZATION ACTIVITIES

•  Organizer of the workshop "Free boundary problems and related topics", DMATH ETHZ, Zurich, Switzerland, 04-08 July 2022 (in collaboration with Prof. A. Figalli and Prof. J. Serra).

THESIS SUPERVISION

•  Master Thesis: A variational problem for the spatial segregation of reaction-diffusion systems, Patrik Rosa Ferreira (co-advised with Prof. J. Serra).