Abstract
We are interested in the effect of the nonlinearities on the emergence of non trivial patterns in several differential models arising in physics and other sciences. Such self organized structures correspond to selected solutions of the differential problem, possessing some special symmetries or shadowing particular shapes. We wish to understand the main analytical mechanisms involved in this process in terms of the common variational structure of the problems . A feature of this project rests indeed in the interchange of attack strategies between different specific applications in the fields of partial differential equations and systems and Hamiltonian systems. There is a remarkable unity in methodology across the different parts of the project. On the other hand, all the proposed issues must be addressed in interdisciplinary spirit and require expertise in several fields of mathematics: variational and topological methods, qualitative and regularity theory for PDE’s and free boundary problems, Morse and Critical Point theory, equivariant topology, geometric measure theory. We intend to address the following strongly interconnected themes: A. Pattern formation in reaction-diffusion systems, phase separation and optimal partition problems , arising in multispecies and multiagent models. Includes the analysis of the interfaces between the different phases in the presence non local diffusions and interactions. B. Shape optimization, free boundary problems , including the dynamics of nodal sets and free boundaries in evolutionary problems as well as spectral domain optimization issues. C. Complex solutions in Celestial Mechanics and Hamiltonian PDEs , where we seek solutions with prescribed behavior to the N-body problem and study their stability properties, with the final intent of detecting the occurrence of controlled chaos. The same paradigms will be applied to the search of entire solutions of several classes of PDEs. D. Complex domains and their effects on solutions to linear and nonlinear equations. We focus on the effects of global geometrical and topological properties, as well as on more local facts, like unique continuation and boundary effects. E. Geometric Variational problems and concentration phenomena , as they appear in Prescribed curvature problems, Conformal Geometry, Mathematical Physics and in the study Partial Differential Equations and systems when critical nonlinearities occur or, for some limit values of a parameter, special solutions exhibiting a singular limiting behaviour appear. Achievement of Objectives and Related Outcomes During the reporting period, the Bologna Research Unit actively contributed to the scientific goals of the project “Pattern Formation in Nonlinear Phenomena”, with specific emphasis on the following research directions: free boundary problems for elliptic operators; free boundary problems for parabolic operators; regularity properties of almost minimizers; properties of nonlocal operators. The activity mainly concerned the regularity of flat free boundaries for solutions of the inhomogeneous Stefan problem. These results are fully coherent with the long-term objectives of the project, particularly those related to shape optimization and free boundary problems. The methodological framework remains strongly aligned with the common asymptotic analysis and perturbative approach shared within the PRIN network. Compliance with DNSH, Open Access and Equal Opportunity Principles The research activities are purely theoretical and therefore fully compliant with the DNSH (Do No Significant Harm) principle. All publications are made available in Open Access through institutional repositories and/or arXiv, in accordance with the project commitments. The unit operates in full respect of gender equality, generational balance and equal opportunity principles, with active involvement of early-career researchers and PhD students. 2. Selected Publications (Bologna Unit) The following publications, authored by members of the Bologna unit during the PRIN period, are directly connected with the project themes. 1. Ferrari, Fausto; Lederman, Claudia. Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side. Calc. Var. Partial Differential Equations 63 (2024), no. 5, Paper No. 132, 43 pp. 2. Ferrari, Fausto; Giovagnoli, Davide. Some counterexamples to Alt-Caffarelli-Friedman monotonicity formulas in Carnot groups. Ann. Mat. Pura Appl. (4) 204 (2025), no. 2, 427-445. 3. Corni, Francesca; Ferrari, Fausto. The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation. J. Geom. Anal. 35 (2025), no. 4, Paper No. 106, 94 pp. 4. Ferrari, Fausto; Forcillo, Nicolo; Merlino, Enzo Maria. Regularity for almost minimizers of a one-phase Bernoullitype functional in Carnot groups of step two. Calc. Var. Partial Differential Equations 64 (2025), no. 4, Paper No. 107, 32 pp. 5. Giovagnoli, Davide; Jesus, David. A fully nonlinear transmission problem degenerating on the interface. Calc. Var. Partial Differential Equations 65 (2026), no. 3, Paper No. 93. 6. Jesus, David; Soria-Carro, Maria. Fully nonlinear parabolic fixed transmission problems. Nonlinear Anal. 264 (2026), Paper No. 114004, 26 pp. 7. Giovagnoli, Davide; Merlino, Enzo Maria; Moreira, Diego. A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems. La Matematica (2026). 8. Ferrari, Fausto; Lederman, Claudia. Lipschitz regularity of solutions to two-phase p-Laplacian free boundary problems with right hand side. Communications in Analysis and Geometry 34(2), pp. 465–503(2026). 9. Fausto Ferrari; Nicolo Forcillo; Davide Giovagnoli; David Jesus. Free boundary regularity for the inhomogeneous one-phase Stefan problem. Communications in Contemporary Mathematics, 2026, 2650048 https://arxiv.org/abs/2404.07535 10. Fausto Ferrari; Enzo Maria Merlino. Lipschitz regularity for almost minimizers of a one-phase p-Bernoulli-type functional in Carnot groups of step two. In press on Adv. Calc. Var. https://doi.org/10.48550/arXiv.2407.00084 11. Fausto Ferrari; Monica Jacob; Claudia Lederman, Two-phase free boundary problems for operators with nonstandard growth 2026, 5(2), 27. Special Volume The head of the Bologna unit edited, in collaboration with Daniela De Silva (Columbia University, New York), a special volume devoted to free boundary problems in honor of the scientific activity of Sandro Salsa. The volume is published in La Matematica (Springer journal): https://link.springer.com/collections/hcafegahbe Preprints Submitted for Publication / Under Review 11. Fausto Ferrari; Davide Giovagnoli; David Jesus. On the Geometry of the Solutions of the Fully Nonlinear Inhomogeneous One-Phase Stefan Problem. https://doi.org/10.48550/arXiv.2504.12912 12. Davide Giovagnoli; David Jesus; Luis Silvestre. C^{1,alpha} regularity for fractional p-harmonic functions. https://doi.org/10.48550/arXiv.2509.26565 13. Fausto Ferrari; Davide Giovagnoli; Enzo Maria Merlino. On a fractional Alt-Caffarelli-Friedman-type monotonicity formula. https://doi.org/10.48550/arXiv.2509.25891 14. David Jesus; Edgard A. Pimentel; David Stolnicki. Boundary regularity for a fully nonlinear free transmission problem. https://arxiv.org/abs/2411.15335 3. Progress of Activities Activities Carried Out The Bologna unit developed the following research lines: Research line Focus Free boundary problems Regularity of the free boundaries of inhomogeneous onephase problems associated with non-standard growth problems. Degenerate free boundary problems Global regularity of almost minimizers of one-phase functionals in non-commutative group structures. Parabolic free boundary problems Regularity of the free boundary of solutions of the inhomogeneous one-phase Stefan problem. The activities were carried out in collaboration with personnel hired by the PRIN unit of Bologna, in particular a postdoctoral researcher, together with other postdoctoral researchers, PhD students and international collaborators with long-standing scientific links to the head of the Bologna unit. This contributed to the overall methodological coherence of the project. Project activities are collected on the Bologna unit webpage: https://site.unibo.it/pattern-formation-in-nonlinear phenomena/en Challenges and Future Developments No substantial deviations from the approved research plan occurred. The main challenge concerned the long time required for the acceptance of submitted papers and the substantial technical effort required to obtain regularity results for solutions of the inhomogeneous Stefan problem. This development is consistent with the high-risk/high-gain nature of the project and has strengthened the theoretical framework for ongoing investigations. Further papers are currently in preparation on: regularity in parabolic transmission problems; regularity of the free boundary in inhomogeneous two-phase Stefan problems; properties of the free boundaries of almost minimizers. 4. Bologna Unit: Personnel and Meetings Postdoctoral Position Name David Joao Brandligt De Jesus Position Research Fellow Period 1 July 2024 - 30 June 2025 Institution University of Bologna Scientific Meetings Free Boundary in Action: Bologna, 10-11 March 2025. https://events.unibo.it/free-boundaries-in-action/poster Special Session (B9): Free Boundary Problems - Viscosity and Variational Approaches: Joint Meeting AMS-UMI 2024, Palermo, 23-26 July 2024. https://umi.dm.unibo.it/jm-umi-ams/special-sessions/special-sessions-and-lecturerooms-2/lecture-rooms-on-july-25-26/ Latinos on Free Boundaries: Bologna, 29 July 2024. https://matematica.unibo.it/it/eventi/latinos-on-free-boundaries-in-bologn
Dettagli del progetto
Responsabile scientifico: Fausto Ferrari
Strutture Unibo coinvolte:
Dipartimento di Matematica
Coordinatore:
Università degli Studi di TORINO(Italy)
Contributo totale Unibo: Euro (EUR) 27.300,00
Durata del progetto in mesi: 24
Data di inizio
28/09/2023
Data di fine:
28/09/2025
