Abstract
The project aims at studying some of the most challenging problems still open in the specific area of SR structures both of variational type and of analytical type. Precisely we will focus on regularity properties of SR abnormal geodesics (a very challenging problem, since this phenomenon does not occur in the Riemannian case), regularity of optimal mass transportation problems and Bernstein’s type theorems, namely the classification of entire graphs that are area minimizing. One of the most important open problem in the second class of problems is regularity at the boundary for PDEs of sub-elliptic type, which is very challenging, due to the presence of singular points, called characteristic on SR surfaces. Finally, we will study operators on differential forms in SR manifolds, following the approach of Rumin. The final part of the project has a more applicative character and deals with models for the functional structure of visual and the motor cortex expressed with tools from SR analysis. Another important application will be to study Alzheimer’s disease, with the scope of testing new therapeutic protocols. This study is of deep mathematical interest, and it will provide a significant improvement with respect to the state of the art. Many areas of mathematics would benefit from our work: PDEs, Harmonic Analysis, Calculus of Variations, Geometric Measure Theory and Control Theory. Our results will also have impact on neurological problems, medical images problems, and robotics. Transfer of knowledge, performed through master thesis, will respect the DNSH criteria, since providing efficient mathematical models efficiently contribute to reduce the need of experimental data and to limit risks of environmental damage. Objectives of the project Regularity and structure of optimal solutions in SR manifolds Regularity of geodesics Fine properties of the distance function Regularity of optimal transport maps with non-quadratic cost Bernstein’s problem on the rigidity of entire minimal graphs Regularity for solutions of non-elliptic PDEs and differential forms New lifting technique for the global analysis of fundamental solutions Boundary regularity of solutions of PDEs, including characteristic points Interior C^{1,alpha} regularity for p-Laplace equations in Carnot groups Analysis of Rumin’s differential forms in SR manifolds Application of SR tools to the study of the human cortex Models describing the interaction of visual and motor cortex Models for the propagation of degenerative neuronal diseases Results Achieved Main objectives of the project The project addressed several open problems in sub-Riemannian (SR) geometry, analysis, and applications. It is organized in three main objectives Objective 1 To study the relation between geodesics and minimal surfaces, optimal mass transport, Objective 2 To study boundary regularity for sub-elliptic PDEs, totally degenerate PDE with p-q growth and differential form operators using Rumin’s approach. Objective 3 To explore applications to models of the visual and motor cortex as well as Alzheimer’s disease.
Results achieved
in the project All the main objectives have been reached. Results related to Objective 1: In the first part of the project, we studied the regularity and structure of optimal objects in sub-Riemannian (SR) manifolds, focusing on geodesics, minimal surfaces, and optimal transport problems. - We studied the fine properties of the SR distance in Carnot groups, including the role of the cut locus - we investigated optimal transport problems in SR settings. We established regularity results for optimal transport maps with non-quadratic costs and studied a congested transport model in the Heisenberg group - We also examined H-minimal surfaces, the prescribed curvature problems and the Bernstein-type rigidity problem in different geometric setting. Minimal surfaces also arise in phase transition problems. We studied the problem for integral functionals with p-growth, and relaxation for functionals depending on vector. Results related to Objective 2: The second part of the project focused on the regularity theory for - We applied properties of the fundamental solutions to the study of mean value formulas, for representation of harmonic or caloric functions - We obtained new results on the boundary regularity of solutions to subelliptic PDEs - We also established interior regularity results for nonlinear p-Laplacian type equations in Euclidean and Carnot groups - Finally, we investigated Rumin’s complex of differential forms in SR manifolds and analyzed related Poincaré, Sobolev, and Hodge-type structures Results related to Objective 3 The third part of the project concerned applications of SR methods to the study of the human cortex. We developed mathematical models describing the propagation of signals in visual and motor cortical areas within a unified SR framework and studied the interaction between these structures. - In particular, models of stereo vision and motor control were integrated using geometric lifting techniques. In addition, we established a link between mean-field equations and sub-Riemannian geometric models of the brain. - We propose models of propagation in totally heterogeneous setting, with application to propagation of Alzheimer’s desease. Pubblication, preprints, organization of conferences - 50 publications in peer-reviewed journals, all accessible as preprints on arXiv. - 9 preprints already posted on arXiv or submitted for publication. - 32 conference proceedings / workshop papers. - A book is in preparation. - 12 conferences organizedDettagli del progetto
Responsabile scientifico: Giovanna Citti
Strutture Unibo coinvolte:
Dipartimento di Matematica
Coordinatore:
ALMA MATER STUDIORUM - Università di Bologna(Italy)
Contributo totale di progetto: Euro (EUR) 165.225,00
Contributo totale Unibo: Euro (EUR) 83.700,00
Durata del progetto in mesi: 24
Data di inizio
28/09/2023
Data di fine:
28/02/2026
