Regularity problems in sub-Riemannian structures

Funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 del 2/2/2022

Analysis in sub-Riemannian (SR) manifolds deals with totally degenerate structures which are not Riemannian at any point. A SR structure is a regular manifold with a sub-bundle of the tangent bundle, called horizontal space, and a metric on it. Local generators of this sub-bundle, called horizontal vector fields, encode the directions of propagation of mechanical, physical, biological or geometrical phenomena. All the relevant differential objects of a SR manifold are expressed in terms of these vector-fields. The degenerate and anisotropic character of the structure calls for new tools in the study of the regularity properties of geodesics, minimal surfaces, solutions to PDEs and to optimal transport problems. 

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  

1. Regularity and structure of optimal solutions in SR manifolds
   1. Regularity of geodesics 
   2. Fine properties of the distance function 
   3.  Regularity of optimal transport maps with non-quadratic cost 
   4. Bernstein’s problem on the rigidity of entire minimal graphs 
2. Regularity for solutions of non-elliptic PDEs and differential forms
   1. New lifting technique for the global analysis of fundamental solutions 
   2. Boundary regularity of solutions of PDEs, including characteristic points 
   3. Interior C^{1,alpha} regularity for p-Laplace equations in Carnot groups 
   4. Analysis of Rumin’s differential forms in SR manifolds 
3. Application of SR tools to the study of the human cortex
   1. Models describing the interaction of visual and motor cortex
   2. Models for the propagation of degenerative neuronal diseases