30 sep 2013
Centro De Giorgi
ERC-School on Geometric Measure Theory and Real Analysis
We study the regularity problem for sub-Riemannian geodesics, i.e., those curves that minimize length among all curves joining two fixed endpoints and whose derivatives are tangent to a given, smooth distribution of planes with constant dimension. We will review necessary conditions for optimality, focusing in particular on Pontryagin Maximum Principle and Goh condition. The regularity problem is non-trivial due to the presence of the so-called abnormal extremals, i.e., certain curves satisfying the necessary conditions that may develop singularities. After reviewing the recent literature on the subject, we will focus on the case of stratified groups and present some recent results obtained in collaboration with E. Le Donne, G. P. Leonardi and R. Monti.