Cacciopoli Sets, Capacities and Quasiconformal Mappings in Carnot-Caratheodory Spaces

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

We propose to investigate three fundamental problems arising in Analysis on Carnot-Caratheodory spaces. The first question is from geometric measure theory. It is related to Cacciopoli sets and their characteristic locus with applications to minimal surfaces. The second problem is from non-linear potential theory and sub-elliptic PDE related to sharp capacity estimates. The third question concerns the integrability and Hausdorff dimension distortion of quasiconformal mappings in Carnot-Caratheodory spaces. The fundamental difficulty in this theory lies in the fact that in Carnot-Caratheodory spaces we can only work with a restricted number of differential operators (corresponding to admissible directions) while in Euclidean analysis all possible directions were allowed. The underlying geometry has a fractal-like character illustrating well the complexity and the need of new ideas for approaching the above problems. Due to their construction by mechanical linkages, robot motions have to satisfy infinitesimal constrains of admissible directions. Similar differential constrains appear in thermodynamics between the characteristic quantities (temperature, pressure, volume, entropy and energy) describing the stage of an ideal gas. The right mathematical context for studying these and related phenomena from engineering and physics is the setting of Carnot-Caratheodory spaces. The present proposal intends to develop the adequate tools of Analysis for approaching various practical questions in this context.