Memoirs of the American Mathematical Society

2 total works

Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-caratheodory Spaces

by Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu

Published 1 June 2006

The object of the present study is to characterize the traces of the Sobolev functions in a sub-Riemannian, or Carnot-Caratheodory space. Such traces are defined in terms of suitable Besov spaces with respect to a measure which is concentrated on a lower dimensional manifold, and which satisfies an Ahlfors type condition with respect to the standard Lebesgue measure. We also study the extension problem for the relevant Besov spaces. Various concrete applications to the setting of Carnot groups are analyzed in detail and an application to the solvability of the subelliptic Neumann problem is presented.

Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem

by Donatella Danielli, Nicola Garofalo, Arshak Petrosyan, and Tung To

Published 1 September 2017

The authors give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.