Seminar

Arguably, one of the most basic results in analysis is Gauss' Divergence Theorem. Its original formulation involves mildly regular domains and sufficiently smooth vector fields (typically both of class C^1), though applications to rougher settings have prompted various generalizations. One famous extension, due to De Giorgi and Federer, lowers the regularity assumptions on the underlying domain to a mere local finite perimeter condition. While geometrically this is in the nature of best-possible, the De Giorgi- Federer theorem still asks that the intervening vector field has Lipschitz components. The latter assumption is, however, unreasonably strong, both from the point of view of the very formulation of the Divergence Formula, and its applications to PDE's which often involve much less regular functions. In my talk I will discuss a refinement which addresses this crucial issue, through the use of tools and techniques from Harmonic Analysis (Whitney decompositions, weighted isoperimetric inequalities, non-tangential maximal operators). In particular, this sharpened form of the Divergence Theorem yields a variety of refined results, from the nature of the Green function, to the behavior of singular integral operators in very general domains.