R201, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 201室)

An Overview on Divergence-measure Fields and Weak Integration by Parts Formulas

Giovanni Comi (Universität Hamburg)

Abstract:

The Gauss--Green and integration by parts formulas are of significant relevance in many areas of analysis and mathematical physics. Their importance motivated several investigations to obtain extensions to more general classes of integration domains and weakly differentiable vector fields, thus ultimately leading to the definition of divergence-measure fields. These are

-summable vector fields on

whose distributional divergence is a Radon measure. It is not difficult to notice that the divergence-measure fields form a new family of function spaces, which generalize the

fields.

In this talk, we shall present an overview of the theory of divergence-measure fields, with a particular focus on the case

. Indeed, for essentially bounded divergence-measure fields, it was proved that the Gauss--Green formulas hold on sets of finite perimeter and that the interior and exterior normal traces are essentially bounded functions on the reduced boundary of the given set.

Subsequent extensions of these results and generalizations to non-Euclidean geometries will be also briefly discussed.