The Gauss—Green formulas for divergence-measure fields and sets of finite perimeter --- Giovanni Eugenio Comi (SNS)

ABSTRACT:

The Gauss--Green formulas are of significant relevance in many areas of analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields, and thus led to the definition of the divergence-measure fields. These are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose distributional divergence is a Radon measure, and so they form a new family of function spaces, which in a sense generalize the $BV$ fields. The divergence-measure fields were introduced at first by Anzellotti in 1983 for the case $p = \infty$, and then they have been rediscovered in the early 2000s by many authors interested in various applications.

In this talk, we shall present an overview of such researches, with a particular focus on the results concerning essentially bounded divergence-measure fields. For such a family of vector fields, Silhavy (2005), Chen, Torres and Ziemer (2009) and Comi and Payne (2017) showed that Gauss--Green formulas hold for sets of finite perimeter, by proving the existence of essentially bounded interior and exterior normal traces on the reduced boundary of the given set.

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