Calculus of Variations and Geometric Measure Theory
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Divergence-measure fields: an overview of the generalizations of the Gauss-Green formulas

Giovanni Eugenio Comi (Scuola Normale Superiore)

created by comi on 09 Jun 2017
modified on 17 Jul 2018

8 jun 2017

Mathematical Institute, Oxford

Abstract.

Divergence-measure fields are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose divergence is a Radon measure. Such vector fields form a new family of function spaces, which in a sense generalize the $BV$ fields, and were introduced at first by Anzellotti, before being rediscovered in the early 2000s by many authors for different purposes.

Chen and Frid were interested in the applications to the theory of systems of conservation laws with the Lax entropy condition and achieved a Gauss-Green formula for divergence-measure fields, for any $1 \le p \le \infty$, on open bounded sets with Lipschitz deformable boundary. We show in this talk that any Lipschitz domain is deformable.

Later, Chen, Torres and Ziemer extended this result to the sets of finite perimeter in the case $p = \infty$, showing in addition that the interior and exterior normal traces of the vector field are essentially bounded functions.

The Gauss-Green formula for $1 \le p \le \infty$ has been also studied by Silhavy on general open sets, and by Schuricht on compact sets. In such cases, the normal trace is not in general a summable function: it may even not be a measure, but just a distribution of order 1. However, we can show that such a trace is the limit of the integral of classical normal traces on (smooth) approximations of the integration domain.


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