A descriptive definition of a variational integral and applications. (English) Zbl 0747.26010

The author has introduced a coordinate free variational integral on bounded sets of finite perimeter, for which a Gauss-Green theorem holds for continuous vector fields which are differentiable outside a set whose codimension one Hausdorff dimension is \(\sigma\)-finite. This paper proposes a descriptive definition of this integral which allows an extension of the result to vector fields which are discontinuous on a set of positive Lebesgue measure. The proof makes use of interesting properties of BV sets. The results are compared to earlier ones of Shapiro and Jurkat.


26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
26B15 Integration of real functions of several variables: length, area, volume
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