The essential boundary of certain sets

Bruckner, A. M.; Davies, Roy O.; Goffman, C.

doi:10.1090/S0002-9939-1986-0826484-5

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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The essential boundary of certain sets
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by A. M. Bruckner, Roy O. Davies and C. Goffman PDF

Proc. Amer. Math. Soc. 96 (1986), 579-584 Request permission

Abstract:

The essential boundary of a measurable set is related to the de Giorgi perimeter and was introduced by Vol’pert in his "improvement" of Federer’s work. For a totally disconnected compact set of positive measure in $n$ space the essential boundary can be of Hausdorff $n - 1$ dimension but cannot have $\sigma$ finite $(n - 1)$-measure.

References

Herbert Federer, A note on the Gauss-Green theorem, Proc. Amer. Math. Soc. 9 (1958), 447–451. MR 95245, DOI 10.1090/S0002-9939-1958-0095245-2
A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338