On the exceptional sets of the flux of a bounded vectorfield. (English) Zbl 1042.49039
Summary: For each purely \(n-1\) unrectifiable compact set \(C \subset \mathbb R^n\) such that \(0 < \mathcal H^{n-1}(C) < \infty\), there is a bounded Borel measurable vectorfield \(v: \mathbb R^n\to \mathbb R^n\) whose flux vanishes in \(\mathbb R^n \sim C\) but not in \(\mathbb R^n\)
MSC:
49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
28A75 | Length, area, volume, other geometric measure theory |
26A39 | Denjoy and Perron integrals, other special integrals |
26B15 | Integration of real functions of several variables: length, area, volume |
References:
[1] | David, G.; Mattila, P., Removable sets for Lipschitz harmonic functions in the plane, Rev. Mat. Iberoamericana, 16, 1, 137-216 (2000) · Zbl 0976.30016 |
[2] | De Pauw, T., On SBV dual, Indiana Univ. Math. J., 47, 1, 99-121 (1998) · Zbl 0915.49003 |
[3] | T. De Pauw, W.F. Pfeffer, The Gauss-Green theorem and removable sets for PDE’s in divergence form, Adv. Math; T. De Pauw, W.F. Pfeffer, The Gauss-Green theorem and removable sets for PDE’s in divergence form, Adv. Math · Zbl 1100.35022 |
[4] | Federer, H., Geometric Measure Theory, Grundlehren Math. Wiss., 153 (1969), Springer-Verlag: Springer-Verlag New York · Zbl 0176.00801 |
[5] | Federer, H., Real flat chains, cochains and variational problems, Indiana Univ. Math. J., 24, 351-407 (1974) · Zbl 0289.49044 |
[6] | Fremlin, D. H., Broad Foundations, Measure Theory, 2 (2001), Torres Fremlin · Zbl 1165.28001 |
[7] | Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Stud. Adv. Math., 44 (1995), Cambridge Univ. Press · Zbl 0819.28004 |
[8] | Mattila, P.; Paramonov, P. V., On geometric properties of harmonic \(Lip_1\)-capacity, Pacific J. Math., 171, 469-491 (1995) · Zbl 0852.31004 |
[9] | Meyers, N. G.; Ziemer, W. P., Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math., 99, 1345-1360 (1977) · Zbl 0416.46025 |
[10] | Pfeffer, W. F., Derivation and Integration, Cambridge Tracts in Mathe., 140 (2001), Cambridge Univ. Press · Zbl 0554.26008 |
[11] | Rudin, W., Functional Analysis (1973), McGraw-Hill · Zbl 0253.46001 |
[12] | Whitney, H., Geometric Integration Theory, Princeton Math. Ser., 21 (1957), Princeton Univ. Press · Zbl 0083.28204 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.