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Some results about the structure of primitivity domains for linear partial differential operators with constant coefficients. (English) Zbl 1501.47084

The purpose of the author is to examine the structure of \(G\)-primitivity domains of \(f\), a locally integrable function defined on \(\Omega\) (an open set of \(\mathbb R^n\)), at points of the \(G\)-nonintegrability set of \(f\).

MSC:

47F05 General theory of partial differential operators
28A75 Length, area, volume, other geometric measure theory

References:

[1] Adams, RA, Sobolev Spaces (1975), London: Academic Press, inc., London · Zbl 0314.46030
[2] Alberti, G., A Lusin type theorem for gradients, J. Funct. Anal., 100, 110-118 (1991) · Zbl 0752.46025 · doi:10.1016/0022-1236(91)90104-D
[3] Balogh, ZM, Size of characteristic sets and functions with prescribed gradient, J. Reine Angew. Math., 564, 63-83 (2003) · Zbl 1051.53024
[4] Balogh, ZM; Pintea, C.; Rohner, H., Size of tangencies to non-involutive distributions, Indiana Univ. Math. J., 60, 6, 2061-2092 (2011) · Zbl 1271.58001 · doi:10.1512/iumj.2011.60.4489
[5] Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac Systems and Computational Algebra. Progress in Mathematical Physics, 39. Birkhäuser Boston, Inc., Boston, MA (2004) · Zbl 1064.30049
[6] Delladio, S., Functions of class \(C^1\) subject to a Legendre condition in an enhanced density set, Rev. Mat. Iberoam., 28, 1, 127-140 (2012) · Zbl 1244.49073 · doi:10.4171/RMI/670
[7] Delladio, S., A note on some topological properties of sets with finite perimeter, Glasg. Math. J., 58, 3, 637-647 (2016) · Zbl 1355.28004 · doi:10.1017/S0017089515000385
[8] Delladio, S.: Structure of prescribed gradient domains for non-integrable vector fields. Ann. Mat. Pura ed Appl. (1923 -) 198(3), 685-691 (2019) · Zbl 1418.53037
[9] Delladio, S., Structure of tangencies to distributions via the implicit function theorem, Rev. Mat. Iberoam., 34, 3, 1387-1400 (2018) · Zbl 1401.28008 · doi:10.4171/RMI/1028
[10] Delladio, S.: The identity \(G(D)f=F\) for a linear partial differential operator \(G(D)\). Lusin type and structure results in the non-integrable case. Proc. R. Soc. Edinb. Sect. 151(6), 1893-1919 (2021) · Zbl 1523.35127
[11] Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI (1998) · Zbl 0902.35002
[12] Evans, L.C., Gariepy, R.F.: Lecture Notes on Measure Theory and Fine Properties of Functions. (Studies in Advanced Math.). CRC Press (1992) · Zbl 0804.28001
[13] Federer, H., Geometric Measure Theory (1969), New York: Springer, New York · Zbl 0176.00801
[14] Francos, G., The Luzin theorem for higher-order derivatives, Michigan Math. J., 61, 3, 507-516 (2012) · Zbl 1256.28006 · doi:10.1307/mmj/1347040255
[15] Hörmander, L.: Linear Partial Differential Operators. Die Grundlehren der mathematischen Wissenschaften, Bd. 116 Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin (1963) · Zbl 0108.09301
[16] Li, S.: A note on Alberti’s Luzin-type theorem for gradients. Ric. Mat. 70(2), 479-488 (2021) · Zbl 1515.28005
[17] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge (1995) · Zbl 0819.28004
[18] Ponce, A.C.: Elliptic PDEs, Measures and Capacities (from the Poisson equation to nonlinear Thomas-Fermi problems). Tracts in Mathematics 23, European Math. Soc. (2016) · Zbl 1357.35003
[19] Rudin, W., Real and Complex Analysis (1970), New York: McGraw-Hill, New York · Zbl 0925.00005
[20] Shakarchi, R.; Stein, EM, Real Analysis (Measure Theory, Integration and Hilbert Spaces) (2005), Princeton: Princeton University Press, Princeton · Zbl 1081.28001
[21] Ziemer, W.P.: Weakly Differentiable Functions. GTM 120, Springer, New York (1989)
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