Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 
 
 

 

Strict interior approximation of sets of finite perimeter and functions of bounded variation


Author: Thomas Schmidt
Journal: Proc. Amer. Math. Soc. 143 (2015), 2069-2084
MSC (2010): Primary 28A75, 26B30, 41A63, 41A30; Secondary 28A78, 26B15
DOI: https://doi.org/10.1090/S0002-9939-2014-12381-1
Published electronically: November 25, 2014
MathSciNet review: 3314116
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate an open set $ \Omega $ of finite perimeter in $ \mathbb{R}^n$ strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the $ (n{-}1)$-dimensional Hausdorff measure of the topological boundary $ \partial \Omega $ equals the perimeter of $ \Omega $. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation of $ BV$-functions from a prescribed Dirichlet class.


References [Enhancements On Off] (What's this?)

  • [1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292 (2003a:49002)
  • [2] Gabriele Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc. 290 (1985), no. 2, 483-501. MR 792808 (87b:49025), https://doi.org/10.2307/2000295
  • [3] G. Anzellotti, BV solutions of quasilinear PDEs in divergence form, Comm. Partial Differential Equations 12 (1987), no. 1, 77-122. MR 869103 (88b:35073), https://doi.org/10.1080/03605308708820485
  • [4] G. Anzellotti and M. Giaquinta, BV functions and traces, Rend. Sem. Mat. Univ. Padova 60 (1978), 1-21 (1979) (Italian, with English summary). MR 555952 (82e:46046)
  • [5] L. Beck and T. Schmidt, Convex duality and uniqueness for $ \rm BV$-minimizers, preprint (2013), 32 pages.
  • [6] Michael Bildhauer, Convex variational problems, Linear, nearly linear and anisotropic growth conditions, Lecture Notes in Mathematics, vol. 1818, Springer-Verlag, Berlin, 2003. MR 1998189 (2004i:49072)
  • [7] Gui-Qiang Chen, Monica Torres, and William P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math. 62 (2009), no. 2, 242-304. MR 2468610 (2009m:49076), https://doi.org/10.1002/cpa.20262
  • [8] Guy David, Singular sets of minimizers for the Mumford-Shah functional, Progress in Mathematics, vol. 233, Birkhäuser Verlag, Basel, 2005. MR 2129693 (2006a:49001)
  • [9] Pavel Doktor, Approximation of domains with Lipschitzian boundary, Časopis Pěst. Mat. 101 (1976), no. 3, 237-255 (English, with Czech and loose Russian summaries). MR 0461122 (57 #1107)
  • [10] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284 (88d:28001)
  • [11] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325 (41 #1976)
  • [12] Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682 (87a:58041)
  • [13] Giovanni Gregori, Generalized solutions for a class of non-uniformly elliptic equations in divergence form, Comm. Partial Differential Equations 22 (1997), no. 3-4, 581-617. MR 1443050 (98e:35048), https://doi.org/10.1080/03605309708821275
  • [14] Jan Kristensen and Filip Rindler, Characterization of generalized gradient Young measures generated by sequences in $ W^{1,1}$ and BV, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 539-598. MR 2660519 (2011f:49072), https://doi.org/10.1007/s00205-009-0287-9
  • [15] J. Kristensen and F. Rindler, Piecewise affine approximations for functions of bounded variation, preprint (2013), 14 pages.
  • [16] Samuel Littig and Friedemann Schuricht, Convergence of the eigenvalues of the $ p$-Laplace operator as $ p$ goes to 1, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 707-727. MR 3148132, https://doi.org/10.1007/s00526-013-0597-5
  • [17] U. Massari and L. Pepe, Sull'approssimazione degli aperti lipschitziani di $ R^{n}$ con varietà differenziabili, Boll. Un. Mat. Ital. (4) 10 (1974), 532-544 (Italian, with English summary). MR 0365318 (51 #1571)
  • [18] Jindřich Nečas, On domains of type $ {\mathfrak{N}}$, Czechoslovak Math. J. 12 (87) (1962), 274-287 (Russian, with French summary). MR 0152734 (27 #2709)
  • [19] Washek F. Pfeffer, The Gauss-Green theorem, Adv. Math. 87 (1991), no. 1, 93-147. MR 1102966 (92b:26024), https://doi.org/10.1016/0001-8708(91)90063-D
  • [20] Thierry Quentin de Gromard, Approximation forte dans BV $ (\Omega )$, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 6, 261-264 (French, with English summary). MR 803213 (86m:26014)
  • [21] Thierry Quentin de Gromard, Strong approximation of sets in $ {\rm BV}(\Omega )$, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), no. 6, 1291-1312. MR 2488060 (2010i:49022), https://doi.org/10.1017/S0308210507000492
  • [22] Thierry Quentin de Gromard, Approximation forte des ensembles dans $ {\rm BV}(\Omega )$ par des ensembles à frontière $ \mathcal {C}^1$, C. R. Math. Acad. Sci. Paris 348 (2010), no. 7-8, 369-372 (French, with English and French summaries). MR 2607021 (2011d:49062), https://doi.org/10.1016/j.crma.2010.02.016
  • [23] Italo Tamanini and Corrado Giacomelli, Approximation of Caccioppoli sets, with applications to problems in image segmentation, Ann. Univ. Ferrara Sez. VII (N.S.) 35 (1989), 187-214 (1990). English, with Italian summary. MR 1079588 (91j:49065)
  • [24] Italo Tamanini and Corrado Giacomelli, Monotone approximation of sets of finite perimeter from inside, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 1 (1990), no. 3, 181-187 (Italian, with English summary). MR 1083246 (91j:49063)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 28A75, 26B30, 41A63, 41A30, 28A78, 26B15

Retrieve articles in all journals with MSC (2010): 28A75, 26B30, 41A63, 41A30, 28A78, 26B15


Additional Information

Thomas Schmidt
Affiliation: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy – and – Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
Email: thomas.schmidt@sns.it, thomas.schmidt@math.uzh.ch

DOI: https://doi.org/10.1090/S0002-9939-2014-12381-1
Received by editor(s): May 24, 2013
Received by editor(s) in revised form: October 9, 2013
Published electronically: November 25, 2014
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement GeMeThnES No. 246923.
Communicated by: Tatiana Toro
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society