Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



The trace of $ BV$-functions on an irregular subset

Authors: Yu. D. Burago and N. N. Kosovskiĭ
Translated by: the authors
Original publication: Algebra i Analiz, tom 22 (2010), nomer 2.
Journal: St. Petersburg Math. J. 22 (2011), 251-266
MSC (2010): Primary 46E35; Secondary 28A75
Published electronically: February 8, 2011
MathSciNet review: 2668125
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Certain basic results on the boundary trace discussed in Maz'ya's monograph on Sobolev spaces are generalized to a wider class of regions. The paper is an extended and supplemented version of a preliminary publication, where some results were presented without proofs or in a weaker form. In Maz'ya's monograph, the boundary trace was defined for regions $ \Omega$ with finite perimeter, and the main results were obtained under the assumption that normals in the sense of Federer exist almost everywhere on the boundary. Instead, now it is assumed that the region boundary is a countably $ (n-1)$-rectifiable set, which is a more general condition.

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

  • 1. J. J. Almgren, The theory of varifolds, Preprint, Princeton, 1965.
  • 2. Yu. Burago and N. Kosovskiy, Boundary trace for $ BV$ functions in regions with irregular boundary, Analysis, Partial Differential Equations and Applications -- The Vladimir Maz'ya Anniversary Volume, Oper. Theory Adv. Appl., vol. 193, Birkhäuser, Basel, 2009, pp. 1-14.
  • 3. Ju. S. Burago and V. G. Maz′ja, Certain questions of potential theory and function theory for regions with irregular boundaries, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 3 (1967), 152 (Russian). MR 0227447
  • 4. E. Giusti, Minimal surfaces and functions of bounded variation, Monogr. in Math., vol. 80, Birkhäuser, Basel, 1984. MR 0775682 (87a:58041)
  • 5. V. G. Maz'ya, Sobolev spaces, Leningrad. Gos. Univ., Leningrad, 1985; English transl., Springer-Verlag, Berlin, 1985. MR 0807364 (87g:46056)
  • 6. V. G. Maz′ja, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960), 882–885. MR 0126152
  • 7. Frank Morgan, Geometric measure theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR 1775760
  • 8. Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • 9. Wendell H. Fleming and Raymond Rishel, An integral formula for total gradient variation, Arch. Math. (Basel) 11 (1960), 218–222. MR 0114892,
  • 10. William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 46E35, 28A75

Retrieve articles in all journals with MSC (2010): 46E35, 28A75

Additional Information

Yu. D. Burago
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia

N. N. Kosovskiĭ
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, 28 Universitetskii Prospekt, Peterhoff, St. Petersburg 198504, Russia

Keywords: Trace, rectifiability, perimeter, embedding theorems
Received by editor(s): May 20, 2009
Published electronically: February 8, 2011
Additional Notes: Partially supported by RFBR (grant no. 08-01-00079a)
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society