Abstract

We discuss the standard relaxed version of a minimization problem for variational integrals of linear growth together with prescribed Dirichlet boundary data u0u_0 and give estimates for the size of the set {x∈∂Ω:u(x)≠u0(x)}\{x \in \partial \Omega: u (x) \not= u_0 (x)\} for BV-minimizers uu which imply Hn−1({x∈∂Ω:u(x)<u0(x)})=Hn−1({x∈∂Ω:u(x)>u0(x)}){\cal{H}}^{n -1} \left(\left\{x \in \partial \Omega: u (x) < u_0 (x)\right\}\right) = {\cal{H}}^{n - 1} \left(\left\{x \in \partial \Omega: u (x) > u_0 (x) \right\}\right) in the case of minimal surfaces uu not attaining the boundary values u0u_0 on a subset of ∂Ω\partial \Omega with positive measure.

Author information

Contact details are reproduced from the original publication and may be historical.

Michael Bildhauer

Fachbereich Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041 Saarbrücken, Germany

bibi@math.uni-sb.de

Martin Fuchs

Fachbereich Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041 Saarbrücken, Germany

fuchs@math.uni-sb.de

Keywords

  • Variational problems of linear growth
  • boundary behaviour
  • minimal surfaces

Mathematics Subject Classification

49J40, 49J45, 49Q05

References

  1. R. A. Adams: Sobolev Spaces, Academic Press, New York (1975).Google Scholar
  2. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  3. L. Beck, T. Schmidt: On the Dirichlet problem for variational integrals in BV, J. Reine Angew. Math. 674 (2013) 113–194.https://doi.org/10.1515/CRELLE.2011.188Google Scholar
  4. M. Bildhauer: Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions, Lect. Notes Math. 1818, Springer, New York (2003).https://doi.org/10.1007/978-3-642-55627-2_18Google Scholar
  5. F. Demengel, R. Temam: Convex functions of a measure and applications, Indiana Univ. Math. J. 33 (1984) 673–709.https://doi.org/10.1512/iumj.1984.33.33036Google Scholar
  6. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, Studies in Mathematics and its Applications 1, North-Holland, Amsterdam (1976).Google Scholar
  7. M. Giaquinta, G. Modica, J. Souček: Functionals with linear growth in the calculus of variations, Comm. Math. Univ. Carolinae 20 (1979) 143–172.Google Scholar
  8. E. Giusti: Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel (1984).https://doi.org/10.1007/978-1-4684-9486-0Google Scholar
  9. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton (1970).Google Scholar
  10. E. Santi: Sul problema al contorno per l’equazione della superfici di area minima su domini limitati qualunque, Ann. Univ. Ferrara 17 (1972) 13–26.https://doi.org/10.1007/BF02826875Google Scholar
  11. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

M. Bildhauer, M. Fuchs. “Some Remarks on the (Non-) Attainment of the Boundary Data for Variational Problems in the Space BV.” Journal of Convex Analysis 25 (2018), No. 1, 219–223. https://doi.org/10.68381/jca25013

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.