Examples of discontinuous, divergence-free solutions to elliptic variational problems

Wenge Hao; Salvatore Leonardi; Mark Steinhauer

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 3, page 511-517
  • ISSN: 0010-2628

Abstract

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We give an example of a bounded discontinuous divergence-free solution of a linear elliptic system with measurable bounded coefficients in ℝ 3 and a corresponding example for a Stokes-like system.

How to cite

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Hao, Wenge, Leonardi, Salvatore, and Steinhauer, Mark. "Examples of discontinuous, divergence-free solutions to elliptic variational problems." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 511-517. <http://eudml.org/doc/247740>.

@article{Hao1995,
abstract = {We give an example of a bounded discontinuous divergence-free solution of a linear elliptic system with measurable bounded coefficients in $\mathbb \{R\}^3$ and a corresponding example for a Stokes-like system.},
author = {Hao, Wenge, Leonardi, Salvatore, Steinhauer, Mark},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {divergence-free solutions; elliptic systems; systems of Stokes-type; regularity; singular solution; systems of Stokes-type; divergence-free solution},
language = {eng},
number = {3},
pages = {511-517},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Examples of discontinuous, divergence-free solutions to elliptic variational problems},
url = {http://eudml.org/doc/247740},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Hao, Wenge
AU - Leonardi, Salvatore
AU - Steinhauer, Mark
TI - Examples of discontinuous, divergence-free solutions to elliptic variational problems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 3
SP - 511
EP - 517
AB - We give an example of a bounded discontinuous divergence-free solution of a linear elliptic system with measurable bounded coefficients in $\mathbb {R}^3$ and a corresponding example for a Stokes-like system.
LA - eng
KW - divergence-free solutions; elliptic systems; systems of Stokes-type; regularity; singular solution; systems of Stokes-type; divergence-free solution
UR - http://eudml.org/doc/247740
ER -

References

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  1. Campanato S., Sistemi ellittici in forma divergenza. Regolarita' all'interno, Quaderno Sc. Norm. Sup. Pisa Pisa (1980). (1980) Zbl0453.35026MR0668196
  2. De Giorgi E., Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. U.M.I. 4 (1968), 135-137. (1968) MR0227827
  3. Frehse J., Una generalizzazione di un controesempio di De Giorgi nella teoria delle equazioni ellitiche, Boll. U.M.I. 6 (1970), 998-1002. (1970) MR0276612
  4. Galdi G.P., An introduction to the mathematical theory of the Navier-Stokes equations, Volume I: Linearized steady problems, Springer tracts in natural philosophy vol.38, Springer-Verlag, New York Berlin Heidelberg et alii, 1994. MR1284205
  5. Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies, vol. 105 Princeton University Press Princeton (1983). (1983) Zbl0516.49003MR0717034
  6. Giaquinta M., Modica G., Non linear systems of the type of the Navier-Stokes system, J. Reine Angewandte Math. 330 (1982), 173-214. (1982) MR0641818
  7. John O., Malý J., Stará J., Nowhere continuous solutions to elliptic systems, Comment. Math. Univ. Carolinae 30 (1989), 33-43. (1989) MR0995699
  8. Koshelev A.I., Chelkak S.I., Regularity of solutions of quasilinear elliptic systems, Teubner Leipzig (1985). (1985) Zbl0581.35003MR0825485
  9. Ladyzhenskaya O. A., The mathematical theory of viscous incompressible flow, Gordon and Breach Science Publishers New York (1969). (1969) Zbl0184.52603MR0254401
  10. Leonardi S., On constants of some regularity theorems. De Giorgi counterexamples, Preprint N.I.U (1993) [Thesis (1995-96)], to appear. 
  11. Munkres J.R., Analysis on manifolds, Addison-Wesley Redwood City (1991). (1991) Zbl0743.26006MR1079066
  12. Nečas J., Introduction to the theory of nonlinear elliptic equations, Teubner, Leipzig, 1983 or J. Wiley, Chichester (1986). (1986) MR0731261
  13. Nečas J., Notes for a talk given at the Equadiff 8 in Bratislava 1993, (personal communication). 
  14. Souček J., Singular solutions to linear elliptic systems,, Comment. Math. Univ. Carolinae 25 (1984), 273-281. (1984) MR0768815
  15. Steinhauer M., Funktionenräume vom Campanato-Typ und Regularitätseigenschaften elliptischer Systeme zweiter Ordnung, Diplomarbeit Univ. Bonn (1993). (1993) 

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