An example of a nonlinear second order elliptic system in three dimension

Josef Daněček; Marek Nikodým

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 3, page 431-442
  • ISSN: 0010-2628

Abstract

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We provide an explicit example of a nonlinear second order elliptic system of two equations in three dimension to compare two C 0 , γ -regularity theories. We show that, for certain range of parameters, the theory developed in Daněček, Nonlinear Differential Equations Appl.9 (2002), gives a stronger result than the theory introduced in Koshelev, Lecture Notes in Mathematics,1614, 1995. In addition, there is a range of parameters where the first theory gives H"older continuity of solution for all γ < 1 , while the Koshelev theory is not applicable at all.

How to cite

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Daněček, Josef, and Nikodým, Marek. "An example of a nonlinear second order elliptic system in three dimension." Commentationes Mathematicae Universitatis Carolinae 45.3 (2004): 431-442. <http://eudml.org/doc/249360>.

@article{Daněček2004,
abstract = {We provide an explicit example of a nonlinear second order elliptic system of two equations in three dimension to compare two $C^\{0,\gamma \}$-regularity theories. We show that, for certain range of parameters, the theory developed in Daněček, Nonlinear Differential Equations Appl.9 (2002), gives a stronger result than the theory introduced in Koshelev, Lecture Notes in Mathematics,1614, 1995. In addition, there is a range of parameters where the first theory gives H"older continuity of solution for all $\gamma <1$, while the Koshelev theory is not applicable at all.},
author = {Daněček, Josef, Nikodým, Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonlinear elliptic systems; regularity; Campanato-Morrey spaces; nonlinear elliptic systems; regularity; Campanato-Morrey spaces},
language = {eng},
number = {3},
pages = {431-442},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An example of a nonlinear second order elliptic system in three dimension},
url = {http://eudml.org/doc/249360},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Daněček, Josef
AU - Nikodým, Marek
TI - An example of a nonlinear second order elliptic system in three dimension
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 3
SP - 431
EP - 442
AB - We provide an explicit example of a nonlinear second order elliptic system of two equations in three dimension to compare two $C^{0,\gamma }$-regularity theories. We show that, for certain range of parameters, the theory developed in Daněček, Nonlinear Differential Equations Appl.9 (2002), gives a stronger result than the theory introduced in Koshelev, Lecture Notes in Mathematics,1614, 1995. In addition, there is a range of parameters where the first theory gives H"older continuity of solution for all $\gamma <1$, while the Koshelev theory is not applicable at all.
LA - eng
KW - nonlinear elliptic systems; regularity; Campanato-Morrey spaces; nonlinear elliptic systems; regularity; Campanato-Morrey spaces
UR - http://eudml.org/doc/249360
ER -

References

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  1. Balanda L., Viszus E., On Liouville theorem and the regularity of weak solutions to some nonlinear elliptic systems of higher order, Comment. Math. Univ. Carolinae 32.4 (1991), 615-625. (1991) Zbl0773.35017MR1159808
  2. Campanato S., Sistemi ellittici in forma divergenza. Regolarita all'interno, Quaderni Scuola Norm. Sup. Pisa Pisa (1980). (1980) Zbl0453.35026MR0668196
  3. Campanato S., A maximum principle for non-linear elliptic systems: Boundary fundamental estimates, Adv. in Math. 48 (1983), 16-43. (1983) 
  4. Daněček J., On the interior regularity of weak solutions to nonlinear elliptic systems of second order, Z. Anal. Anwendungen 9.6 (1990), 535-544. (1990) MR1119297
  5. Daněček J., The interior B M O -regularity for a weak solution of a nonlinear second order elliptic systems, NoDEA Nonlinear Differential Equations Appl. 9 (2002), 385-396. (2002) MR1941264
  6. Daněček J., John O., Stará J., The interior C 1 , γ -regularity for a weak solution of nonlinear second order elliptic systems, Math. Nachr., to appear. MR2100046
  7. Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies 105 Princeton University Press Princeton, NJ (1983). (1983) Zbl0516.49003MR0717034
  8. Koshelev A.I., Regularity Problem for Quasilinear Elliptic and Parabolic System, Lecture Notes in Mathematics 1614 Springer Heidelberg (1995). (1995) MR1442954
  9. Kufner A., John O., Fučík S., Function Spaces, Academia Prague (1977). (1977) MR0482102
  10. Nečas J., Introduction to the Theory of Nonlinear Elliptic Equations, Teubner-Texte zur Mathematik Band 52 Leipzig (1983). (1983) MR0731261
  11. Nikodým M., Thesis, FSI VUT Brno, 2003, 25 pp. (in Czech), . 

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