# Article

 Title: An example of a nonlinear second order elliptic system in three dimension (English) Author: Daněček, Josef Author: Nikodým, Marek Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 45 Issue: 3 Year: 2004 Pages: 431-442 . Category: math . Summary: 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 {\it Dan\v{e}\v{c}ek, Nonlinear Differential Equations Appl.\/} {\bf 9} (2002), gives a stronger result than the theory introduced in {\it Koshelev, Lecture Notes in Mathematics,} {\bf 1614}, 1995. In addition, there is a range of parameters where the first theory gives H"{o}lder continuity of solution for all $\gamma<1$, while the {\it Koshelev} theory is not applicable at all. (English) Keyword: nonlinear elliptic systems Keyword: regularity Keyword: Campanato-Morrey spaces MSC: 35B65 MSC: 35D10 MSC: 35J45 MSC: 35J60 idZBL: Zbl 1105.35035 idMR: MR2103138 . Date available: 2009-05-05T16:46:20Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119471 . Reference: [BV] 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. Zbl 0773.35017, MR 1159808 Reference: [C1] Campanato S.: Sistemi ellittici in forma divergenza. Regolarita all'interno.Quaderni Scuola Norm. Sup. Pisa Pisa (1980). Zbl 0453.35026, MR 0668196 Reference: [C2] Campanato S.: A maximum principle for non-linear elliptic systems: Boundary fundamental estimates.Adv. in Math. 48 (1983), 16-43. Reference: [D1] Daněček J.: On the interior regularity of weak solutions to nonlinear elliptic systems of second order.Z. Anal. Anwendungen 9.6 (1990), 535-544. MR 1119297 Reference: [D2] Daněček J.: The interior $BMO$-regularity for a weak solution of a nonlinear second order elliptic systems.NoDEA Nonlinear Differential Equations Appl. 9 (2002), 385-396. MR 1941264 Reference: [DJS] Daněček J., John O., Stará J.: The interior $C^{1,\gamma}$-regularity for a weak solution of nonlinear second order elliptic systems.Math. Nachr., to appear. MR 2100046 Reference: [Gia] Giaquinta M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems.Annals of Mathematics Studies 105 Princeton University Press Princeton, NJ (1983). Zbl 0516.49003, MR 0717034 Reference: [Ko] Koshelev A.I.: Regularity Problem for Quasilinear Elliptic and Parabolic System.Lecture Notes in Mathematics 1614 Springer Heidelberg (1995). MR 1442954 Reference: [KJF] Kufner A., John O., Fučík S.: Function Spaces.Academia Prague (1977). MR 0482102 Reference: [Ne] Nečas J.: Introduction to the Theory of Nonlinear Elliptic Equations.Teubner-Texte zur Mathematik Band 52 Leipzig (1983). MR 0731261 Reference: [Nik] Nikodým M.: Thesis, FSI VUT Brno, 2003, 25 pp. (in Czech).. .

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