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nonlinear elliptic systems; regularity; Campanato-Morrey spaces
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.
[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. MR 1159808 | Zbl 0773.35017
[C1] Campanato S.: Sistemi ellittici in forma divergenza. Regolarita all'interno. Quaderni Scuola Norm. Sup. Pisa Pisa (1980). MR 0668196 | Zbl 0453.35026
[C2] Campanato S.: A maximum principle for non-linear elliptic systems: Boundary fundamental estimates. Adv. in Math. 48 (1983), 16-43.
[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
[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
[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
[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). MR 0717034 | Zbl 0516.49003
[Ko] Koshelev A.I.: Regularity Problem for Quasilinear Elliptic and Parabolic System. Lecture Notes in Mathematics 1614 Springer Heidelberg (1995). MR 1442954
[KJF] Kufner A., John O., Fučík S.: Function Spaces. Academia Prague (1977). MR 0482102
[Ne] Nečas J.: Introduction to the Theory of Nonlinear Elliptic Equations. Teubner-Texte zur Mathematik Band 52 Leipzig (1983). MR 0731261
[Nik] Nikodým M.: Thesis, FSI VUT Brno, 2003, 25 pp. (in Czech).
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