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An example of a nonlinear second order elliptic system in three dimension. (English) Zbl 1105.35035
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 J. Daněček [NoDEA, Nonlinear Differ. Equ. Appl. 9, No. 4, 385–396 (2002; Zbl 1055.35042)] gives a stronger result than the theory introduced in A. Koshelev [Regularity problem for quasilinear elliptic and parabolic systems. (Lecture Notes in Mathematics. 1614. Berlin: Springer-Verlag) (1995; Zbl 0847.35023)]. In addition, there is a range of parameters where the first theory gives Hölder continuity of solution for all $$\gamma <1$$, while the Koshelev theory is not applicable at all.
##### MSC:
 35J60 Nonlinear elliptic equations 35J45 Systems of elliptic equations, general (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs
##### Citations:
Zbl 1055.35042; Zbl 0847.35023
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