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.


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
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