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 {Dan\v {e}\v {c}ek, 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\"{o}lder continuity of solution for all $\gamma <1$, while the {Koshelev} theory is not applicable at all.
Keywords: nonlinear elliptic systems, regularity, Campanato-Morrey spaces
AMS Subject Classification: 35J60