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We provide an explicit example of a nonlinear second order elliptic system of two equations in three dimension to compare two -regularity theories. We show that, for certain range of parameters, the theory developed in
(2002), gives a stronger result than the theory introduced in
, 1995. In addition, there is a range of parameters where the first theory gives H"older continuity of solution for all , while the theory is not applicable at all.
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