On the regularity of solutions in the Pucci-Serrin identity

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Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators (2003, methodology): “Various extensions of this identity have been given under various assumptions on the structure of the difierential operator in (4.1) and on the regularity of the solution u (see, for instance, [30], [17], [ 9 ], [14]).”
Liouville results for m-Laplace equations of Lane―Emden―Fowler type (2009, unknown): “Squassina [10].”
Remarks on an overdetermined boundary value problem (2008, methodology): “However, as shown in Theorem 2 of [3] Eq.”
Existence and nonexistence results for anisotropic quasilinear elliptic equations (2004, background): “cisely, in order to have well-defined boundary terms, see [7,8,12].”