Stability of classes of mappings and Hölder continuity of higher derivatives of elliptic solutions to systems of nonlinear differential equations

Nirenberg published the following well-known result in 1954: Let a function $z$ be a twice continuously differentiable solution to a nonlinear second-order elliptic equation. Suppose that the function $F$ defining the equation is continuous and has continuous first-order partial derivatives with respect to all of its arguments (i.e. independent together with $z$ and the symbols of all first- and second-order partial derivatives of $z$). Then the partial derivatives of $z$ are locally Holder continuous. Simultaneously with Nirenberg, Morrey obtained an analogous result for elliptic systems of second-order nonlinear equations. In this article, we get the same result for the higher derivatives of elliptic solutions to systems of nonlinear partial differential equations of arbitrary order and a rather general shape. The proof is based on the results of the author's recent research on the study of the stability phenomena in the $C^l$ –norm of classes of mappings.