Abstract
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.
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Kopylov, A.P. Stability of Classes of Mappings and Holder Continuity of Higher Derivatives of Elliptic Solutions to Systems of Nonlinear Differential Equations. Siberian Mathematical Journal 43, 68–82 (2001). https://doi.org/10.1023/A:1013824605070
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DOI: https://doi.org/10.1023/A:1013824605070