Abstract
Solutions of general, uniformly elliptic systems of quasilinear second order partial differential equations in divergence form are studied under the assumption that the right hand side of the system grows at most quadratically with respect to the gradient of the solution. A partial regularity result is obtained, asserting that in the general case, any weak solution with sufficiently small modulus (an explicit bound is given) has hölder continuous first order derivatives in a neighbourhood of almost every point of the domain of definition. For diagonal systems where the coefficients depend on the modulus, but in no other way, of the gradient of the unknown function, it is shown that regularity in fact holds throughout the domain.
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Ivert, PA. Regularitätsuntersuchungen von Lösungen Elliptischer Systeme von Quasilinearen Differentialgleichungen Zweiter Ordnung. Manuscripta Math 30, 53–88 (1979). https://doi.org/10.1007/BF01305990
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DOI: https://doi.org/10.1007/BF01305990