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We establish $C^{2,\alpha}$ estimates for PDE of the form convex $+$ a sum ofweakly concave functions of the Hessian, thus generalising a recent result ofCollins which is in turn inspired by a theorem of Caffarelli and Yuan.Independently, we also prove an existence result for a certain generalisedMonge-Amp\`ere PDE.

$C^{2,\alpha}$ estimates and existence results for certain nonconcave PDE