Differentiability of implicit functions Beyond the implicit function theorem

The implicit function theorem (IFT) can be used to deduce the differentiability of an implicit mapping S : u -> y given by the equation e(y, u) = 0. However, the IFT is not applicable when different norms are necessary for the differentiation of e w.r.t. y and the invertibility of the partial derivative e(y) (y, u). We prove theorems ensuring the (twice) differentiability of the mapping S which can be applied in this case. We highlight the application of our results to quasilinear partial differential equations whose principal part depends nonlinearly on the gradient of the state del y. (C) 2014 Elsevier Inc. All rights reserved.