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↦yS:u↦y given by the equation e(y,u)=0e(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 ey(y,u)ey(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 ∇y.