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Beyond the Implicit Function Theorem

Typically, the implicit function theorem can be used to deduce the differentiability of an implicit mapping S : u 7→ y given by the equation e(y, u) = 0. However, the implicit function theorem is not applicable if different norms are necessary for the differentiation of e w.r.t. y and the invertibility of the partial derivative ey(y, u). We prove theorems ensuring the (twice) differentiability of the mapping S which can be applied in this case. We highlight the particular application to quasilinear partial differential equations whose principal part depends nonlinearly on the gradient of the state ∇y.