Tangential manifolds and conditional extrema in mathematical physics (Q2719444)

According to \textit{L. A. Ljusternik} [Rec. Math. Moscow 41, No. 3, 390-401 (1934; Zbl 0011.07401)] it is known that, if \(f\) is a continuously differentiable mapping of an open subset \(W\) of the Banach space \(X\) into the Banach space \(Y\) and \(f'(x_0)X=Y\) for some point \( x_0\in W\), then \( x_0+\text{ker} f'(x_0)\) is a tangential manifold to the manifold \(\{x\in W\mid f(x) = f(x_0)\}\) at the point \(x_0\). The author presents a generalization of this result and discusses its application to nonlinear partial differential equations as well as to the stationary and evolution Navier-Stokes systems.