Morse lemma for functionals of variational calculus (Q1178099)

An approach which permits to prove Morse's lemma for the functional f in the space \(W_{0,p}^ 2(\Omega ) = W_ p^ 2(\Omega )\cap{\overset\circ W_ 2^ 1}(\Omega)\) is proposed. The approach is based on the consideration of the functional f in two spaces -- a Banach space in which the functional f possesses the needed differential properties and in a Hilbert space in which the spectral properties of the derivatives of f are investigated. The obtained results are applied to the analysis of degenerate extremals of variational problems.