Morse theory and the calculus of variations.

We prove for the first time that classical Morse theory applies to functionals of the form 풥 ⁢ (u) = 1 2 ⁢ ∫ Ω A α ⁢ β i ⁢ j ⁢ (x) ⁢ ∂ ⁡ u i ∂ ⁡ x α ⁢ ∂ ⁡ u j ∂ ⁡ x β ⁢ 푑 x + ∫ Ω G ⁢ (x , u) ⁢ 푑 x where u : Ω → ℝ N , Ω ⊂ ℝ n compact with C ∞ boundary ∂ ⁡ Ω , u | ∂ ⁡ Ω = φ , and we argue that this is the largest class to which Morse theory applies. [ABSTRACT FROM AUTHOR]