Morse theory and the calculus of variations

We prove for the first time that classical Morse theory applies to functionals of the form 𝒥 ⁢ ( u ) = 1 2 ⁢ ∫ Ω A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ∂ ⁡ u i ∂ ⁡ x α ⁢ ∂ ⁡ u j ∂ ⁡ x β ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x \displaystyle\mathcal{J}(u)=\frac{1}{2}\int_{\Omega}A^{ij}_{\alpha\beta}(x)% \frac{\partial u^{i}}{\partial x^{\alpha}}\frac{\partial u^{j}}{\partial x^{% \beta}}\,dx+\int_{\Omega}G(x,u)\,dx where u : Ω → ℝ N {u:\Omega\to\mathbb{R}^{N}} , Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} compact with C ∞ {C^{\infty}} boundary ∂ ⁡ Ω {\partial\Omega} , u | ∂ ⁡ Ω = φ {u|_{\partial\Omega}=\varphi} , and we argue that this is the largest class to which Morse theory applies.