Necessary Condition for Integral Functional to have Extremum for given function/Dependent on n Variables

Theorem
Let $ \mathbf x $ be an n-dimensional vector.

Let $ u \left ( { \mathbf x } \right ) $ be a real function.

Let $ J $ be a functional such that


 * $ \displaystyle J \left [ { u } \right ] = \idotsint_R F \left ( { \mathbf x, u, u_{ \mathbf x } } \right ) \mathrm d x_1 \dots \mathrm d x_n $

Then a necessary condition for $ J \left [ { u } \right ] $ to have an extremum (strong or weak) for a given function $ u \left ( { \mathbf x } \right )$ is that $ u \left ( { \mathbf x } \right )$ satisfies Euler's equation:


 * $ \displaystyle F_u - \frac{ \partial }{ \partial \mathbf x } F_{ u_{ \mathbf x } } = 0 $

Proof

 * $ \displaystyle u^* \left ( { \mathbf x } \right ) = u \left ( { \mathbf x } \right ) + \epsilon \psi \left ( { \mathbf x } \right ) + \cdots $


 * $ \displaystyle \idotsint_R \frac{ \partial }{ \partial \mathbf x } \left [ { F_{ u_{ \mathbf x } } \psi \left ( { x } \right ) } \right ] \mathrm d x_1 \dots \mathrm d x_n = \idotsint_\Gamma \psi \left ( { \mathbf x } \right ) F_{ u_{ \mathbf x } } \boldsymbol \nu \mathrm d \sigma$


 * $ \displaystyle \delta J = \epsilon \idotsint_R \left ( { F_u - \frac{ \partial F_{ u_{ \mathbf x } } }{ \partial \mathbf x } } \right ) \psi \left ( { x } \right ) \mathrm d x_1 \dots \mathrm d x_n$