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 $\map u {\mathbf x}$ be a real function.

Let $R$ be a fixed region.

Let $J$ be a functional such that


 * $\displaystyle J\sqbrk u=\idotsint_R \map F {\mathbf x,u,u_{\mathbf x} }\rd x_1\dots\rd x_n$

Then a necessary condition for $J\sqbrk u$ to have an extremum (strong or weak) for a given mapping $\map u {\mathbf x}$ is that $\map u {\mathbf x}$ satisfies Euler's equation:


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

Proof
By definition of increment of the functional:

Use multivariate Taylor's Theorem on $F$ around the point $\paren{\mathbf x,u,u_{\mathbf x} }$:


 * $\displaystyle F\sqbrk{\mathbf x,u+h,u_{\mathbf x}+h_{\mathbf x} }=F\sqbrk{\mathbf x,u,u_{\mathbf x} }+\frac{\partial F\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial u}h+\frac{\partial F\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial u_{\mathbf x} }h_{\mathbf x}+\mathcal O\paren{h^2,hh_{\mathbf x},h_{\mathbf x}^2}$

where $\mathcal O$ stands for Big-O.

Then:


 * $\displaystyle\Delta J\sqbrk{u,h}=\idotsint_R \paren{\frac{\partial F\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial u}h+\frac{ \partial F\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial u_{\mathbf x} }h_{\mathbf x}+\mathcal O\paren{h^2,hh_{\mathbf x},h_{\mathbf x}^2} }\rd x_1\dots\rd x_n$

By definition of variation of the functional:

By Green's theorem:


 * $\displaystyle\idotsint_R\frac{\partial}{\partial\mathbf x}\sqbrk{\frac{\partial F\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial u_{\mathbf x} }\map h {\mathbf x} }\rd x_1\dots\rd x_n=\idotsint_\Gamma \map h {\mathbf x}F_{u_{\mathbf x} }\sqbrk{\mathbf x,u,u_{\mathbf x} }\boldsymbol\nu\rd\sigma$

where $\Gamma$ denotes boundary of $R$, and $\boldsymbol\nu$ is an outward normal to $\Gamma$.

Since the region is fixed, so are its boundary points.

Hence, the difference function $h$ has to vanish at the boundary.

In other words:


 * $\displaystyle\forall\mathbf x\in\Gamma:\map h {\mathbf x}=0$

This leaves only the first integral.


 * $\displaystyle\delta J=\idotsint_R \paren{\frac{\partial F\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial u}-\frac{\partial F_{ u_{ \mathbf x } }\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial\mathbf x} }\map h {\mathbf x}\rd x_1\dots\rd x_n$

For arbitrary $h$ the first variation vanishes if the term in brackets vanishes:


 * $\displaystyle\frac{\partial F\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial u}-\frac{\partial F_{u_{\mathbf x} }\sqbrk{\mathbf x,u,u_{\mathbf x} } }{\partial\mathbf x}=0$