Minimizing a functional in 1-D

In 1-D, the minimization problem can be stated as

Find ${\displaystyle u(x)}$ such that

${\displaystyle U[u(x)]=\int _{x_{0}}^{x_{1}}F(x,u,u^{'})dx}$

is a minimum.

We have seen that the minimization problem can be reduced down to the solution of an Euler equation

${\displaystyle {\frac {\partial F}{\partial u}}-{\frac {d}{dx}}\left({\frac {\partial F}{\partial u^{'}}}\right)=0}$

with the associated boundary conditions

${\displaystyle \eta (x_{0})=0~{\text{and}}~\eta (x_{1})=0}$

or,

${\displaystyle \left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{0}}=0~{\text{and}}\left.{\frac {\partial F}{\partial u^{'}}}\right|_{x_{1}}=0}$

Minimizing a Functional in 3-D

In 3-D, the equivalent minimization problem can be stated as

Find ${\displaystyle \mathbf {u} (\mathbf {x} )}$ such that

${\displaystyle U[\mathbf {u} (\mathbf {x} )]=\int _{\mathcal {R}}F(\mathbf {x} ,\mathbf {u} ,{\boldsymbol {\nabla }}\mathbf {u} )~dV}$

is a minimum.

We would like to find the Euler equation for this problem and the associated boundary conditions required to minimize ${\displaystyle U}$.

Let us define all our quantities with respect to an orthonormal basis ${\displaystyle ({\widehat {\mathbf {e} }}_{i})}$.

Then,

${\displaystyle \mathbf {x} =x_{i}{\widehat {\mathbf {e} }}_{i}~~;~~~\mathbf {u} =u_{i}{\widehat {\mathbf {e} }}_{i}~~;~~~{\boldsymbol {\nabla }}\mathbf {u} =u_{i,j}{\widehat {\mathbf {e} }}_{i}\otimes {\widehat {\mathbf {e} }}_{j}}$

and

${\displaystyle U[\mathbf {u} (\mathbf {x} )]=\int _{\mathcal {R}}{\tilde {F}}(x_{i},u_{i},u_{i,j})~dV}$

Taking the first variation of ${\displaystyle U}$, we get

${\displaystyle \delta U=\int _{\mathcal {R}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i}}}\delta u_{i}+{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i,j}\right)dV}$

All the nine components of ${\displaystyle \delta u_{i,j}}$ are not independent. Why ?

The variation of the functional ${\displaystyle U}$ needs to be expressed entirely in terms of ${\displaystyle \delta u_{i}}$. We do this using the 3-D equivalent of integration by parts - the divergence theorem.

Thus,

${\displaystyle {\begin{aligned}\int _{\mathcal {R}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i,j}~dV&=\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}\right)dV-\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\delta u_{i}~dV\\&=\int _{\partial {\mathcal {R}}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}~n_{j}~dA-\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}{}{}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\delta u_{i}~dV\end{aligned}}}$

Substituting in the expression for ${\displaystyle \delta U}$, we have,

${\displaystyle {\begin{aligned}\delta U&=\int _{\mathcal {R}}{\frac {\partial {\tilde {F}}}{\partial u_{i}}}\delta u_{i}~dV+\int _{\partial {\mathcal {R}}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}~n_{j}~dA-\int _{\mathcal {R}}{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\delta u_{i}~dV\\&=\int _{\mathcal {R}}\left[{\frac {\partial {\tilde {F}}}{\partial u_{i}}}-{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)\right]\delta u_{i}~dV+\int _{\partial {\mathcal {R}}}{\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\delta u_{i}~n_{j}~dA\end{aligned}}}$

For ${\displaystyle U}$ to be minimum, a necessary condition is that ${\displaystyle \delta U=0}$ for all variations ${\displaystyle \delta \mathbf {u} }$.

Therefore, the Euler equation for this problem is

${\displaystyle {\frac {\partial {\tilde {F}}}{\partial u_{i}}}-{\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}\right)=0~~~~\forall ~~\mathbf {x} \in {\mathcal {R}}}$

and the associated boundary conditions are

${\displaystyle {\frac {\partial {\tilde {F}}}{\partial u_{i,j}}}=0~~~{\text{or,}}~~~\delta u_{i}=0~~~~\forall ~~\mathbf {x} \in \partial {\mathcal {R}}}$