Discussion
Consider the functional :
\[ I(u(x, y)) = \iint_{G} F(x, y, u, u_{x}, u_{y})\, dx\, dy \]
over the region of integration G, where \(u\) is continuous and has continuous derivatives upto second order. Further \(u\) takes prescribed values on the boundary of G. Then the differential equation for the extremization of the functional is :
Answer: D
The equation is \(F_{u} - \partial_{x}F_{u_{x}} - \partial_{y}F_{u_{y}} = 0\), option D.
This is the Euler-Ostrogradsky equation. Vary \(u\) to \(u + \epsilon\eta\) with \(\eta = 0\) on the boundary. The first variation is
\[ \begin{aligned} \iint (&F_{u}\eta + F_{u_{x}}\eta_{x} \\ &+ F_{u_{y}}\eta_{y})\,dx\,dy \end{aligned} \]
Integrating the last two terms by parts (Green's theorem), the boundary terms vanish and
\[ \begin{aligned} \iint \eta\,(F_{u} &- \partial_{x}F_{u_{x}} \\ &- \partial_{y}F_{u_{y}}) = 0 \end{aligned} \]
for every \(\eta\), which forces the bracket to vanish. Each derivative must be paired with its own variable: \(\partial/\partial x\) with \(F_{u_{x}}\) and \(\partial/\partial y\) with \(F_{u_{y}}\). Options A, B and C each mismatch one pair.
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