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\begin{document}

{\bf Question}

Write down the Euler-Lagrange equation $u(x,y)$ must satisfy on an
area $S$ of the $x,\ y$-plane if $u(x,y)$ takes prescribed values
on the closed curve $C$ bounding $S$ and

$$I=\ds\int_SdS(\bigtriangledown u)^r=\ds\int \!\!\! \ds\int_S
\{u^2_x+u^2_y\}^{\frac{r}{2}} dx dy$$

is to be stationary, where $r$ is a given real constant ($\ne 0$).
(It may be assumed that $\bigtriangledown u \ne 0$ on $S$.


\medskip

{\bf Answer}

If $I=\ds\int \!\!\! \ds\int(u_x^2+u_y^2)^{\frac{r}{2}}dxdy$ the
E-L equation is

$\ds\frac{\pl F}{\pl u}-\ds\frac{\pl}{\pl x}\left(\ds\frac{\pl
F}{\pl u_x}\right)-\ds\frac{\pl}{\pl y}\left(\ds\frac{\pl F}{\pl
u_y}\right)=0$ with $F=(u_x^2+u_y^2)^{\frac{r}{2}}$

$\Rightarrow \ds\frac{\pl}{\pl
x}(ru_x(u_x^2+u_y^2)^{\frac{r}{2}-1})+\ds\frac{\pl}{\pl
y}(ru_y(u_x^2+u_y^2)^{\frac{r}{2}-1})=0$

which, after a bit of tedious algebra can be written

$$[(r-1)u_x^2+u_y^2]u_{xx}+2(r-2)u_xu_yu_{xy}+[(r-1)u_y^2+u_x^2]u_{yy}=0$$

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