Given the Lagrangian
$$ J(u)= \int_{V} \sqrt{1+|\operatorname{grad}(u)|^{2})} $$
with the constraint $ \int_{V}udx =1 $
(1) Why is the volume constraint there ?
(2) For the case of $\mathbb R^{3}$, I know this must satisfy the equation
$$ (1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0 $$
however how could i generalize this equation to arbitrary $\mathbb R^{n} $ ?
From the Euler-Lagrange equation I get (plus Lagrange parameter)
$$ \sum_{i} \partial_{x_{i}} \frac{u_{x_{i}}}{\sqrt{1+|gra(u)|^{2}}}= \lambda $$
I think this equation is correct but I need to order this a bit more. :)