Evans asks:
Assume $u$ is a smooth minimizer of the area integral $$I[w] = \int_U(1+|Dw|^2)^{1/2})\,dx$$ subject the boundary conditions $w=g$ on $\partial U$ and the constraint $$J[w] = \int_U w\,dx = 1$$
My issue is I'm getting 2 solutions in two different ways. So either one (or both) are incorrect.
Up to constant, the Lagrangian works out to be $$\int_U(1+|Dw|^2)^{1/2}) + w \,dx$$
Strategy 1: Consider $r=w-g$ -- this forces r to go to zero on the boundaries. Then use EL equations to obtain: $$\sum_{i=1}^{n} \bigg(\frac{r_{x_{i}}}{(1+|Dr|^2)^{1/2}}\bigg)_{x_i} = 1$$ Then substitute $w-g$ back in everywhere.
Strategy 2: Vary the Lagrangian above with respect to $w$ to solve for the EL equation and do not eliminate boundary term. The resulting expression then looks like: $$\sum_{i=1}^{n} \bigg(\frac{w_{x_{i}}}{(1+|Dw|^2)^{1/2}}\bigg)_{x_i} - \bigg(\frac{g_{x_{i}}}{(1+|Dg|^2)^{1/2}}\bigg) = 1$$
These expressions look different - what am I missing?
The variational PDE normally does not involve the boundary conditions... setting $r=w-g$ makes little sense if $g$ is only defined on $\partial U$. As far as calculus of variations is concerned, the role of boundary condition is to force us to use compactly supported perturbations only, as is done in the derivation of the Euler-Lagrange equation.
There is an integral constraint here: handle it with a Lagrange multiplier, letting $$ L[w] = I[w]+\lambda J[w] = \int_U ((1+|Dw|^2)^{1/2} + \lambda w) $$ The Euler-Lagrange equation is $$ - \operatorname{div}\left( \frac{Dw}{(1+|Dw|^2)^{1/2}} \right) + \lambda = 0 $$ So, the vector field $Dw/\sqrt{1+|Dw|^2}$ must be of constant divergence. We won't know $\lambda$ until the boundary value problem is solved, which isn't going to happen with abstract boundary data $g$.