I'm stuck on this question,
Let $V$ be an open ball in $\mathbb R^3$ in $ { x,y,z \text{ st }x^2 +y^2 +z^2<1}$
I need to minimise the integral $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$
when subject to constraints $$ \iiint_V u \, dx \, dy \, dz = 4 \pi $$ and $$ u=1 \text{ on } \partial V$$
I've got down to this
$$ F= u_x^2+u_y^2+u_z^2 \text{ and } G=u$$ The Euler-Lagrange $$ {\partial\over \partial x}\left( {\partial (F- \lambda G)\over \partial u_x } \right) + {\partial\over \partial y}\left( {\partial (F- \lambda G)\over \partial u_y } \right)+{\partial\over \partial z}\left( {\partial (F- \lambda G)\over \partial u_z } \right)- {\partial\over \partial u}( F- \lambda G) =0 $$
I have worked out the above to be as follows and I know you simplify it to
$$ 2 u_{xx} + 2u_{yy}+ 2u_{zz} - \lambda =0$$ $$ \nabla^2 (u)= {\lambda \over 2} $$
Any suggestions on what to do next?