Find function $u(x,y,z)$ so that the function $I(u)$ to be optimized
$$I(u)=\iiint_{\Omega}(u_x^2+u_y^2+u_z^2-xyzu)\,\mathrm dx\,\mathrm dy\,\mathrm dz $$
that
$$\Omega=\{(x,y,z):0\le x,y,z\le1\}$$
with
$$u(0,y,z)=y,\quad u(1,y,z)=z,\quad u_y(x,0,z)=x,\quad u(x,1,z)=z,\quad u(x,y,0)=y,\quad u(x,y,1)=x$$
Who can advise me? I tried to change four item of boundary conditions to homogeneous form , but i found that i can't do this and it's impossible, then i changed just two of them but in this case , i recived so much difficulities ...