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Now a cubic box gave me some complicated so I switched the problem to be that of a sphere of radius R, for which I want to find the charge density that minimizes the total electrostatic energy. Mathematically I am therefore trying to solve the problem of finding a ρ(r) that minimizes:

E = 1/4πε

_{0}4π∫

_{0}

^{R}ρ(r) V(ρ(r)) dr

with the boundary conditions:

4π∫

_{0}

^{R}ρ(r)dr = ρ

_{0}

ρ(R)=0.

Now V(r) I can calculate using the symmetry of the sphere, but it will depend on ρ(r), which explains the notation above.

Is it possible to use variational calculus or other means to find ρ(r) such that E is minimized? And is it possible for other geometries?