The Dirichlet's principle in Folland's PDE book enunciate that 3 conditions related to harmonic functions in the Sobolev space $H_1(\Omega)$ ($\Omega$ is a domain) are equivalent, but there is no proof about it. The statement says:
If $f$ and $g$ are in $H_1(\Omega)$, the following three conditions are equivalent:
(a) $w$ is harmonic in $\Omega$ and $w-f\in H_1^0(\Omega)$.
(b) $D(w,w)\leq D(u,u)$ for all $u\in H_1(\Omega)$ such that $u-f$ is harmonic in $\Omega$.
(c) $D(w-f,w-f)\leq D(u,u)$ for all $u\in H_1(\Omega)$ such that $u-f$ is harmonic in $\Omega$.
Here $H_1$ is the completion of $C^1$, $H_1^0$ is the closure of $C_c^\infty$ in $H_1$, and $D(u,v)=\int_\Omega \nabla(u)\bar{\nabla(v)}$.
I trying to do it in the following way: (a) if and only if (b) and (a) is and only if (c), but I don't know how to start. Could someone give me a hint for example for the equivalence (a) if and only if (c)?
