= Solution
Write $a(v,w)=\int_\Omega a\nabla v\cdot\nabla w$ and $A[v]=a(v,v)$. The admissible set is the closed <affine subspace> $H_g^1=g+H_0^1$. Symmetry of the coefficient matrix gives
$$
A[u+t\varphi]=A[u]+2t\,a(u,\varphi)+t^2A[\varphi],\qquad\varphi\in H_0^1.
$$
If $u$ minimizes the energy, its <first variation> vanishes: differentiating at $t=0$ gives $a(u,\varphi)=0$. Hence $u$ is a <weak solution>.
Conversely, if $u$ is a <weak solution>, every admissible $v$ has $v-u\in H_0^1$ and therefore
$$
A[v]=A[u]+2a(u,v-u)+A[v-u]=A[u]+A[v-u]\geq A[u],
$$
by <uniform ellipticity>. This is the <Dirichlet principle> for the stated symmetric energy.
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