= Solution
{c}
For any $v\in H^1(\Omega)$, expand $J(u+\epsilon v)$ in $\epsilon$. Its linear term must vanish at a minimizer, giving
$$
\int_\Omega(\nabla u\cdot\nabla v+uv)=\int_\Omega fv.
$$
Thus the required functionals are
$$
\boxed{a(u,v)=\int_\Omega(\nabla u\cdot\nabla v+uv),\qquad
\ell(v)=\int_\Omega fv.}
$$
The form is continuous, symmetric, and <coercive>, with $a(v,v)=\|v\|_{H^1}^2$. Conversely, this weak equation implies $J(u+w)=J(u)+\frac12a(w,w)$, so its unique solution is indeed the global minimizer.
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