= Solution
Assume each link delay $\ell_e$ is continuous and nondecreasing. Define the <Beckmann potential> on feasible route flows by
$$
\Psi(f)=\sum_e\int_0^{(Af)_e}\ell_e(u)\,du.
$$
The integral of a nondecreasing function is <convex>, so $\Psi$ is a <convex function>. The <chain rule> gives
$$
\frac{\partial\Psi}{\partial f_p}=L_p(f).
$$
At a minimum, move a small amount of flow from a used route $p$ to any other route $q$ for the same source-sink pair. This is a feasible direction, and its one-sided derivative is $L_q-L_p\geq0$. Thus every used route has minimum cost, establishing the <Wardrop equilibrium> conditions.
Conversely, if $f$ satisfies the <Wardrop equilibrium> conditions, then for any feasible $g$,
$$
\nabla\Psi(f)\cdot(g-f)=\sum_k\sum_{p\in\mathcal P_k}L_p(f)(g_p-f_p)\geq\sum_k\lambda_k\sum_{p\in\mathcal P_k}(g_p-f_p)=0.
$$
Here $\sum_pL_pf_p=\lambda_kd_k$ on each pair, whereas $\sum_pL_pg_p\geq\lambda_kd_k$. The supporting inequality for a <convex function> then gives $\Psi(g)\geq\Psi(f)$. Therefore
$$
\boxed{\text{Wardrop equilibria are precisely the minimizers of the Beckmann potential.}}
$$
The feasible route-flow set is compact, and the <Beckmann potential> is continuous, so a minimizer exists. If each $\ell_e$ is strictly increasing, the potential is strictly convex as a function of link loads. Two minimizers with distinct link-load vectors would have a midpoint with strictly smaller potential, a contradiction. Thus equilibrium link loads are unique in this case; <route-flow nonuniqueness at a Wardrop equilibrium> can still occur because different route decompositions can have the same link loads.
The optimization of selfish route choice must be distinguished from the minimization of total travel time
$$
D(f)=\sum_p f_pL_p(f)=\sum_ev_e\ell_e(v_e).
$$
For differentiable delays, the marginal social cost on link $e$ is $\ell_e(v_e)+v_e\ell_e'(v_e)$, rather than just $\ell_e(v_e)$. The <marginal external cost toll> $\tau_e(v_e)=v_e\ell_e'(v_e)$ makes each traveler's perceived link cost equal to this derivative. If $D$ is <convex>, the tolled <Wardrop equilibria> minimize $D$: the corresponding <Beckmann potential> is exactly $D$, up to its zero reference value. Convexity of $D$ is an additional assumption and does not follow merely from nondecreasing delays.
An elastic-demand extension assigns pair $k$ an <inverse demand function> $P_k(d_k)$ and allows demand to vary. Subtracting $\sum_k\int_0^{d_k}P_k(u)\,du$ from the <Beckmann potential> produces the optimization formulation: its marginal condition equates minimum route delay with marginal willingness to travel on positive-demand pairs. With decreasing inverse demands, the subtracted utility is concave and the resulting objective remains convex. This explains why both fixed-demand route choice and an <elastic-demand Wardrop equilibrium> fit a common optimization framework.
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