2.2. LAPLACE’S EQUATION 43
and there is no boundary term since u − w = g − g ≡ 0 on ∂U. Hence
U
|Du|2
− uf dx =
U
Du · Dw − wf dx
≤
U
1
2
|Du|2
dx +
U
1
2
|Dw|2
− wf dx,
where we employed the estimates
|Du · Dw| ≤ |Du||Dw| ≤
1
2
|Du|2
+
1
2
|Dw|2,
following from the Cauchy–Schwarz and Cauchy inequalities (§B.2). Rear-
ranging, we conclude
(48) I[u] ≤ I[w] (w ∈ A).
Since u ∈ A, (47) follows from (48).
2. Now, conversely, suppose (47) holds. Fix any v ∈ Cc
∞(U)
and write
i(τ) := I[u + τv] (τ ∈ R).
Since u + τv ∈ A for each τ, the scalar function i(·) has a minimum at zero,
and thus
i (0) = 0 =
d
dτ
,
provided this derivative exists. But
i(τ) =
U
1
2
|Du +
τDv|2
− (u + τv)f dx
=
U
1
2
|Du|2
+ τDu · Dv +
τ
2
2
|Dv|2
− (u + τv)f dx.
Consequently
0 = i (0) =
U
Du · Dv − vf dx =
U
(−Δu − f)v dx.
This identity is valid for each function v ∈ Cc
∞(U)
and so −Δu = f in
U.
Dirichlet’s principle is an instance of the calculus of variations applied
to Laplace’s equation. See Chapter 8 for more.
