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] R).
Since u + τv A for each τ, the scalar function i(·) has a minimum at zero,
and thus
i (0) = 0 =
d

,
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.
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