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.