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.
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