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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