42 2. FOUR IMPORTANT LINEAR PDE THEOREM 16 (Uniqueness). There exists at most one solution u ∈ C2( ¯ U ) of (46). Proof. Assume ˜ u is another solution and set w := u − ˜. u Then Δw = 0 in U, and so an integration by parts shows 0 = − U wΔw dx = U |Dw|2 dx. Thus Dw ≡ 0 in U, and, since w = 0 on ∂U, we deduce w = u − ˜ u ≡ 0 in U. b. Dirichlet’s principle. Next let us demonstrate that a solution of the boundary-value problem (46) for Poisson’s equation can be characterized as the minimizer of an appropriate functional. For this, we define the energy functional I[w] := U 1 2 |Dw|2 − wf dx, w belonging to the admissible set A := {w ∈ C2( ¯ U ) | w = g on ∂U}. THEOREM 17 (Dirichlet’s principle). Assume u ∈ C2( ¯ U ) solves (46). Then (47) I[u] = min w∈A I[w]. Conversely, if u ∈ A satisfies (47), then u solves the boundary-value problem (46). In other words if u ∈ A, the PDE −Δu = f is equivalent to the statement that u minimizes the energy I[ · ]. Proof. 1. Choose w ∈ A. Then (46) implies 0 = U (−Δu − f)(u − w) dx. An integration by parts yields 0 = U Du · D(u − w) − f(u − w) dx,
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