. (The Neumann problem) If r 2 D 0 in D and @ =@n D 0 on S,

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. (The Neumann problem) If r 2 D 0 in D and @ =@n D 0 on S, show that r .x; y; z/ D 0 on D. The Neumann problem for the

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. (The Neumann problem) If r 2 D 0 in D and @ =@n D 0 on S, show that r .x; y; z/ D 0 on D. The Neumann problem for the Laplacian operator is the boundary-value problem 8 < : r 2u.x; y; z/ D f .x; y; z/ on D @ @nu.x; y; z/ D g.x; y; z/ on S,where f and g are given functions defined on D and S, respectively. Show that, if D is connected, then any two solutions of the Neumann problem must differ by a constant on D.

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