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[Step-by-Step] Let D be a region in R^3 with smooth boundary. Suppose that f is a biharmonic function; that is, f is smooth, real-valued, and Δ(Δ

Question: Let \(D\) be a region in \(\mathbb{R}^{3}\) with smooth boundary. Suppose that \(f\) is a biharmonic function; that is, \(f\) is smooth, real-valued, and \(\Delta(\Delta f)=0\) inside \(D\). Show that if both \(f=0\) and \(\partial f / \partial n=0\) on \(\partial D\), then \(f=0\) everywhere inside \(D\).

Hint: find a way to use the functions \(f\) and \(\Delta f\) in Green's identity:

\[\int_{\partial D}\left(u \frac{\partial v}{\partial n}-v \frac{\partial u}{\partial n}\right) d s=\int_{D}(u \Delta v-v \Delta u) d A\]

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