9.1 Green’s representation formula
Throughout, \(\Omega \) is a bounded open set in \(\mathbb R^n\) with smooth boundary, \(\nu \) the outward unit normal, and \(\Phi \) the fundamental solution of Section 8, normalised by \(-\Delta \Phi = \delta \). Green’s second identity is \[ \int _\Omega (u\Delta v - v\Delta u)\,dy = \int _{\partial \Omega }(u\,\partial _\nu v - v\,\partial _\nu u)\,dS , \] from the divergence theorem applied to \(u\nabla v - v\nabla u\).
Theorem 9.1.1 (Representation formula). If \(u \in C^2(\overline \Omega )\) and \(x \in \Omega \), then \[ u(x) = \int _\Omega \Phi (y - x)\bigl (-\Delta u(y)\bigr )\,dy + \int _{\partial \Omega }\bigl (\Phi (y - x)\,\partial _\nu u(y) - u(y)\,\partial _\nu \Phi (y - x)\bigr )\,dS(y) . \]
Proof. Apply Green’s identity with \(v = \Phi (\cdot - x)\) on \(\Omega _\varepsilon = \Omega \setminus \overline {B_\varepsilon (x)}\), where \(\Delta v = 0\): \[ -\int _{\Omega _\varepsilon }\Phi \,\Delta u\,dy = \int _{\partial \Omega }(u\,\partial _\nu \Phi - \Phi \,\partial _\nu u)\,dS + \int _{\partial B_\varepsilon (x)}(u\,\partial _\nu \Phi - \Phi \,\partial _\nu u)\,dS . \] On the small sphere the outward normal of \(\Omega _\varepsilon \) points towards \(x\), so \(\partial _\nu \Phi = 1/(\omega _n\varepsilon ^{n - 1})\) (the flux normalisation), and \(\int _{\partial B_\varepsilon }u\,\partial _\nu \Phi \) is the spherical mean of \(u\), which tends to \(u(x)\). The other term is at most \(\max |\nabla u|\) times \(|\Phi (\varepsilon )|\,\omega _n\varepsilon ^{n - 1}\), which is \(O(\varepsilon \log \varepsilon )\) in the plane and \(O(\varepsilon )\) in space. Since \(\Phi \) is integrable, letting \(\varepsilon \to 0\) and rearranging gives the formula. □
The formula expresses \(u\) through \(\Delta u\) in \(\Omega \) and both \(u\) and \(\partial _\nu u\) on the boundary. But a Dirichlet problem prescribes only \(u\) on the boundary, and the two cannot be prescribed independently (that would be a Cauchy problem for an elliptic equation, ill posed by Hadamard’s example). The Green’s function removes the unknown \(\partial _\nu u\).
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