9.2 The Green’s function
Definition 9.2.1. The Green’s function of \(\Omega \) for the Dirichlet problem is \(G(x, y) = \Phi (y - x) - \varphi ^x(y)\), where for each \(x \in \Omega \) the corrector \(\varphi ^x\) is harmonic in \(\Omega \) with \(\varphi ^x(y) = \Phi (y - x)\) on \(\partial \Omega \). Thus \(-\Delta _yG = \delta _x\) in \(\Omega \) and \(G = 0\) on \(\partial \Omega \).
Theorem 9.2.2. If \(u \in C^2(\overline \Omega )\) solves \(-\Delta u = f\) in \(\Omega \) with \(u = g\) on \(\partial \Omega \), then \[ u(x) = \int _\Omega G(x, y)f(y)\,dy - \int _{\partial \Omega }g(y)\,\partial _{\nu _y}G(x, y)\,dS(y) . \] Moreover \(G(x, y) = G(y, x)\), and \(G(x, y) > 0\) for \(x \neq y\) in \(\Omega \).
Proof. Green’s identity for \(u\) and the harmonic \(\varphi ^x\) gives \(\int _\Omega \varphi ^x(-\Delta u) = \int _{\partial \Omega }(u\,\partial _\nu \varphi ^x - \varphi ^x\partial _\nu u)\). Subtract this from the representation formula; on \(\partial \Omega \), \(\varphi ^x = \Phi \), so the terms in \(\partial _\nu u\) cancel, leaving the stated formula. For symmetry, apply Green’s identity to \(G(x, \cdot )\) and \(G(z, \cdot )\) on \(\Omega \) with small balls about \(x\) and \(z\) removed; both vanish on \(\partial \Omega \), and the small spheres contribute \(G(z, x)\) and \(-G(x, z)\) in the limit. For positivity, \(G(x, \cdot )\) is harmonic in \(\Omega \setminus \{x\}\), zero on \(\partial \Omega \), and tends to \(+\infty \) at \(x\), so by the minimum principle on \(\Omega \) minus a small ball it is positive. □
The function \(-\partial _{\nu _y}G(x, y)\) is the Poisson kernel of \(\Omega \); it is positive and integrates to \(1\) over the boundary (take \(g = 1\), \(f = 0\)). Symmetry is a form of reciprocity: the potential at \(y\) due to a unit charge at \(x\) equals the potential at \(x\) due to a unit charge at \(y\), in any grounded enclosure.
In one dimension everything is explicit. For \(-u'' = f\) on \((0, 1)\) with \(u(0) = u(1) = 0\), the Green’s function is \[ G(x, y) = \begin {cases} x(1 - y), & x \leq y, \\ y(1 - x), & x \geq y , \end {cases} \] continuous, linear on each side of \(y\), with a jump of \(-1\) in slope at \(y\): it is the shape of a taut string under a unit point load at \(y\), and \(u = \int _0^1G(x, y)f(y)\,dy\) is the shape under the distributed load \(f\).
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