Green's Identities and Potential Theory

Green's identities, derived from the divergence theorem, form the foundation of potential theory and are essential for solving boundary value problems involving the Laplace and Poisson equations.

Green's First and Second Identities

Theorem12.5Green's First Identity

Let $E$ be a bounded region with smooth boundary $S = \partial E$ , and let $f, g$ be $C^2$ functions. Then $\iiint_E (f\nabla^2 g + \nabla f \cdot \nabla g)\,dV = \oiint_S f \frac{\partial g}{\partial n}\,dS$ where $\frac{\partial g}{\partial n} = \nabla g \cdot \hat{\mathbf{n}}$ is the outward normal derivative.

Theorem12.6Green's Second Identity

Under the same hypotheses, $\iiint_E (f\nabla^2 g - g\nabla^2 f)\,dV = \oiint_S \left(f\frac{\partial g}{\partial n} - g\frac{\partial f}{\partial n}\right) dS$

Green's second identity expresses the self-adjointness of the Laplacian: the inner product $\langle f, \nabla^2 g \rangle$ equals $\langle \nabla^2 f, g \rangle$ plus boundary terms.

Applications to Harmonic Functions

ExampleMean Value Property

If $u$ is harmonic ( $\nabla^2 u = 0$ ) in a region containing a ball $B_R(\mathbf{a})$ , then using Green's second identity with $g = 1/|\mathbf{x} - \mathbf{a}|$ (the fundamental solution): $u(\mathbf{a}) = \frac{1}{4\pi R^2} \oiint_{S_R(\mathbf{a})} u\,dS$ The value of a harmonic function at any point equals its average over any surrounding sphere. This mean value property in fact characterizes harmonic functions.

ExampleUniqueness of solutions

Green's first identity with $f = g = u_1 - u_2$ (the difference of two solutions to the Dirichlet problem $\nabla^2 u = f$ with $u|_S = h$ ) gives: $\iiint_E |\nabla(u_1 - u_2)|^2\,dV = 0$ Therefore $u_1 - u_2$ is constant, and since it vanishes on $S$ , we have $u_1 = u_2$ . This proves uniqueness for the Dirichlet problem.

RemarkGreen's function and representation formula

Green's second identity leads to the Green's representation formula: for a harmonic function $u$ , $u(\mathbf{x}) = -\oiint_S \left(u\frac{\partial G}{\partial n} - G\frac{\partial u}{\partial n}\right) dS$ where $G(\mathbf{x}, \mathbf{y})$ is the Green's function of the domain. This reduces the problem of finding harmonic functions to finding the Green's function, a cornerstone technique in mathematical physics and engineering.