Green's functions and the fundamental solution

The fundamental solution of the Laplacian is the potential of a unit point source: \[ \Phi(x) = \frac{1}{4\pi|x|}\ (n = 3), \qquad \Phi(x) = -\frac{1}{2\pi}\ln|x|\ (n = 2). \] Away from the origin it is harmonic, which the example checks for \( 1/r \). At the origin all of its "Laplacian" is concentrated: \( \nabla\Phi = -x/(4\pi|x|^3) \), and its flux through the sphere of radius \( r \) is \( -\frac{1}{4\pi r^2}\cdot 4\pi r^2 = -1 \) whatever \( r \) is. In the language of distributions, \( -\Delta\Phi = \delta \).

Superposing point sources gives the Newtonian potential \( u(x) = \int\Phi(x - y)f(y)\,dy \), which solves \( -\Delta u = f \) in all of space for nice \( f \). Gravity, electrostatics and the Biot-Savart law (velocity recovered from vorticity) are all this convolution.

On a bounded domain \( \Omega \) we need a Green's function \( G(x, y) = \Phi(x - y) - h(x, y) \), where for each \( y \) the corrector \( h \) is harmonic in \( x \) and equals \( \Phi(x - y) \) on \( \partial\Omega \), so \( G = 0 \) on the boundary. Then the solution of \( -\Delta u = f \) in \( \Omega \), \( u = g \) on \( \partial\Omega \), is \[ u(x) = \int_\Omega G(x,y)f(y)\,dy - \int_{\partial\Omega}g(y)\,\partial_{\nu_y}G(x,y)\,dS_y. \] For a half-space the corrector is an image charge: subtract the potential of a source reflected across the boundary. In one dimension, \( -u'' = f \) on \( (0,1) \) with \( u(0) = u(1) = 0 \) has \( G(x,y) = x(1 - y) \) for \( x \le y \) and \( y(1 - x) \) for \( y \le x \); with \( f = 1 \) this gives \( u = \int_0^1 G(x,y)\,dy = x(1 - x)/2 \).

Picture it: \( G(x, y) \) is the shape of a taut string pushed up by a unit point force at \( y \): a tent with straight sides pinned at both ends. Any load is a sum of point forces, so any deflection is a sum of tents.

Think it: a Green's function is the kernel of the inverse operator, \( (-\Delta)^{-1} \), just as a matrix inverse has entries. Its symmetry \( G(x,y) = G(y,x) \) reflects the symmetry of \( -\Delta \). In incompressible flow the pressure is \( p = (-\Delta)^{-1}\partial_i\partial_j(u_iu_j) \), a nonlocal function of the velocity, and it is this operator that couples every point of the fluid to every other.