Green's function
/ greenz /
Suppose you want to solve a problem inside a region with a wall around it — heat in a room, voltage in a box, a membrane clamped at its rim — driven by some source. Green's function is the master key: it is the response inside the region to a unit point source placed at one interior point, computed so that it already respects the wall. Once you know how the region responds to a poke at every single point, you know its response to any source whatsoever, just by adding up the pokes.
Named after George Green, the function G(x, y) solves the operator equation with a point source at y and satisfies the boundary condition on the wall. For the Laplacian with a Dirichlet condition, G(x, y) satisfies -Laplacian_x G = delta(x - y) inside the region and G = 0 on the boundary. It splits into two pieces: G(x, y) = E(x - y) + h(x, y), where E is the free-space fundamental solution carrying the singularity, and h is a smooth boundary corrector (a homogeneous solution, L h = 0 inside) chosen so that the sum vanishes (or has the right normal derivative) on the boundary. Then the solution of L u = f with that boundary condition is u(x) = integral of G(x, y) f(y) dy, possibly plus boundary terms that bring in the prescribed boundary data.
Green's functions are how you write the solution explicitly, and they reveal structure: the symmetry G(x, y) = G(y, x) (reciprocity) is a deep physical statement that source and observation points are interchangeable; the eigenfunction expansion of G connects it to the operator's spectrum; the method of images builds G in simple geometries. Two honest limits: an explicit Green's function is usually available only for special, symmetric domains (half-space, disk, ball) — for a general blobby region you know it exists but cannot write it down in closed form; and the whole construction relies on the problem being linear.
On the unit disk, the Dirichlet Green's function for the Laplacian is G(x, y) = -(1/(2 pi))[ log|x - y| - log( |y| |x - y*| ) ], where y* = y/|y|^2 is the image of y reflected across the circle. The image term is the boundary corrector that makes G = 0 on the rim.
Free-space log plus an image term that cancels it on the boundary.
An explicit Green's function is usually only obtainable for special symmetric domains; for a general region it exists but generally cannot be written in closed form, and it is specific to one operator AND one boundary condition.