Green's identities
/ greenz /
Integration by parts is the trick that moves a derivative off one factor and onto another, leaving a boundary term behind. Green's identities are integration by parts for the Laplacian: they let you trade second derivatives between two functions over a region, paying for the trade with integrals over the boundary. They are the algebraic engine behind nearly everything in potential theory — uniqueness proofs, energy estimates, and the representation formula all run on them.
There are three, in increasing power. Green's first identity is integral over region of (u Laplacian v) = - integral of (grad u . grad v) + integral over boundary of (u dv/dn): one integration by parts, where dv/dn is the outward normal derivative. Green's second identity is the antisymmetric combination integral over region of (u Laplacian v - v Laplacian u) = integral over boundary of (u dv/dn - v du/dn): apply the first identity twice and subtract, and the grad-grad terms cancel. The third Green's identity is the second one with v taken to be the fundamental solution E(x - y); the delta hidden in Laplacian E plucks out the value u(x), giving a formula that writes u inside the region in terms of its source Laplacian u and its boundary values and boundary normal derivatives. That third identity IS the representation formula in disguise.
Why they are indispensable: the first identity, with u = v, gives integral of (u Laplacian u) = -integral of |grad u|^2 + boundary, the foundation of energy methods and of Dirichlet's principle. The second identity, with the boundary conditions of two Green's functions, gives reciprocity. The third gives the solution as a boundary-integral, the starting point of boundary-integral-equation methods. One caveat: these identities assume enough smoothness for the integrals and the divergence theorem to make sense (a reasonable boundary, functions with continuous second derivatives up to it); the singularity of E at the source point must be handled by excising a tiny ball and taking a limit.
Green's first identity with u = v and Laplacian u = 0 (u harmonic) gives integral of |grad u|^2 = integral over boundary of (u du/dn). If u = 0 on the boundary, the right side vanishes, so grad u = 0 everywhere and u is constant — a one-line uniqueness proof for the Dirichlet problem.
Integration by parts turns a uniqueness question into a one-liner.
These identities are just integration by parts (the divergence theorem) and need enough smoothness; the third identity uses the singular fundamental solution, so it is derived by excising a small ball around the source and letting its radius shrink.