Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 3 / b / Solution

This is the Dirichlet principle. Put $v=w-\psi$, so $v$ vanishes on the boundary. For real, sufficiently regular functions of finite Dirichlet energy, Green's first identity givesThe surface term is zero because $v=0$, and the volume term is zero because $\psi$ is a harmonic function. Expanding the square of $\nabla w=\nabla \psi +\nabla v$ now gives the stronger identityThus the harmonic function minimizes Dirichlet energy among functions with the same boundary values. Equality requires $\nabla v=0$, hence $v$ is constant on each connected component; the zero boundary value makes that constant zero on every component having boundary. For an unbounded domain one additionally needs the boundary term at infinity to vanish; the usual bounded regular-domain setting needs no such extra condition.