Dirichlet's Principle (Harmonic Functions)

Theorem
Let the function $\map u x$ be the particular solution to Poisson's equation:


 * $\Delta u + f = 0$

on a domain $\Omega$ of $\R^n$ with boundary condition:


 * $u = g$ on $\partial \Omega$

Then $u$ can be obtained as the minimizer of the Dirichlet's energy:


 * $\displaystyle E \sqbrk {\map v x} = \int_\Omega \paren {\frac 1 2 \cmod {\nabla v}^2 - v f} \rd x$

amongst all twice differentiable functions $v$ such that $v = g$ on $\partial \Omega$.

This result holds provided that there exists at least one function which makes the Dirichlet integral finite.

Also known as
Some sources give this as the Dirichlet principle.