Dirichlet's Principle for 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:

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

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.

Proof

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Also known as

is also known just as Dirichlet's Principle.

However, there is more than one theorem named such, so its full name is recommended on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources give this as the Dirichlet Principle.

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.

Historical Note

was based on work done by Carl Friedrich Gauss during the course of his investigations into electromagnetism.

Riemann used this principle to good effect, and it was he who named it for Dirichlet.

It was finally proved by David Hilbert in $1899$.