Dirichlet's Principle

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Dirichlet's Principle may refer to:

Dirichlet's Box Principle

Let $S$ be a finite set whose cardinality is $n$.

Let $S_1, S_2, \ldots, S_k$ be a partition of $S$ into $k$ subsets.

Then:

at least one subset $S_i$ of $S$ contains at least $\ceiling {\dfrac n k}$ elements

where $\ceiling {\, \cdot \,}$ denotes the ceiling function.


Dirichlet's Principle for Harmonic Functions

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.


Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.