Category:Dirichlet's Principle
This category contains pages concerning Dirichlet's Principle:
Disambiguation
This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.
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:
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.
Source of Name
This entry was named for Johann Peter Gustav Lejeune Dirichlet.
Subcategories
This category has the following 2 subcategories, out of 2 total.
D
Pages in category "Dirichlet's Principle"
The following 3 pages are in this category, out of 3 total.