Dirichlet's Principle

Also known as Thomson's Principle. There exists a function $u$ that minimizes the functional

$$D[u] =\int_\Omega \vert\nabla u\vert^2\,dV$$

(called the Dirichlet Integral) for $\Omega\subset\Bbb{R}^2$ or $\Bbb{R}^3$ among all the functions $u\in C^{(1)}(\Omega)\cap C^{(0)}(\bar\Omega)$ which take on given values $f$ on the boundary $\partial\Omega$ of $\Omega$, and that function $u$ satisfies $\nabla^2=0$ in $\Omega$, $u\vert _{\partial \Omega} =f$, $u\in C^{(2)}(\Omega)\cap C^{(0)}(\bar\Omega)$. Weierstraß showed that Dirichlet's argument contained a subtle fallacy. As a result, it can be claimed only that there exists a lower bound to which $D[u]$ comes arbitrarily close without being forced to actually reach it. Kneser, however, obtained a valid proof of Dirichlet's principle.

See also Dirichlet's Box Principle, Dirichlet Integrals