# Dirichlet integral

A functional connected with the solution of the Dirichlet problem for the Laplace equation by the variational method. Let $\Omega$ be a bounded domain in $\mathbf R ^ {n}$ with boundary $\Gamma$ of class $C ^ {1}$, let $x = ( x _ {1} \dots x _ {n} )$ and let the function $u \in W _ {2} ^ {1} ( \Omega )$( cf. Sobolev space). The Dirichlet integral for the function $u$ is the expression

$$D [ u] = \int\limits _ \Omega \sum _ {i = 1 } ^ { n } \left ( \frac{\partial u }{\partial x _ {i} } \right ) ^ {2} dx .$$

For a certain given function $\phi$ on $\Gamma$ one considers the set $\pi _ \phi$ of functions from $W _ {2} ^ {1} ( \Omega )$ which satisfy the boundary condition $u \mid _ \Gamma = \phi$. If the set $\pi _ \phi$ is non-empty, there exists a unique function $u _ {0} \in \pi _ \phi$ for which

$$D [ u _ {0} ] = \inf _ {u \in \pi _ \phi } D [ u] ,$$

and this function is harmonic in $\Omega$. The converse theorem is also true: If a harmonic function $u _ {0}$ belongs to the set $\pi _ \phi$, then $\inf D [ u]$ is attained on it. Thus, $u _ {0}$ is a generalized solution from $W _ {2} ^ {1} ( \Omega )$ of the Dirichlet problem for the Laplace equation. However, not for every function $\phi$ it is possible to find a function $u _ {0}$. There exists even continuous functions on $\Gamma$ for which the set $\pi _ \phi$ is empty, i.e. the space $W _ {2} ^ {1} ( \Omega )$ contains no functions $u$ satisfying the condition $u \mid _ \Gamma = \phi$. The classical solution of the Dirichlet problem for the Laplace equation with such boundary function $\phi$ cannot have a finite Dirichlet integral and is not a generalized solution from the space $W _ {2} ^ {1} ( \Omega )$.

How to Cite This Entry:
Dirichlet integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_integral&oldid=46718
This article was adapted from an original article by A.K. Gushchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article