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 $W _ {2} ^ {1} ( \Omega )$.

References

[1]	V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)

Comments

The restriction of a function (distribution) $u$ to a set (in this case the boundary) $\Gamma$ is also called the trace of $u$ on $\Gamma$ in this setting.

See [a1] for a well-known additional reference. Note that the Hilbert space obtained by completion of the set of all $C ^ \infty$- functions with compact support with respect to the scalar product

$$( u , v ) \mapsto \int\limits _ \Omega \sum _ { i= } 1 ^ { n } \frac{\partial u }{\partial x _ {i} } \frac{\partial v }{\partial x _ {i} }$$

can be continuously imbedded into $L ^ {2}$. This observation leads to the introduction of the axiomatic theory of Dirichlet spaces, explaining larger parts of classical potential theory (see, e.g., [a2] or [a3], and Potential theory).

See also Dirichlet principle; Dirichlet variational problem.

References

[a1]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[a2]	J. Deny, "Méthodes Hilbertiennes et théorie du potential" M. Brelot (ed.) H. Bauer (ed.) J.-M. Bony (ed.) J. Deny (ed.) G. Mokobodzki (ed.) , Potential theory (CIME, Stresa, 1969) , Cremonese (1970)
[a3]	M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980)