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)