Dirichlet integral

A functional connected with the solution of the Dirichlet problem for the Laplace equation by the variational method. Let be a bounded domain in with boundary of class , let and let the function (cf. Sobolev space). The Dirichlet integral for the function is the expression For a certain given function on one considers the set of functions from which satisfy the boundary condition . If the set is non-empty, there exists a unique function for which and this function is harmonic in . The converse theorem is also true: If a harmonic function belongs to the set , then is attained on it. Thus, is a generalized solution from of the Dirichlet problem for the Laplace equation. However, not for every function it is possible to find a function . There exists even continuous functions on for which the set is empty, i.e. the space contains no functions satisfying the condition . The classical solution of the Dirichlet problem for the Laplace equation with such boundary function cannot have a finite Dirichlet integral and is not a generalized solution from the space .