The Laplace operator on a domain can be understood as the operator associated with the quadratic form defined by
In fact, it is possible to develop a theory for quadratic forms on Hilbert spaces, characterizing those forms having a "good" associated operator. We call this operator A.
It turns out that properties of the quadratic form are reflected from properties of the solution of the equation
Most notably, invariance of convex sets of the Hilbert space can be characterized in terms of properties of the quadratic form. To be more specific: if S is closed convex set, then there is an algebraic characterization in terms of the form of the fact that solutions that start in S also stay in S - this is known as Ouhabaz's criterion.
If the Hilbert space has a product structure and it is infinite dimensional (as C²=C x C, C³= C x C x C,... in the finite dimensional case), then it is possible to write the quadratic form as a kind of matrix of quadratic forms. More interesting: the properties of the solutions are obtained applying finite dimensional arguments to the properties of the infinite dimensional forms, and this is what we discuss in the paper.
Further readings: the preprint on arxiv, an introduction to the theory of forms, the home page of the book of Ouhabaz, my PhD thesis.