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
du(t)/dt=Au(t).
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.
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