The Hodge Decomposition Theorem for General Three-Dimensional Vector Fields, without Cuts

Preprint. The Hodge decomposition of the space $L^2(\Omega)^3$ of square sommable vector fields defined on a domain $\Omega$ of the ordinary three–dimensional space is a tool used in a wide theoretic and applied literature concerning Electromagnetism and Fluid dynamics. In order to obtain an effective Hodge decomposition, suitable also for numerical computations, the classical strategy, which goes back to Helmholtz and which is used nowadays as well, is to consider only bounded domains with sufficiently regular boundary, which contain connected and pairwise disjoint surfaces, called cut surfaces of $\Omega$, whose complement $C$ in $\Omega$ is simply connected. The advantage of this situation is that the union of such cut surfaces does not have singularities, every curl–free vector field in $\Omega$ admits potential in $C$ and hence it is possible to apply standard variational methods to obtain the Hodge decomposition via scalar potential formulations. Recently, R. Benedetti, R. Frigerio and the author gave an exhaustive topological description of bounded domains, called Helmholtz domains, which admit such cut surfaces. We proved that the Helmholtz domains are truly special three–dimensional domains. In fact, their topology is elementary. An evidence of this claim is given, for example, by the fact that the complement of any non-trivial thickened knot in a box domain is not a Helmholtz domain. In this way, we realize that the range of application of the classical strategy is quite limited. In this paper, we announce several results concerning an effective Hodge decomposition of $L^2(\Omega)^3$ valid for every bounded domain $\Omega$ with Lipschitz–regular boundary, where $\Omega$ describes a spatial medium formed by a possible inhomogeneous and/or anisotropic material.