A continuous interval and a finite set of discrete points, joined into a single domain. The energy functional has three components: a nonlocal term on the continuous part, a nonlocal term on the discrete nodes, and an interface term coupling the two. The variational problem minimizes this combined energy.
The interface term is the key. Without it, the continuous and discrete problems decouple — two independent variational problems that happen to share a label. With it, the two phases are coupled, and the coupling is coercive: it generates a hybrid norm that controls both the continuous and discrete components simultaneously. The coupled system has a unique weak solution characterized by a hybrid Euler-Lagrange system — a nonlocal integral equation on the interval joined to a nonlocal algebraic system on the nodes.
The hybrid domain is not an approximation. It's not a discretization of a continuous problem or a continuum limit of a discrete one. It's a domain with both continuous and discrete parts, and the equation that lives on it respects both structures. The continuous part has integral operators; the discrete part has matrix operators; and the interface enforces compatibility without forcing either part to pretend it's the other.
The construction models any system where a continuum interacts with finitely many discrete agents at specific points — neurons synapsing onto a continuous fiber, sensors along a beam, data points coupled to a continuous field. The mathematics treats the hybrid nature as fundamental rather than as a defect to be resolved by choosing one scale or the other.