Boundariesconvergent

Boundariesconvergent is a term used in geometry and analysis to describe a mode of convergence for a sequence of domains based on the convergence of their boundaries. In informal terms, a sequence of domains E_n in a metric space is boundariesconvergent to a domain E when the boundaries ∂E_n approach ∂E in a precise metric sense, and the domains themselves converge to E in a compatible interior sense.

Definition: Let X be a metric space and let {E_n} and E be nonempty closed bounded subsets of X with boundaries ∂E_n and ∂E. The sequence {E_n} is boundariesconvergent to E if: (i) the boundaries converge in the Hausdorff metric: d_H(∂E_n, ∂E) → 0; and (ii) E_n converges to E in the Kuratowski sense, i.e., every x ∈ E is a limit of some sequence x_n ∈ E_n, and every limit of a sequence x_n ∈ E_n lies in E. Some authors also require that the indicator functions 1_{E_n} converge to 1_E in L^1 to reinforce interior convergence.

Examples: If E_n are closed disks with radii r_n → r, then ∂E_n converges to ∂E (a circle of radius r) in the Hausdorff metric, so E_n is boundariesconvergent to E. More complex cases include sequences of polygons or domains with boundaries that approximate a smooth curve, provided the boundary approximation error tends to zero.

Relation and applications: This notion relates to Hausdorff convergence of boundaries and to Kuratowski convergence of sets. It is distinct from mere convergence of area or perimeter, though perimeter convergence can accompany boundary convergence under regularity. In practice, boundariesconvergence appears in numerical PDEs, geometric modeling, and mesh refinement to ensure stable boundary representation and boundary conditions.

See also: Hausdorff convergence; Kuratowski convergence; boundary regularity.