Hausdorff compactness and regularity for classes of open sets under geometric constraints

Abstract: This article introduces innovative classes of open sets in (\mathbb{R}^{N}), where (N=2, 3), characterized by a geometric property associated with the inward normal. The focus lies on proving compactness results for the Hausdorff topology within these classes. Furthermore, the paper establishes the equivalence of convergences, encompassing Hausdorff, compact, and characteristic functions, for select classes. We also investigate the regularity of the thickness function associated with these domains and analyze how the regularity of the fixed convex set (C) influences the boundary regularity of the admissible shapes.

• The paper establishes Hausdorff compactness for new classes of open sets defined by a geometric normal property relative to a fixed convex set.
• It rigorously demonstrates that boundary regularity is transferred through thickness functions and bi-Lipschitz mappings.
• The analysis provides existence results for shape optimization, extending to non-connected and locally constrained domains.

Hausdorff Compactness and Regularity for Classes of Open Sets Under Geometric Constraints

Introduction and Motivation

This work addresses fundamental issues of compactness and regularity for admissible sets in shape optimization, focusing especially on classes of open subsets of Euclidean space $(\mathbb{R}^N,~N=2,3)$ characterized by geometric normal properties with respect to a fixed compact convex set $C$. The primary objective is to construct and analyze new classes of admissible domains where the inward normal directions of the boundary satisfy intersection criteria with $C$—the so-called $C$-GNP (Geometric Normal Property). Establishing compactness in the Hausdorff topology for these classes is crucial for demonstrating existence results in shape optimization and PDE-constrained geometric problems.

The novelty lies in the generalized handling of connected and non-connected domains, including rigorous analysis of boundary regularity as influenced by both the base convex set $C$ and auxiliary control functions such as a thickness map.

Definition and Structure of Admissible Classes

The paper formalizes several classes of open sets with progressively more general geometric constraints:

• $C$-GNP Class: Domains $\Omega \subset D$ (a reference large ball) satisfy: $C \subset \Omega$, local Lipschitz regularity of $\partial\Omega$ outside $C$, connected intersection of outward normals from $C$0 with $C$1, and the fundamental condition that any inward normal ray from $C$2 meets $C$3.
• Non-connected Classes: The sets $C$4 and $C$5 generalize admissibility to domains comprising unions of non-overlapping open sets, each respecting a $C$6-GNP (for disjoint convex reference sets $C$7, $C$8), with a uniform separation constraint (distance or projected distance $C$9).
• Local Property Classes: The sets $C$0 and $C$1 formalize local geometric admissibility wherein boundary components are confined to neighborhoods of possibly several compact balls and satisfy either $C$2-GNP or analogous normal cone criteria only locally.

These generalizations allow the admissible classes to encompass physically relevant configurations, including multiply connected domains or those with regulatory “thickness” characteristics.

Compactness in the Hausdorff Topology

A central technical achievement is the establishment of Hausdorff compactness for all presented classes:

• Any sequence of sets in $C$3 (and analogously for multi-component and local classes) admits a subsequence converging in the Hausdorff sense to a limit set that remains within the class.
• The proof leverages the normal cone and projection structures, the local compactness of the ambient ball, and geometric stability lemmas regarding intersection and distance constraints under Hausdorff convergence.
• For non-connected sets, compactness is retained provided the minimal distance or projected distance constraints are enforced, and exemplified sequences demonstrate failure of compactness if this is relaxed.

Moreover, the equivalence of Hausdorff, compact, and $C$4 (characteristic function) convergence is rigorously demonstrated for these classes, under specified regularity and negligible boundary measure hypotheses.

Regularity of the Boundary and Thickness Function

A notable contribution of this work is the detailed analysis of how boundary regularity of admissible sets is governed by:

• The geometric regularity (smoothness, Lipschitz continuity, $C$5 properties) of the reference convex set $C$6,
• The regularity of the associated thickness function $C$7, which records the shift along outward normals from $C$8 to points of $C$9.

Explicit regularity transfer theorems are proven:

• If $C$0 is $C$1 and $C$2 is $C$3, then local portions of $C$4 inherit regularity $C$5.
• Conversely, if $C$6 is regular, so is the corresponding $C$7.
• Under smallness or regularity of $C$8 and moderate curvature of $C$9, the mapping from $C$0 to $C$1 is bi-Lipschitz, and otherwise this fails in sharp examples.

Additionally, the invariance of the $C$2-GNP property and boundary regularity under bi-Lipschitz mappings (preserving convexity of $C$3) is established, providing robustness for subsequent geometric analysis.

Extremal Problems and Applications

The analysis extends to non-classical variational problems, including perimeter minimization under $C$4-GNP constraints, characterizing extremal shapes and their boundary representations. Notably, specific geometric solutions (e.g., union of disks with tangent constraints) are identified as saturating the geometric normal constraint.

The practical motivations and applications explicitly discussed include:

• Thermal insulation design: Optimizing an insulating layer around a core $C$5 while ensuring compactness and regularity of the admissible class for existence.
• Modeling droplets with rigid inclusions: Perimeter minimization under $C$6-GNP reflects realistic surface-tension constraints.
• Acoustic cavity design: Leveraging sets with only local normal constraints for more flexible, possibly nonconvex, soundproof shapes.
• Optimal design with disjoint components: Maintaining physical separations for layouts involving heat sources/sinks.

These examples demonstrate the direct relevance of the new classes and compactness/regularity theory to diverse engineering, geometric, and analytic problems.

Theoretical Implications and Future Directions

The formalization and rigorous compactness results for non-standard, possibly nonconnected, or locally constrained classes of open sets set the stage for robust existence results in PDE-constrained shape optimization well beyond traditional smooth or connected settings. The controlled trade-off between boundary regularity of $C$7 and admissible sets via the thickness function offers a precise analytic handle on regularity questions.

Potential future directions include further generalization to higher dimensions, incorporation of more complex local geometric or functional inequalities, and systematic study of minimizers for non-convex and non-smooth objective functions within these classes. The framework also opens doors to numerically tractable shape optimization algorithms given its explicit geometric construction and compactness guarantees.

Conclusion

This work provides a comprehensive and mathematically rigorous foundation for classes of open sets constrained by geometric normal properties and associated compactness and regularity analysis in Hausdorff topology. By introducing both global and local admissibility classes, establishing equivalence of natural modes of convergence, and analyzing regularity trade-offs, the paper significantly generalizes previously considered admissible sets for shape optimization. These tools are highly relevant for applications in engineering design, material science, and analytical PDEs, and suggest concrete lines for further exploration within geometric analysis and optimization theory.

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Citation: "Hausdorff compactness and regularity for classes of open sets under geometric constraints" (2604.01813)