Geometric Properties of Level Sets for Domains under Geometric Normal Property

Abstract: This work is devoted to the study of the geometric properties of level sets for solutions of elliptic boundary value problems in domains satisfying the geometric normal property with respect to a convex set $C$ ($C$-GNP class). We prove that, for the classical Dirichlet problem as well as for the coupled system $\mathcal{B}(f,g)$ (related to the biharmonic plate equation), the level sets inherit the $C$-GNP structure. We establish their star-shapedness property, exact formulas for their mean curvature, and characterize their asymptotic behavior near singular contact points (cusps). We also study the stability of these level sets under the Hausdorff convergence of domains, establishing their convergence in the Hausdorff sense, in the compact sense, and in $L^1$. The analysis relies on adapted coarea formulas, leading to Faber-Krahn, Szegö-Weinberger, and Payne-Rayner type isoperimetric inequalities. In order to go beyond the purely qualitative framework of the $C$-GNP class, we introduce and analyze two new quantitative geometric measures: the thickness function $τ_Ω$ and the convexity gap $γ(Ω)$. We rigorously study their behavior, regularity, and continuity under Hausdorff convergence. These theoretical tools open up new perspectives for shape optimization under geometric constraints, the study of free boundary problems, and the geometric control of latent spaces in machine learning.

• The paper demonstrates that level sets in C-GNP domains inherit a structured radial parametrization and maintain star-shapedness, facilitating variational analysis.
• The research introduces a novel thickness function and convexity gap to quantitatively measure domain regularity, isoperimetric properties, and deviations from convexity.
• The study establishes stability and convergence of level set properties under Hausdorff convergence, underpinning applications in shape optimization and free boundary problems.

Geometric Properties of Level Sets under the Geometric Normal Property

Context and Motivation

The paper addresses the geometric analysis of level sets arising from solutions to elliptic boundary value problems in domains $\Omega$ that satisfy the Geometric Normal Property with respect to a convex set $C$ ($C$-GNP). This property ensures domains possess a structured radial parametrization centered on $C$, which is crucial for regularity, compactness, and analytical control in free boundary problems, overdetermined boundary value problems, and variational applications such as shape optimization.

A major focus is on whether the inheritance of geometric regularity (star-shapedness, radial structure, regularity of boundaries) by level sets carries over from $\Omega$ to sublevel domains $\Omega^t = \{x\in\Omega : u_\Omega(x) > t\}$ for solution $u_\Omega$ to $-\Delta u = f$ in $\Omega$. An analogous extension is considered for the coupled biharmonic system $\mathcal{B}(f,g)$, i.e., for biharmonic plate equations relevant in both classical PDE and applied contexts.

Main Results: Inheritance and Structure of Level Sets

The paper rigorously demonstrates that the level sets of solutions in $C$0-GNP domains preserve the $C$1-GNP structure. For both the classical Dirichlet problem and the biharmonic coupled system, the level sets:

• Remain in the $C$2-GNP class;
• Admit a well-defined, Lipschitz thickness function $C$3 along the convex core;
• Are star-shaped with respect to $C$4;
• Possess explicit descriptions for their radial parametrizations and thickness evolution.

Star-shapedness is shown via the strong maximum principle and Hopf lemma, ensuring that segments from any point in $C$5 to the boundary of the level set remain inside the set, a property critical for variational comparison and symmetrization techniques.

A key equation derived is the ODE for the thickness function along normal rays: $C$6 where $C$7 locates the intersection of the level surface with the normal ray at $C$8; this differential relationship encodes the geometry and PDE solution structure into the evolution of the level set.

An explicit consequence is the monotonicity of the thickness with respect to the parameter $C$9, which is instrumental in the analysis of solution nesting and comparison.

Quantitative Measures: Thickness and Convexity Gap

The paper introduces and analyzes two nontrivial geometric measures, which move analysis from a qualitative to a quantitative setting:

• The thickness function $C$0 at the boundary (distance from $C$1 to $C$2 along the inward normal);
• The convexity gap $C$3, quantifying the deviation from convexity away from $C$4.

Both measures are proven to be stable under Hausdorff convergence, locally regular (Lipschitz), and suitable for finer analysis of domains, with tight control near cusp/contact points. These properties make them suitable as constraints or objectives in shape optimization problems.

Asymptotic and Curvature Analysis

The manuscript supplies explicit asymptotic expansions at singularities, such as cusp (contact) points on $C$5. There, the detachment of level lines is shown to be quadratic in the tangential direction, and the level set boundary is tangent to $C$6 at these points. Curvature formulas for level sets (and their biharmonic analogues) are established, giving precise mean curvature in terms of the gradient and Laplacian of the solution, and, for the biharmonic case, incorporating the value of the coupled field as well.

Stability and Convergence

A sequence of technical results establishes that if a sequence of $C$7-GNP domains converges in the Hausdorff sense, then:

• The associated solutions converge in $C$8 and locally uniformly;
• The level sets converge in Hausdorff, compact, and $C$9 sense;
• The thickness functions and convexity gaps converge pointwise and in $C$0.

This stability is essential for both variational analysis and numerical approximation schemes.

Isoperimetric and Variational Implications

Via adapted coarea formulas and symmetrization, sharp isoperimetric-type inequalities of Faber-Krahn, Szegő-Weinberger, and Payne-Rayner type are proven for level sets of both the Laplacian and biharmonic problems:

• The radial (constant thickness) domains minimize Dirichlet eigenvalues and integrals among all $C$1-GNP domains of a given volume;
• The upper bounds for integrals of $C$2 and $C$3 are given explicitly in terms of geometric integrals over the thickness.

These inequalities underline the optimality and rigidity provided by the $C$4-GNP structure, and enable precise variational statements about shape derivatives.

Applications and Potential Extensions

Shape Optimization

The regularity and compactness properties of the $C$5-GNP class, combined with the lower semicontinuity and convergence of the new geometric measures $C$6 and $C$7, guarantee the well-posedness of optimization problems involving geometric constraints---important for structural engineering, PDE-constrained optimization, and control.

Machine Learning

The paper speculates on applications in machine learning, where the geometry of latent spaces realized by deep models may be governed or regularized through $C$8-GNP-type properties, and where the introduced measures can control representational depth and model complexity.

Free Boundary and Higher Order Problems

The precise description of level set geometry opens perspectives for new analysis of free boundary regularity, extension to non-convex $C$9, study for higher-order or nonlinear PDEs ($\Omega$0-Laplacian), and evolution under geometric flows.

Conclusion

This work rigorously advances the geometric understanding of level sets for PDEs in $\Omega$1-GNP domains, establishing detailed inheritance of structure, quantitative geometric control, isoperimetric inequalities, and convergence properties under domain perturbation. The mathematical framework developed enables precise application to shape optimization, offers insight into high-dimensional geometric control (potentially relevant for modern AI), and suggests future studies in free boundary regularity and geometric flows. The introduction of the thickness and convexity gap as quantitative metrics bridges geometric analysis and applications in both pure and applied mathematics.

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Reference: "Geometric Properties of Level Sets for Domains under Geometric Normal Property" (2603.30026)