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Optimality and stability of the radial shapes for the Sobolev trace constant

Simone Cito

TL;DR

The paper addresses optimality and stability of Sobolev trace constants for the p-Laplacian on bounded domains, proving that the ball maximizes the trace constant \\sigma_{p,q}(\\Omega) among convex domains with fixed perimeter and providing a quantitative deficit tied to the Hausdorff asymmetry. It develops a rearrangement-based, web-function method to obtain sharp isoperimetric-type bounds and extends these ideas to domains with holes, proving that the spherical shell maximizes the exterior trace constant \\tilde{\\sigma}_{p,q} under outer-perimeter and volume constraints, with a stability estimate governed by a novel hybrid asymmetry that couples outer boundary and hole geometry. The analysis relies on variational characterizations, the theory of quermassintegrals, quantitative Aleksandrov-Fenchel inequalities, and careful treatment of inner parallel sets and web-functions. The results offer quantitative shape-optimization insights for nonlinear Sobolev trace inequalities and have implications for related Robin–Neumann spectral problems with negative boundary parameters.

Abstract

In this work we establish the optimality and the stability of the ball for the Sobolev trace operator $W^{1,p}(Ω)\hookrightarrow L^q(\partialΩ)$ among convex sets of prescribed perimeter for any $1< p <+\infty$ and $1\le q\le p$. More precisely, we prove that the trace constant $σ_{p,q}$ is maximal for the ball and the deficit is estimated from below by the Hausdorff asymmetry. With similar arguments, we prove the optimality and the stability of the spherical shell for the Sobolev exterior trace operator $W^{1,p}(Ω_0\setminus\overlineΘ)\hookrightarrow L^q(\partialΩ_0)$ among open sets obtained removing from a convex set $Ω_0$ a suitably smooth open hole $Θ\subset\subsetΩ_0$, with $Ω_0\setminus\overlineΘ$ satisfying a volume and an outer perimeter constraint.

Optimality and stability of the radial shapes for the Sobolev trace constant

TL;DR

The paper addresses optimality and stability of Sobolev trace constants for the p-Laplacian on bounded domains, proving that the ball maximizes the trace constant \\sigma_{p,q}(\\Omega) among convex domains with fixed perimeter and providing a quantitative deficit tied to the Hausdorff asymmetry. It develops a rearrangement-based, web-function method to obtain sharp isoperimetric-type bounds and extends these ideas to domains with holes, proving that the spherical shell maximizes the exterior trace constant \\tilde{\\sigma}_{p,q} under outer-perimeter and volume constraints, with a stability estimate governed by a novel hybrid asymmetry that couples outer boundary and hole geometry. The analysis relies on variational characterizations, the theory of quermassintegrals, quantitative Aleksandrov-Fenchel inequalities, and careful treatment of inner parallel sets and web-functions. The results offer quantitative shape-optimization insights for nonlinear Sobolev trace inequalities and have implications for related Robin–Neumann spectral problems with negative boundary parameters.

Abstract

In this work we establish the optimality and the stability of the ball for the Sobolev trace operator among convex sets of prescribed perimeter for any and . More precisely, we prove that the trace constant is maximal for the ball and the deficit is estimated from below by the Hausdorff asymmetry. With similar arguments, we prove the optimality and the stability of the spherical shell for the Sobolev exterior trace operator among open sets obtained removing from a convex set a suitably smooth open hole , with satisfying a volume and an outer perimeter constraint.
Paper Structure (14 sections, 24 theorems, 156 equations, 3 figures)

This paper contains 14 sections, 24 theorems, 156 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminary results
  3. Notations and basic tools of measure theory
  4. The nonlinear problem: properties of the solutions
  5. Nearly sperical and nearly annular sets; Hausdorff asymmetry indices
  6. Some useful tools involving quermassintegrals
  7. Some results on inner parallel sets and web-functions
  8. The hybrid asymmetry for domains with holes
  9. Proof of the Main Results: convex domains
  10. Proof of the main results: domains with holes
  11. Reduction to nearly annular sets
  12. Further remarks and open problems
  13. Stability of the spherical shell for the Robin-Neumann problems with negative boundary parameter
  14. Open problems

Key Result

Theorem 2.2

Let $\Omega\subset\mathbb{R}^d$ be a measurable set, $f:\Omega\to\mathbb{R}$ be a Lipschitz function and let $u:\Omega\to\mathbb{R}$ be a measurable function. Then,

Figures (3)

  • Figure 1: $\mathcal{A}_\mathcal{H}(\Omega_0)$ is drawn in orange; the green set is the integration domain in the definition of $\tilde{\mathcal{A}}(\Theta;\Omega_0)$.
  • Figure 2: The darker green set is the actual integration domain in the definition of $\tilde{\mathcal{A}}(\Theta;\Omega_0)$.
  • Figure 3: The blue set on the left is clearly more symmetric than the blue set on the right. The hybrid asymmetry recognizes their difference; nevertheless their canonical Hausdorff and Fraenkel distances from the gray spherical shell are the same.

Theorems & Definitions (49)

  • Definition 2.1
  • Theorem 2.2: Coarea formula
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5: DoTo, Theorem 2
  • Remark 2.6: further radial cases
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • Remark 2.9: Immediate geometric upper bounds
  • ...and 39 more