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On the uniqueness of the critical point of $ψ_Ω$

Junyuan Liu, Shuangjie Peng, Fulin Zhong

Abstract

We prove that for any bounded convex domain $Ω\subset \mathbb{R}^n$, the function \begin{equation*} ψ_Ω(ξ) = \int_{\mathbb{R}^n\setminusΩ} \frac{\mathrm{d}x}{|x-ξ|^{2n}}, \quad ξ\inΩ, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Saldaña in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write $ψ_Ω$ as an integral of the distance function $ρ(ξ,ω)$. This approach is not limited to $ψ_Ω$. Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts.

On the uniqueness of the critical point of $ψ_Ω$

Abstract

We prove that for any bounded convex domain , the function \begin{equation*} ψ_Ω(ξ) = \int_{\mathbb{R}^n\setminusΩ} \frac{\mathrm{d}x}{|x-ξ|^{2n}}, \quad ξ\inΩ, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Saldaña in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write as an integral of the distance function . This approach is not limited to . Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts.

Paper Structure

This paper contains 8 sections, 22 theorems, 148 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Illustrative examples and the role of convexity
  3. Notation and preliminaries
  4. Strict convexity and uniqueness
  5. Convexity of $F$
  6. Uniqueness of the critical point
  7. Further properties of $\rho$
  8. The computation of $A_k$

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^n$ be a bounded convex domain (no further regularity is required). Then the function has a unique critical point in $\Omega$. This critical point is the global minimizer of $\psi_\Omega$. Moreover, if in addition $\Omega$ is symmetric with respect to some point $p\in\Omega$ (for example, $\Omega$ is a ball or an ellipsoid centered at $p$), then $p$ is the unique cri

Figures (7)

  • Figure 1: A non-convex domain may exhibit multiple critical points.
  • Figure 2: Two convex domains with symmetry.
  • Figure 3: Two examples of non-convex domains.
  • Figure 4: The disk $\Omega_{\text{disk}}$ and the ellipse $\Omega_{\text{ellipse}}$ are convex domains.
  • Figure 5: Left: a single annular domain with a critical sphere. Right: multiple concentric annular domains, each with a critical sphere.
  • ...and 2 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 39 more