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Continuous symmetrizations and uniqueness of solutions to nonlocal equations

Matias G. Delgadino, M. Vaughan

TL;DR

This work extends symmetry and uniqueness results to nonlocal equations governed by fractional seminorm energies. By establishing that continuous Steiner symmetrization strictly decreases the fractional energy for non-radial data, the authors prove radial symmetry and monotone decay of critical points; they further show strict convexity of the energy along a height-function interpolation, enabling uniqueness within the class of radially symmetric decreasing profiles. The results apply to fractional thin film dynamics and related higher-order nonlocal models, with a rigorous framework that combines ε-regularization, explicit good-function calculus, and truncated symmetrizations to preserve support. Collectively, the findings provide a robust variational route to symmetry and uniqueness for a broad family of nonlocal gradient flows and stationary states, including explicit profiles in the compact-support regime and applications to long-time dynamics.

Abstract

We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and decreasing. Moreover, we show uniqueness of solutions by exploiting the convexity of the energies under a tailored interpolation in the space of radially symmetric and decreasing functions. As an application, we consider the long time dynamics of a higher order nonlocal equation which models the growth of symmetric cracks in an elastic medium.

Continuous symmetrizations and uniqueness of solutions to nonlocal equations

TL;DR

This work extends symmetry and uniqueness results to nonlocal equations governed by fractional seminorm energies. By establishing that continuous Steiner symmetrization strictly decreases the fractional energy for non-radial data, the authors prove radial symmetry and monotone decay of critical points; they further show strict convexity of the energy along a height-function interpolation, enabling uniqueness within the class of radially symmetric decreasing profiles. The results apply to fractional thin film dynamics and related higher-order nonlocal models, with a rigorous framework that combines ε-regularization, explicit good-function calculus, and truncated symmetrizations to preserve support. Collectively, the findings provide a robust variational route to symmetry and uniqueness for a broad family of nonlocal gradient flows and stationary states, including explicit profiles in the compact-support regime and applications to long-time dynamics.

Abstract

We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and decreasing. Moreover, we show uniqueness of solutions by exploiting the convexity of the energies under a tailored interpolation in the space of radially symmetric and decreasing functions. As an application, we consider the long time dynamics of a higher order nonlocal equation which models the growth of symmetric cracks in an elastic medium.
Paper Structure (15 sections, 27 theorems, 213 equations, 5 figures)

This paper contains 15 sections, 27 theorems, 213 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Symmetric decreasing rearrangements
  3. Uniqueness
  4. Application to fractional thin film equations
  5. Organization of the paper
  6. Continuous Steiner symmetrization
  7. On Theorem \ref{['thm:main2']}
  8. Explicit representations for good functions
  9. Good functions and local energies
  10. Explicit derivative computation for good functions
  11. Proof of Proposition \ref{['lem:deriv for good']}
  12. Interpolation between symmetric decreasing functions
  13. On Theorem \ref{['thm:uniqueness']}
  14. Truncated symmetrizations
  15. Proof of Theorem \ref{['thm:uniqueness']}

Key Result

Theorem 1.1

Let $0 < s < 1$ and $1 < p < \infty$. For any positive $f\in L^1(\mathbb{R}^n)\cap C(\mathbb{R}^n)$ that is not radially decreasing about any center, there are constants $\gamma = \gamma(n,s,p,f)>0$, $\tau_0 = \tau_0(f)>0$, and a hyperplane $H$ such that where $f^{\tau}$ is the continuous Steiner symmetrization of $f$ about $H$.

Figures (5)

  • Figure 1: The continuous Steiner symmetrization $f^\tau$ for $0 < \tau_1 < \tau_2 < \infty$ as it interpolates between the function $f$ and its Steiner symmetrization $Sf$.
  • Figure 2: Graph of $f$ and $f^\tau$ in Example \ref{['ex:zero']} with $\tau = .25$, $x=0$
  • Figure 3: Decomposition of $\operatorname{supp} f$ based on the center of mass of the level sets.
  • Figure 4: The height function $H(m)$ associated to $f(x)$
  • Figure 5: The graphs of $f$, $f^\tau$, and $\tilde{f}^\tau$ in Example \ref{['ex:zero']} at $x=0$ with $h_0 =\tau= .25$, illustrating how the level sets below the line $h=h_0$ have dropped

Theorems & Definitions (61)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Lemma 2.1 in CHVY
  • Remark 2.4
  • ...and 51 more