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On a weighted version of the BBM formula

Giorgio Stefani

TL;DR

The paper develops a weighted extension of the BBM formula, proving pointwise and Γ-convergence for weighted nonlocal energies under mild kernel and weight convergence assumptions. It establishes a robust compactness criterion, a weighted Poincaré inequality, and spectral stability results for the first eigenvalue and eigenfunction relative to the weighted energies. A non-local analogue of the weighted BBM formula is also derived, broadening applicability to weighted and anisotropic interactions. Together, these results provide a variational toolkit for analyzing weighted, nonlocal functionals in Sobolev and BV settings with potential applications to fractional operators and spectral problems.

Abstract

We prove a weighted version of the Bourgain-Brezis-Mironescu (BBM) formula, both in the pointwise and $Γ$-convergence sense, together with a compactness criterion for energy-bounded sequences. The non-negative weights need only be $L^\infty$ convergent to a bounded and uniformly continuous limit. We apply the BBM formula to show a Poincaré-type inequality and the stability of the first eigenvalues relative to the energies. Finally, we discuss a non-local analogue of the weighted BBM formula.

On a weighted version of the BBM formula

TL;DR

The paper develops a weighted extension of the BBM formula, proving pointwise and Γ-convergence for weighted nonlocal energies under mild kernel and weight convergence assumptions. It establishes a robust compactness criterion, a weighted Poincaré inequality, and spectral stability results for the first eigenvalue and eigenfunction relative to the weighted energies. A non-local analogue of the weighted BBM formula is also derived, broadening applicability to weighted and anisotropic interactions. Together, these results provide a variational toolkit for analyzing weighted, nonlocal functionals in Sobolev and BV settings with potential applications to fractional operators and spectral problems.

Abstract

We prove a weighted version of the Bourgain-Brezis-Mironescu (BBM) formula, both in the pointwise and -convergence sense, together with a compactness criterion for energy-bounded sequences. The non-negative weights need only be convergent to a bounded and uniformly continuous limit. We apply the BBM formula to show a Poincaré-type inequality and the stability of the first eigenvalues relative to the energies. Finally, we discuss a non-local analogue of the weighted BBM formula.

Paper Structure

This paper contains 14 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Weighted functionals
  3. Local framework
  4. Compactness
  5. Poincaré inequality
  6. Spectral stability
  7. Non-local framework
  8. Proofs of the results
  9. Proof of Theorem 1.1
  10. Proof of Theorem 1.5
  11. Proof of Theorem 1.9
  12. Proof of Theorem 1.10
  13. Proof of Theorem 1.2
  14. Proof of Corollary 1.10

Key Result

Theorem 1.1

Let $p\in[1,\infty)$ and $(\rho_k)_{k\in\mathbb{N}}$ be as above. The following hold: As a consequence, as $k\to\infty$, the functionals $(\mathscr F_{k,p}^{w_k})_{k\in\mathbb{N}}$ converge to $\mathscr D_{p,w}^{\mu}$ pointwise on $\mathcal{S}^{p}(\mathbb{R}^N)$ and in the $\Gamma$-sense on $\mathcal{S}^{p}_R(\mathbb{R}^N)$ for every $R>0$.

Theorems & Definitions (27)

  • Theorem 1.1: Weighted BBM formula
  • Definition 1.2
  • Theorem 1.3: Compactness
  • Definition 1.4
  • Corollary 1.5
  • Remark 1.6: A generalization of KSS25*Th. 1.2
  • Definition 1.7
  • Theorem 1.8: Poincaré inequality
  • Theorem 1.9: Spectral stability
  • Remark 1.10: A generalization of BPS16*Th. 1.2
  • ...and 17 more