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Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms

Massimo Gobbino, Nicola Picenni

Abstract

We consider a family of non-local and non-convex functionals, and we prove that their Gamma-liminf is bounded from below by a positive multiple of the Sobolev norm or the total variation. As a by-product, we answer some open questions concerning the limiting behavior of these functionals. The proof relies on the analysis of a discretized version of these functionals.

Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms

Abstract

We consider a family of non-local and non-convex functionals, and we prove that their Gamma-liminf is bounded from below by a positive multiple of the Sobolev norm or the total variation. As a by-product, we answer some open questions concerning the limiting behavior of these functionals. The proof relies on the analysis of a discretized version of these functionals.
Paper Structure (21 sections, 4 theorems, 92 equations)

This paper contains 21 sections, 4 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Previous literature and Brezis' open problems
  3. Our contribution
  4. Overview of the technique
  5. Comparison with similar discretization arguments for non-local functionals
  6. Structure of the paper
  7. The dyadic functional
  8. Proof of Proposition \ref{['prop:dyadic_repr']}
  9. Proof of Proposition \ref{['prop:est_DF']}
  10. Properties of visible and important dyadic intervals
  11. Estimate from above for $B-A$
  12. Conclusion
  13. Proof of the main result
  14. Case $k=0$
  15. Case $k\geq 1$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\gamma>0$ and $p\geq 1$ be real numbers, let $N$ be a positive integer, and let $\Omega\subseteq \mathbb{R}^N$ be any open set. Then the families of functions defined by (defn:Fgpl) and (defn:Fp) satisfy where $C_{N,p}$ is the geometric constant defined by (defn:C_N), and More precisely, for every $u\in L^1 _{\mathrm{loc}}(\Omega)$ and every family of functions $\{u_\lambda\}\subseteq L^1_{

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2: Toward the pointwise limit
  • Remark 1.3: Toward the Gamma-limit
  • Proposition 2.1: Dyadic representation
  • Proposition 2.2: Asymptotic cost of oscillations for the dyadic functional
  • Proposition 3.1: Lower bound for the cell problem
  • proof