Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Second-order asymptotics of fractional Gagliardo seminorms as $s\to1^-$ and convergence of the associated gradient flows

Andrea Kubin, Valerio Pagliari, Antonio Tribuzio

Abstract

We study the second-order asymptotic expansion of the $s$-fractional Gagliardo seminorm as $s\to1^-$ in terms of a higher order nonlocal functional. We prove a Mosco-convergence result for the energy functionals and that the $L^2$-gradient flows of the energies converge to the $L^2$-gradient flows of the Mosco-limit.

Second-order asymptotics of fractional Gagliardo seminorms as $s\to1^-$ and convergence of the associated gradient flows

Abstract

We study the second-order asymptotic expansion of the -fractional Gagliardo seminorm as in terms of a higher order nonlocal functional. We prove a Mosco-convergence result for the energy functionals and that the -gradient flows of the energies converge to the -gradient flows of the Mosco-limit.

Paper Structure

This paper contains 16 sections, 18 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. The subcritical rate of convergence of the Gagliardo seminorms
  3. Preliminary (known) results and first-order limits
  4. The main limit result
  5. The candidate Mosco-limit
  6. Characterization of the domain via Fourier transform
  7. Second-order $\Gamma$-convergence of Gagliardo seminorms
  8. Coercivity and lower semicontinuity of $\mathscr{G}_s^1$
  9. Proof of the strong limsup inequality
  10. Proof of the compactness
  11. Proof of the weak liminf inequality
  12. Stability of $\mathscr{G}_s^1$ gradient flows under $\Gamma$-convergence
  13. First variation of $\mathscr{G}_s^1$ and $\mathscr{G}^1$
  14. Fourier symbols of $\mathfrak{L}$
  15. Abstract stability results for gradient flows.
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\{s_n\}_{n \in \mathbb{N}}$ be a sequence such that $s_n \to 0^+$ as $n \to + \infty$. Then, the following hold.

Theorems & Definitions (33)

  • Theorem 2.1: Theorem 1.2 in crismale2023variational
  • Theorem 2.2: Theorem 2.1 in crismale2023variational
  • Theorem 2.3: Theorem 1.4 in crismale2023variational
  • Remark 1: alternative form of $\mathscr{G}^1$
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3.1
  • Proposition 1
  • ...and 23 more