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Density estimates for a nonlocal variational model with a degenerate double-well potential via the Sobolev inequality

Serena Dipierro, Alberto Farina, Giovanni Giacomin, Enrico Valdinoci

TL;DR

The paper proves density estimates for level sets of minimizers of a nonlocal energy with a degenerate double-well potential, formulated as $\mathcal{E}_s^p(u,\Omega)$ with a fractional $p$-Laplacian interaction and $s\in(0,1/p)$. A barrier-based approach, combined with the fractional Sobolev inequality, yields sharp lower bounds on the measure of transition layers, accommodating the degeneracy $m\ge p$ of the potential and extending to quasilinear nonlocal equations, including the classical case $p=2$. Consequences include uniform convergence of interfaces in rescaled problems and Hausdorff convergence of limiting interfaces, supported by a compactness framework and Hölder regularity results. The findings clarify how degenerate double-well potentials influence phase density and provide robust tools for nonlocal phase-transition analysis.

Abstract

We provide density estimates for level sets of minimizers of the energy $\frac{1}{2} \int_Ω\int_Ω \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_Ω\int_{\mathbb{R}^n\setminusΩ} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_ΩW(u(x))dx$ where $p\in(1,+\infty)$ and $s\in\left(0,\frac{1}{p}\right)$ and $W$ is a double-well potential with polynomial growth $m\in [p,+\infty)$ from the minima. These kinds of potentials are ''degenerate'', since they detach ''slowly'' from the minima, therefore they provide additional difficulties if one wishes to determine the relative sizes of the ''layers'' and the ''pure phases''. To overcome these challenges, we introduce new barriers allowing us to rely on the fractional Sobolev inequality and on a suitable iteration method. The proofs presented here are robust enough to consider the case of quasilinear nonlocal equations driven by the fractional $p$-Laplacian, but our results are new even for the case $p=2$.

Density estimates for a nonlocal variational model with a degenerate double-well potential via the Sobolev inequality

TL;DR

The paper proves density estimates for level sets of minimizers of a nonlocal energy with a degenerate double-well potential, formulated as with a fractional -Laplacian interaction and . A barrier-based approach, combined with the fractional Sobolev inequality, yields sharp lower bounds on the measure of transition layers, accommodating the degeneracy of the potential and extending to quasilinear nonlocal equations, including the classical case . Consequences include uniform convergence of interfaces in rescaled problems and Hausdorff convergence of limiting interfaces, supported by a compactness framework and Hölder regularity results. The findings clarify how degenerate double-well potentials influence phase density and provide robust tools for nonlocal phase-transition analysis.

Abstract

We provide density estimates for level sets of minimizers of the energy where and and is a double-well potential with polynomial growth from the minima. These kinds of potentials are ''degenerate'', since they detach ''slowly'' from the minima, therefore they provide additional difficulties if one wishes to determine the relative sizes of the ''layers'' and the ''pure phases''. To overcome these challenges, we introduce new barriers allowing us to rely on the fractional Sobolev inequality and on a suitable iteration method. The proofs presented here are robust enough to consider the case of quasilinear nonlocal equations driven by the fractional -Laplacian, but our results are new even for the case .

Paper Structure

This paper contains 11 sections, 16 theorems, 343 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Density estimates
  3. Degenerate double-well potentials
  4. Mathematical framework and main results
  5. Some auxiliary results
  6. A general estimate
  7. Hölder regularity for minimizers
  8. Construction of a suitable barrier
  9. Proofs of Theorem \ref{['th:fracp>=2']} and Corollary \ref{['coro-09']}
  10. Proofs of Theorem \ref{['gliutgbedgty']} and Corollary \ref{['rgeb']}
  11. Some technical results towards the proof of Theorem \ref{['lem:1']}

Key Result

Theorem 1.3

Let $p\in (1,+\infty)$, $s\in \left(0,\frac{1}{p}\right)$ and $\theta_1$, $\theta_2\in (-1,\theta]$. Let also Assume that $u$ is a minimizer for $\mathcal{E}_{s}^p$ in $\Omega$ and that, for some $c_0$, $r_0\in (0,+\infty)$ such that $B_{r_0}\subset \Omega$, Then, there exist $R^*:=R_{s,n,p,m,\theta_*,r_0}^*\in [r_0,+\infty)$, $\tilde{c}:=\tilde{c}_{s,n,p,m,\Lambda,c_1,\theta_*,r_0,c_0}\in (0,1)$

Figures (1)

  • Figure 1: Plots of $\chi_{[-1/2,1/2]}(x)$, $(1-x^2)^2$ and $(1-x^2)^8$ respectively in green, blue and red.

Theorems & Definitions (34)

  • Definition 1.1: Minimizer for $\mathcal{E}_s^p$ and $\mathcal{F}_{s,\epsilon}^p$
  • Remark 1.2
  • Theorem 1.4: DFVERPP
  • Corollary 1.5
  • Theorem 1.6: Compactness
  • Corollary 1.7: Hausdorff convergence
  • Lemma 2.1: Lemma 3.2. in 3
  • Theorem 2.2: Theorem 6.4 and Proposition 7.5 in cozzi2017regularity
  • Theorem 3.1
  • Proposition 3.2
  • ...and 24 more