Nonlocal self-improving properties
type: Item
instance of: scholarly article
title: Nonlocal self-improving properties
zbMATH Open document ID: 1317.35284
author: Tuomo Kuusi
author: Yannick Sire
author: Giuseppe Mingione
published in: Analysis \& PDE
publication date: 2015-05-12T00:00:00Z
review text: In this extremely interesting paper, the authors prove that solutions of nonlocal equations with measurable coefficients are higher differentiable. More precisely, consider nonlocal integrodifferential equations with measurable coefficients of the form \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} (u(x)-u(y)) (\eta(x)-\eta(y)) K(x,y)\,dx\,dy= \int_{\mathbb{R}^n} f\eta\,dx\quad \text{ for all } \eta \in C_{\text{c}}^\infty(\mathbb{R}^n), \] where the kernel \(K(\,\cdot\,,\cdot\,)\) is a measurable function and satisfies \[ \dfrac{1}{\Lambda |x-y|^{n+2\alpha}}\leq K(x,y)\leq \dfrac{\Lambda }{|x-y|^{n+2\alpha}} \] with \(\alpha\in(0,1),\) \(\Lambda>1,\) and \(f\in L^q_{\text{loc}}(\mathbb{R}^n)\) for some \(q>2n/(n+2\alpha)\). The main result of the paper states that there exists a positive, universal exponent \(\delta=\delta(n,\alpha,\Lambda,q)\) such that each weak solution \(u\) of the above nonlocal equation possesses the self-improving property \[ u\in W^{\alpha,2}(\mathbb{R}^n) \Rightarrow u\in W^{\alpha+\delta,2+\delta}_{\text{loc}}(\mathbb{R}^n). \] It is to be noted that this differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear, divergence-form equations with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result of the paper is achieved by proving a fractional version of the Gehring lemma that involves certain families of lifted reverse Hölder-type inequalities in \(\mathbb{R}^{2n}\) and which is implied by delicate covering and exit-time arguments. In turn, such reverse Hölder inequalities are based on the concept of dual pairs, that is, pairs \((\mu,U)\) of measures and functions in \(\mathbb{R}^{2n}\) which are canonically associated to solutions. The authors allow also for more general equations that involve as a source term an integrodifferential operator with kernel which does not necessarily have to be of order \(\alph…
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 6435282
zbMATH Keywords: elliptic equations
zbMATH Keywords: fractional differentiability
zbMATH Keywords: nonlocal operators
zbMATH Keywords: self-improving property
zbMATH Keywords: reverse Hölder inequalities
zbMATH Keywords: fractional Gehring lemma
MaRDI profile type: Publication
OpenAlex ID: W2016171153
Wikidata QID: Q58880708
DOI: 10.2140/APDE.2015.8.57
description: scientific article