Gradient Lq theory for a class of non-diagonal nonlinear elliptic systems. (June 2018)

• Record Type: Journal Article
• Authors: Bulíček, Miroslav; Kalousek, Martin; Kaplický, Petr; Mácha, Václav
• Is Part Of: Nonlinear analysis. Volume 171(2018)
• Page Start: 156
• Page End: 169
• Publication Date: 2018-06
• Subjects: 35B65 -- 35J60; Regularity -- Gradient estimates -- Non-diagonal elliptic systems -- Non-Uhlenbeck systems -- Splitting condition; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.02.004
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a class of nonlinear non-diagonal elliptic systems with p -growth and establish the L q -integrability for all q ∈ [ p, p + 2 ] of any weak solution provided the corresponding right hand side belongs to the corresponding Lebesgue space and the involved elliptic operator asymptotically satisfies the p -uniform ellipticity, the so-called splitting condition and it is continuous with respect to the spatial variable. For operators satisfying the uniform p -ellipticity condition the higher integrability is known for q ∈ [ p, d p ∕ ( d − 2 ) ] and for operators having the so-called Uhlenbeck structure, the theory is valid for all q ∈ [ p, ∞ ) . The key novelty of the paper is twofold. First, the statement uses only the information coming from the asymptotic operator and second, and more importantly, by using the splitting condition, we are able to extend the range of possible q 's significantly whenever p < d − 2 .