Local Lipschitz continuity for p, q−PDEs with explicit u−dependence. (January 2023)

• Record Type: Journal Article
• Authors: Marcellini, Paolo
• Is Part Of: Nonlinear analysis. Volume 226(2023)
• Publication Date: 2023-01
• Subjects: Elliptic equations -- Calculus of Variations -- Regularity of solutions -- Local Lipschitz continuity -- p, q−growth conditions -- General growth conditions; 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.2022.113066
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This article is dedicated to Emmanuele Di Benedetto, great mathematician, colleague, friend. In the spirit to treat a subject that in the last years attracted the interest of several mathematicians, and the attention of Emmanuele too, in this paper we give a first approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type ∑ i = 1 n ∂ ∂ x i a i x, u, D u = b x, u, D u under p, q − growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient D u and on the x variable.