Lorentz improving estimates for the p-Laplace equations with mixed data. (November 2020)

• Record Type: Journal Article
• Authors: Nguyen, Thanh-Nhan; Tran, Minh-Phuong
• Is Part Of: Nonlinear analysis. Volume 200(2020)
• Publication Date: 2020-11
• Subjects: p-Laplace -- Mixed data -- Dirichlet boundary data -- Regularity -- Fractional maximal functions -- Lorentz spaces; 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.2020.111960
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet p -Laplace equation of type div ( | ∇ u | p − 2 ∇ u ) = f + div ( | F | p − 2 F ) in Ω, u = g on ∂ Ω, in Lorentz space, with given data F ∈ L p ( Ω ; R n ), f ∈ L p p − 1 ( Ω ), g ∈ W 1, p ( Ω ) for 1 < p < ∞ and Ω ⊂ R n ( n ≥ 2 ) satisfying a Reifenberg flat domain condition or a p -capacity uniform thickness condition, which are considered in several recent papers. To better specify our result, the proofs of regularity estimates involve fractional maximal operators and valid for a more general class of quasilinear nonhomogeneous elliptic equations with mixed data. This paper not only deals with the Lorentz estimates for a class of more general problems with mixed data but also improves the good- λ approach technique proposed in our preceding works (Tran, 2019; Tran and Nguyen, 2019; Tran and Nguyen, 2020; Tran and Nguyen (in press)), to achieve the global Lorentz regularity estimates for gradient of weak solutions in terms of fractional maximal operators.