Lorentz improving estimates for the [formula omitted]-Laplace equations with mixed data

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−2F)inΩ,u=gon∂Ω, in Lorentz space, with given data F∈Lp(Ω;Rn), f∈Lpp−1(Ω), g∈W1,p(Ω) for 1<p<∞ and Ω⊂Rn (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.