New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

Abstract

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions:{div(A(x,∇u))=div(|F|p−2F)inΩ,u=σon∂Ω, where Ω⊂Rn (n≥2), the nonlinearity A is a monotone Carathéodory vector valued function defined on W01,p(Ω) for p>1 and the p-capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F∈Lp(Ω;Rn), the problem is set up with general Dirichlet boundary data σ∈W1,p(Ω). In this paper, the optimal good-λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45,52,53] and references therein.