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We study a class of non-homogeneous quasilinear elliptic equations with measure data to obtain an optimal regularity estimate. We prove that the gradient of a weak solution to the problem is as integrable as the first order maximal function of the associated measure in the Orlicz spaces up to a correct power. [ABSTRACT FROM AUTHOR]

In this paper we obtain the following local Calderón–Zygmund estimates B ( | f | ) ∈ L l o c q ( Ω ) ⇒ B ( | ∇ u | ) ∈ L l o c q ( Ω ) for any q ≥ 1 of weak solutions for a class of quasilinear elliptic equations div ( a ( | ∇ u | ) ∇ u ) = div ( a ( | f | ) f ) in Ω , where B ( t ) = ∫ 0 t τ a ( τ ) d τ for t ≥ 0 . [ABSTRACT FROM AUTHOR]

Abstract: In this paper we obtain a new global gradient estimates in weighted Lorentz spaces for weak solutions of -Laplacian type equation with small BMO coefficients in a δ-Reifenberg flat domain. The modified Vitali covering lemma, the maximal function technique and the appropriate localization method are the main analytical tools. Our results improve the known results for such equations. [Copyright &y& Elsevier]

In this paper we mainly prove the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the general nonlinear elliptic equations in Musielak-Orlicz spaces. Moreover, we also obtain the equivalence of entropy solutions and renormalized solutions in the present conditions. [ABSTRACT FROM AUTHOR]

In this paper we use the Hardy–Littlewoodmaximal functions to obtain the following global BMO estimates f ∈ B M O (R n) ⇒ ∇ u ∈ B M O (R n) for the weak solutions of a class of quasilinear elliptic equations div a ∇ u ∇ u = div f in R n , where B (t) = ∫ 0 t τ a (τ) d τ for t ≥ 0. Meanwhile, we use the iteration-covering procedure to prove that B f ∈ L q (R n) ⇒ B ∇ u ∈ L q (R n) for any q > 1 for the weak solutions of div a ∇ u ∇ u = div a f f in R n. Moreover, we remark that a (t) = t p − 2 (p -Laplace equation) and a (t) = t p − 2 log (1 + t) satisfy the given conditions in this work. [ABSTRACT FROM AUTHOR]