Search

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 consider the global gradient estimates for weak solutions of the parabolic -Laplacian type equation in a -Reifenberg flat domain. [Copyright &y& Elsevier]

Abstract: We develop a unifying method to obtain the interior and boundary estimates for the weak solution of a nonlinear elliptic partial differential equation of p-Laplacian type with BMO coefficients in a δ-Reifenberg flat domain. Our results greatly improve the known results for such equations. [Copyright &y& Elsevier]

In this paper, we investigate the optimal C 1 , α estimates for the elliptic p (⋅) -Laplace equation: div (a (x) | ∇ u | p (x) − 2 ∇ u) = div h (x) + f (x) in Ω with f ∈ L q (⋅) (Ω) and a , h ∈ C σ (Ω ¯). Based on a certain geometric oscillation estimate, the scaling arguments and appropriate localization technique as well as the careful analysis on the variable exponents, we exhibit how the optimal Hölder exponent of ∇ u is influenced by p (⋅) , q (⋅) and σ. This work can be regarded as a natural follow up to the paper by Araújo and Zhang (in press). [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]