Abstract
In this paper we study subquadratic elliptic systems in divergence form with VMO leading coefficients in \begin{document}$ \mathbb{R}^{n} $\end{document} . We establish pointwise estimates for gradients of local weak solutions to the system by involving the sharp maximal operator. As a consequence, the nonlinear Calderon-Zygmund gradient estimates for \begin{document}$ L^{q} $\end{document} and BMO norms are derived.
Highlights
The objective of this paper is to study pointwise estimates for gradients of local weak solutions to the following subquadratic elliptic systems in divergence form div a(x, ∇u) = div F(x) in Rn (1)with a discontinuous nonlinearity a and n ≥ 3
U : Rn → RN (N ≥ 1) is a vector-valued unknown function, ∇u : Rn → RN×n denotes its gradient, and div stands for the RN -valued divergence operator
Inspired by the works [10] and [17], Breit, et al [2] developed the regularity of the solutions for the p -Laplacian system, and established interior pointwise estimates for the gradients of local weak solutions by using the sharp maximal operators
Summary
We assume that b(x) = {bkij (x)} is measurable, uniformly bounded, and satisfies the strong ellipticity condition, i.e. there exist universal constants 0 < ν ≤ 1 ≤ L such that for almost all x ∈ Rn and every z ∈ RN×n, ν |z|2 ≤ b(x) z, z ≤ L |z|2. Note that a standard example of such a nonlinearity a(x, z) satisfying this condition is the p -Laplacian, if b(x) are unitary matrices. In this case, the system (1) can be converted to the p -Laplace elliptic system div |∇u|p−2∇u = div F(x). Subquadratic elliptic system, gradient estimate, sharp maximal operator, nonlinear Calderon-Zygmund estimate.
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