BMO estimates for the [formula omitted]-Laplacian

Abstract

We prove BMO estimates of the inhomogeneous p -Laplace system given by − div ( | ∇ u | p − 2 ∇ u ) = div f . We show that f ∈ BMO implies | ∇ u | p − 2 ∇ u ∈ BMO , which is the limiting case of the nonlinear Calderón–Zygmund theory. This extends the work of DiBenedetto and Manfredi (1993) [2], which was restricted to the super-quadratic case p ≥ 2 , to the full case 1 < p < ∞ and even more general growth. Moreover, we prove that A ( ∇ u ) inherits the Campanato and VMO regularity of f .