A stability result for p-harmonic systems with discontinuous coefficients

Abstract

The present paper is concerned with p-harmonic systems div(⟨A(x) Du(x), Du(x)⟩ p-2/2 A(x) Du(x)) = div(√A(x) F(x)), where A(x) is a positive definite matrix whose entries have bounded mean oscillation (BMO), p is a real number greater than 1 and F ∈ > r/p-1 is a given matrix field. We find a-priori estimates for a very weak solution of class W1,r, provided r is close to 2, depending on the BMO norm of √A, and p close to r. This result is achieved using the corresponding existence and uniqueness result for linear systems with BMO coefficients [St], combined with nonlinear commutators.