Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations

Abstract

We prove partial regularity for the vector-valued differential forms solving the system δ(A(x, ω))=0, dω=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) ≈ (1+ + |ω|2)(p - 2)/2 ω (p≥2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, ω)=Dωf(x, ω) of a real-valued function f, is analogous to but stronger than quasiconvexity for f. The case 1