Regularity Results for Vector Fields of Bounded Distortion and Applications

In this paper we prove higher integrability results for vector fields B,E, (B,E) ∈ L2− (Ω, R) × L2−ε(Ω, R), ε small, such that div B = 0, curl E = 0 satisfying a “reverse” inequality of the type |B| + |E| ≤ ( K + 1 K ) 〈B,E〉+ |F | with K ≥ 1 and F ∈ L(Ω, R), r > 2 − ε. Applications to the theory of quasiconformal mappings and partial differential equations are given. In particular, we prove regularity results for very weak solutions of equations of the type div a(x,∇u) = div F. If |a(x, z)| + |z| ≤ (K + 1/K) 〈a(x, z), z〉, in the homogeneous case, our method provides a new proof of the regularity result u ∈ W 1,2−ε loc (Ω)⇒ u ∈ W 1,2+ε loc (Ω) where ε is sufficiently small. A result of higher integrability for functions verifying a reverse integral inequality is used, and its optimality is proved.