Mean Oscillation Gradient Estimates for Elliptic Systems in Divergence Form with VMO Coefficients

Abstract

We consider gradient estimates for H¹ solutions of linear elliptic systems in divergence form \(\partial _{\alpha }(A_{ij}^{\alpha \beta } \partial _{\beta } u^{j}) = 0\). It is known that the Dini continuity of coefficient matrix \(A = (A_{ij}^{\alpha \beta }) \) is essential for the differentiability of solutions. We prove the following results:

(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L² mean oscillation ω_A,2 of A satisfies

$ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int \limits }_{r}^{2}} \frac {\omega _{A,2}(t)}{t^{2}} \exp \left (C_{*} {{\int \limits }_{t}^{R}} \frac {\omega _{A,2}(s)}{s} ds\right ) dt < \infty , $

where C_∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.

(b) If X_A,2 = 0, then ∇u ∈ V MO.

(c) Finally, examples satisfying X_A,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when \(\nabla u \in L^{\infty } \cap VMO\).