Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients

TL;DR

This paper establishes gradient estimates for elliptic systems with coefficients that are more regular than VMO but less than Dini continuous, expanding understanding of regularity under weaker conditions.

Contribution

It introduces a new condition on the mean oscillation of coefficients that guarantees BMO and VMO regularity of solutions' gradients, extending previous results.

Findings

Gradient of solutions is in BMO under the new oscillation condition.

If the oscillation measure tends to zero, the gradient is in VMO.

Examples show limits of regularity results under weaker coefficient conditions.

Abstract

We consider gradient estimates for $H^1$ 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 $L^2$ mean oscillation $\omega_{A,2}$ of $A$ satisfies \[ X_{A,2} := \limsup_{r\rightarrow 0} r \int_r^2 \frac{\omega_{A,2}(t)}{t^2} \exp\Big(C_* \int_{t}^R \frac{\omega_{A,2}(s)}{s}\,ds\Big)\,dt < \infty, \] where $C_*$ is a positive constant depending only on the dimensions and the ellipticity, then $\nabla u \in BMO$. (b) If $X_{A,2} = 0$, then $\nabla u \in VMO$. (c) If $A \in VMO$ and if $\nabla u \in L^\infty$, then $\nabla u \in VMO$. (d) Finally, examples satisfying $X_{A,2} = 0$ are given showing that it is not possible to prove the boundedness of $\nabla u$ in statement (b), nor the continuity of $\nabla u$ in statement (c).