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

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 \[ X_{A,2} := \limsup_{r\rightarrow 0} r \int_r^2 \frac{ω_{A,2}(t)}{t^2} \exp\Big(C_* \int_{t}^R \frac{ω_{A,2}(s)}{s}\,ds\Big)\,dt < \infty, \] where $C_{\ast }$ 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).

References in corpus (3)

• Elliptic equations in divergence form with partially BMO coefficients
• On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators
• On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: Part II