论文标题
以VMO系数为单位的椭圆形系统的平均振荡梯度估计值
Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients
论文作者
论文摘要
我们考虑$ h^1 $的梯度估计，以差异形式的线性椭圆系统的解决方案$ \partial_α（a_ {ij}^{αβ} \partial_βu^j）= 0 $。众所周知，系数矩阵$ a =（a_ {ij}^{αβ}）$的DINI连续性对于解决方案的不同性至关重要。我们证明了以下结果： （a）如果$ a $满足条件比DINI连续性稍弱，但比属于VMO更强，即$ l^2 $均值振荡$ω_{a，2} $ of $ a $ a $ a $ a $满足\ [x__ {x_ {a，2} \ frac {ω_{a，2}（t）} {t^2} \ exp \ big big（c_* \ int_ {t}^r \ frac {ω__________{a，2}（s）}} {s} {s} {s} {s} \，ds \ big）椭圆度，然后在bmo $中$ \ nabla u \。 （b）如果$ x_ {a，2} = 0 $，则在vmo $中$ \ nabla u \。 （c）如果在vmo $中$ a \，并且如果$ \ nabla u \ in l^\ infty $，则在vmo $中$ \ nabla u \。 （d）最后，给出了满足$ x_ {a，2} = 0 $的示例，表明语句中不可能证明$ \ nabla u $的界限，也无法证明语句中$ \ nabla u $的连续性（c）。
We consider gradient estimates for $H^1$ solutions of linear elliptic systems in divergence form $\partial_α(A_{ij}^{αβ} \partial_βu^j) = 0$. It is known that the Dini continuity of coefficient matrix $A = (A_{ij}^{αβ}) $ 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 $ω_{A,2}$ 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 $C_*$ 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).
