QUANTITATIVE PROPAGATION OF SMALLNESS FOR SOLUTIONS OF ELLIPTIC EQUATIONS
QUANTITATIVE PROPAGATION OF SMALLNESS FOR SOLUTIONS OF ELLIPTIC EQUATIONS
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椭圆方程解的小性的定量传播
DOI：
10.1142/9789813272880_0143
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
E. Malinnikova
中科院分区：
文献类型：
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作者：
A. Logunov;E. Malinnikova
设u是椭圆方程div（A\nabla u）=0的解，Lipschitz系数在R ^n中.
假设$|u| $在球$B=\{|X|\leq 1\}$.我们证明，如果$|u| 0，\gamma \in（0，1）$不依赖于u$，而只依赖于A$和E$的测度。
我们以Remez型不等式的形式指定了对E的测度的依赖性。对于Hausdorff维数大于$n-1$的集合E$，类似的估计也成立.
对于梯度的解决方案，我们表明，一个类似的传播小持有的Hausdorff维数大于$n-1-c$，其中$c>0$是一个小的数值常数，仅取决于尺寸。
Let $u$ be a solution to an elliptic equation $\text{div}(A\nabla u)=0$ with Lipschitz coefficients in $\mathbb{R}^n$.
Assume $|u|$ is bounded by $1$ in the ball $B=\{|x|\leq 1\}$. We show that if $|u| 0, \gamma \in (0,1)$ do not depend on $u$ and depend only on $A$ and the measure of $E$.
We specify the dependence on the measure of $E$ in the form of the Remez type inequality. Similar estimate holds for sets $E$ with Hausdorff dimension bigger than $n-1$.
For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than $n-1-c$, where $c>0$ is a small numerical constant depending on the dimension only.