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Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity
Potential Analysis ( IF 1.1 ) Pub Date : 2018-10-25 , DOI: 10.1007/s11118-018-9741-3
Niles Armstrong

We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as$$a^{ij}_{k}(x):= \left\{\begin{array}{llll} \alpha_{k} \delta^{ij}(x) &x_{n}>0 \\ \beta_{k} \delta^{ij}(x) &x_{n}<0. \end{array}\right. $$In particular, we will show that the mean value sets associated to such an operator need not be convex as αk and βk converge to 1. This example then leads to an example of nonconvex mean value sets for smooth aij(x).

中文翻译:

平均值集的属性:角度条件,爆破解和非凸性

我们研究二阶散度形式椭圆算子的均值集,其主系数定义为$$ a ^ {ij} _ {k}(x):= \ left \ {\ begin {array} {llll} \ alpha_ { k} \ delta ^ {ij}(x)&x_ {n}> 0 \\ \ beta_ {k} \ delta ^ {ij}(x)&x_ {n} <0。\ end {array} \ right。$$特别地,我们将证明相关联到这样的操作者需要的平均值集不是凸如α ķβ ķ收敛到1。本实施例然后导致的非凸平均值集的示例为平滑一个Ĵx)。
更新日期:2018-10-25
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