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arXiv · 1801.06627

Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity

Niles Armstrong
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Abstract

We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as $$a^{ij}_k(x):= \begin{cases} \alpha_k \delta^{ij}(x) &x_n>0 \beta_k \delta^{ij}(x) &x_n<0. \end{cases}$$ In particular, we will show that the mean value sets associated to such an operator need not be convex as $\alpha_k$ and $\beta_k$ converge to 1. This example then leads to an example of nonconvex mean value sets for smooth $a^{ij}(x)$.

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Niles Armstrong. 2018-01-20. Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity. https://arxiv.org/abs/1801.06627

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