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 $a^{ ij}(x)$.
Categories: math.AP