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

TL;DR

This paper investigates the geometric properties of mean value sets for certain elliptic operators, revealing conditions under which these sets are nonconvex, including examples with smooth coefficients.

Contribution

It demonstrates that mean value sets for elliptic operators can be nonconvex even with smooth coefficients, challenging prior assumptions about their convexity.

Findings

Mean value sets need not be convex when coefficients converge to identity.

Constructs examples of nonconvex mean value sets with smooth coefficients.

Provides angle conditions and blowup solutions related to nonconvexity.

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 $a^{ij}(x)$.