Pure monotonicity of the Green-sublevel ACF functional

Establish whether the functional \(\Phi\), defined by integrating the phase energies over sublevel sets of a Neumann Green's function in the considered planar convex-domain and positively curved manifold settings, satisfies the pure monotonicity inequality \(\Phi'(r)\ge 0\), rather than only the proved almost-monotonicity estimates.

Background

The paper introduces a Green-sublevel analogue of the Alt–Caffarelli–Friedman functional and proves almost-monotonicity estimates for convex planar domains and two-dimensional complete manifolds with nonnegative Gaussian curvature. The resulting bounds include geometric error terms involving the sector parameter or total curvature parameter.

In the sharpness discussion, the authors show that the lower bound produced by the standard ACF/Friedland–Hayman reduction can be negative for suitable sector examples, although this does not itself prove that $\Phi'$ is negative. Thus, determining whether a different argument can establish genuine monotonicity of $\Phi$ remains unresolved.