Geometric Harnack inequalities for level lines

Prove geometric Harnack inequalities for level lines arising from solutions of the elliptic problems considered on domains in O_C, quantifying how geometric features propagate across level sets.

Background

Beyond classical analytic Harnack inequalities, the paper’s geometric viewpoint suggests potential geometric analogues tailored to level sets. Such inequalities would control geometric variation or comparability across different parts of a level set.

The authors explicitly raise the possibility of establishing geometric Harnack-type inequalities in this setting.