Convexity of positivity sets for fractional obstacle problems

Determine which structural assumptions on a general forcing function imply convexity of the positivity set for solutions of the fractional obstacle problem \(\min\{u,(-\Delta)^s u-f\}=0\) in \(\mathbb R^n\).

Background

The paper proves ellipsoidal positivity sets for the fractional obstacle problem with positive definite quadratic forcing. It then observes that the nonlocal theory beyond this special forcing is much less complete and asks whether analogous geometric rigidity can follow from broader structural hypotheses on the forcing.

This problem seeks conditions under which the positivity set is convex for general forcing, extending the explicit ellipsoidal geometry established for quadratic data.

References

For the obstacle problem eq:obstacle, the related question is whether structural assumptions on a general forcing imply convexity of the positivity set.

eq:obstacle:

min⁡{u, (−Δ)su−f}=0in Rn.\min\{u,\,(-\Delta)^s u-f\}=0 \qquad\text{in }\mathbb R^n.

— Ellipsoidal Positivity Sets for Fractional Obstacle Problems with Quadratic Forcing  (2609.00703 - Kitano, 1 Sep 2026) in Section 6, "Beyond quadratic forcing"