Power concavity under structural assumptions on fractional Poisson forcing

Determine which structural assumptions on a spatially dependent forcing function ensure power concavity of the solution to the fractional Poisson problem \((-)^s u=f\) in a convex domain \(\Omega\), with \(u=0\) in \(\mathbb R^n\setminus\Omega\).

Background

The paper explains that its explicit ellipsoidal construction depends essentially on quadratic forcing and that a general exact power-concavity theory for fractional Poisson equations with spatially dependent forcing is not currently available. It notes that only limited results are known, including concavity of the restricted half-Laplacian torsion function in bounded convex planar domains and perturbative interior concavity estimates for fractional p-Laplacian eigenfunctions near the local limit.

The unresolved problem is to identify assumptions on the forcing function that guarantee power concavity for the nonlocal Dirichlet problem in convex domains, thereby extending the quadratic model treated in the paper to more general data.

References

To our knowledge, a general exact power-concavity theory for fractional Poisson equations with spatially dependent forcing is not available. It is therefore natural to ask which structural assumptions on f ensure power concavity of the solution of

(-\Delta)su=f \quad\text{in }\Omega, \qquad u=0 \quad\text{in }\mathbb Rn\setminus\Omega,

when \Omega is convex.

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