Extend level-set results to higher-order and nonlinear operators

Establish whether analogues of the geometric and stability results for level sets proved in the paper extend to higher-order or nonlinear operators, including the p-Laplacian and related operators.

Background

The paper develops a comprehensive geometric theory for level sets associated with linear second-order problems and a coupled biharmonic system, proving star-shapedness, curvature identities, asymptotics near contact points, and stability under Hausdorff convergence.

The authors explicitly raise the question of whether these methods and conclusions carry over to more general operators, notably the p-Laplacian and other higher-order PDEs.

References

Open questions remain:

  • Do versions of these results exist for higher-order operators ($p$-Laplacian, etc.)?
— Geometric Properties of Level Sets for Domains under Geometric Normal Property  (2603.30026 - Barkatou, 31 Mar 2026) in Conclusion

The same framework should apply to master equations with fractional $p$-Laplacian diffusion and cooperative systems. We leave these extensions to future work and focus here on establishing the core phenomenon: radial symmetry persists even under local damping at the origin.

— Radial Symmetry and Strict Radial Decrease for Master Equations with Decreasing Radial Potentials  (2609.20476 - Niu, 17 Sep 2026) in Section 1, subsection “Broader outlook”

Whether a sharper two-point modulus of Andrews--Clutterbuck type holds for p\neq2 remains unclear, since the nonlinear p-flux introduces gradient-dependent degenerate or singular coefficients, and it is not evident how to identify a compatible sharp one-dimensional comparison inequality.

— Fundamental Gaps for the Dirichlet \(p\)-Laplacian with Convex Potentials: Sharp One-Dimensional Bounds and a Higher-Dimensional Dichotomy  (2608.13443 - Chen et al., 13 Aug 2026) in Section 1, immediately following Theorem 1 (Introduction and Main Results)

A natural extension is to study $r$-homogeneous invariance objectives and their associated weighted $r$-Laplacians, which formally lead to generalized-Gaussian targets with heavier-than-Gaussian tails. Establishing the corresponding link between nonlinear spectral structure and linear recovery remains an interesting direction for future work.

— Beyond Gaussian Worlds: Latent Geometry Matters for JEPAs  (2609.21656 - Nicollier et al., 18 Sep 2026) in Section 7, Conclusion

This recovery mechanism also suggests that the limiting solution property assumption in the critical point non-degeneracy estimate of might be removable, provided a similar degeneracy set comparison principle and slice uniqueness property can be obtained in their more general setting.

— Degeneracy Set Comparison Principle and Free Boundary Estimates for Hénon-type Infinity Laplace equations  (2609.02612 - Chen et al., 2 Sep 2026) in Section 5, subsection “The homogeneous case,” immediately before Proposition 5.1