Density of smooth functions vanishing on the outflow boundary

Establish whether smooth functions that vanish on the outflow boundary are dense in the graph space associated with kinetic Fokker–Planck equations, thereby supporting uniqueness arguments for such equations.

Background

The introduction explains that uniqueness arguments for kinetic Fokker–Planck equations require the existence of traces for functions in kinetic Sobolev spaces. They also rely on the density of smooth functions satisfying a homogeneous outflow-boundary condition in the corresponding transport graph space.

The paper develops trace and density results for spherical kinetic Sobolev spaces, including density of smooth functions on bounded Lipschitz domains, but it does not establish the specifically stated density assertion for smooth functions vanishing on the outflow boundary. The problem is therefore included as an unresolved issue explicitly identified by the authors.

References

The corresponding uniqueness arguments, however, require the existence of traces for functions in kinetic Sobolev spaces and ultimately rely on the assertion that smooth functions vanishing on the outflow boundary are dense in the associated graph space, which remains an open problem; see Appendix~A.

— A trace theorem for spherical kinetic Sobolev spaces on $ρ$-convex domains  (2608.19027 - Egger et al., 19 Aug 2026) in Section 1, Introduction