Existence of bounded-domain solutions for the nonlocal 2-Hessian equation

Establish existence of solutions to the nonlocal 2-Hessian equation K^s u=f on bounded domains with prescribed right-hand side f, for the full range of fractional orders s∈(0,1), while accounting for the role of boundary-domain admissibility.

Background

The paper proves uniform ellipticity and interior regularity for the nonlocal 2-Hessian operator Ks under a nonlocal semiconcavity condition and a strictly positive right-hand side. It discusses an all-space problem with prescribed asymptotic profile, for which existence and uniqueness are expected to follow from comparison and Perron-method arguments modeled on earlier work for the fractional Monge–Ampère operator.

The unresolved issue is the corresponding Dirichlet-type theory on bounded domains with prescribed f. The authors emphasize that this requires handling boundary geometry and admissibility, and that the difficulty persists even for the nonlocal Monge–Ampère operator.

References

Existence for problems on bounded domains with a prescribed right-hand side f, and for the full range s∈(0,1), is a nontrivial challenge (it remains largely open even for the nonlocal Monge-Ampère operator), and the geometry of the boundary is expected to play a major role, as in the local case; see for a discussion of domain admissibility.

— A geometric approach to nonlocal 2-Hessian equations  (2609.10195 - Charro et al., 9 Sep 2026) in Section 1, subsection “Further remarks and open problems” (Section 1.4)