Prove existence and uniqueness for the PDE-constrained inverse problem

Prove the existence and uniqueness, within perturbative accuracy, of solutions to the PDE-constrained functional optimization problem that reconstructs proton parton distribution functions from DIS world data under the DGLAP evolution constraint.

Background

The paper formulates global PDF inference as a coupled system of integral equations constrained by the DGLAP integro-differential equations. A successful reconstruction must both reproduce the experimental world data and satisfy the evolution constraint to the accuracy permitted by truncated perturbative QCD.

The authors identify a proof of existence and uniqueness for this constrained optimization problem as an important mathematical test of whether the proposed reconstruction framework can be viable in practice. They explicitly leave this proof unresolved.

References

A proof of the existence and uniqueness of such a solution---within perturbative accuracy---of the problem~eq:pde-constrained-world-data-optimization would be a promising indication for the viability of practical reconstruction of the PDFs from real world data. Such a proof is left for future work.

— Mathematical inverse problem for the world data inference of the parton distribution functions of the proton  (2609.11896 - Hänninen, 10 Sep 2026) in Section 5, “Parton distribution function reconstruction inverse problem at NLO”