Generality of the potential-theoretic reduction to ODE method

Prove that the Potential Theoretic Reduction to ODE method applies to singularities and blow-downs in previously inaccessible partial differential equations of different or mixed type and in integro-differential equations, extends to non-axially-symmetric settings, and can establish flatness-implies-regularity results and selected De Giorgi conjectures.

Background

The paper develops the Potential Theoretic Reduction to ODE approach for three-dimensional axially symmetric steady incompressible cavity flows. The method combines a Green-type representation, integral estimates, a perturbed integral identity, and asymptotic analysis of an associated ordinary differential equation. Beyond the cavity application, the authors explicitly conjecture that the method has substantially broader scope, including non-axially-symmetric free boundary problems, equations of different or mixed type, integro-differential equations, regularity consequences, and selected De Giorgi conjectures. These extensions are not established in the paper.

References

We conjecture that the method ---apart from being more precise than for example an analysis by frequency formulas--- is so general and robust that it is applicable to the analysis of various (so far inaccessible) singularities and blow-downs in PDE of different or even mixed type and in integro-differential equations, that it can be extended to non-axially-symmetric settings, and that it may even be used to show flatness-implies-regularity and certain De Giorgi conjectures.

— Complete Rigidity at infinity and Existence of the Levinson Cavity  (2608.18913 - Li et al., 19 Aug 2026) in Section 1, subsection “The Potential Theoretic Reduction to ODE approach”