Viscosity 1-eigenpairs via a 1-limit eigen-equation

Establish the existence (or non-existence) of viscosity 1-eigenpairs defined by a 1-limit eigenvalue equation that is sharper than the generalized subgradient (Clarke) formulation, and, if they exist, characterize such solutions.

Background

For p→∞, limits of p-eigenpairs lead to viscosity ∞-eigenpairs satisfying a specific limit equation, and these are known to be generalized eigenpairs. An analogous viscosity theory for p→1 has not been established beyond the generalized (subgradient-based) definition of 1-eigenpairs.

A refined 1-limit eigen-equation would parallel the ∞-case and potentially yield new structural results for 1-Laplacian eigenfunctions.

References

The existence or not existence of viscosity 1-eigenpairs defined via some $1$-limit eigenequation sharper than the generalized eigenequation given in terms of subgradients (see \Cref{Def:generalized_p-eigenpair}), is currently an open problem.

— Nonlinear spectral graph theory (2504.03566 - Deidda et al., 4 Apr 2025) in Subsubsection 3.3.1, “Viscosity p-eigenpairs”