Unique solvability for nonlinear eigenvalue local minimization problems

Determine whether the local minimization problem in the energy-minimizing domain decomposition algorithm is uniquely solvable for nonlinear eigenvalue problems, beyond the linear and unconstrained cases.

Background

The paper proves that fixed points of the energy-minimizing domain decomposition (EMDD) algorithm are critical points of the underlying energy under several assumptions, including unique solvability of each local minimization problem. The authors explain that this assumption is clear for unconstrained problems and, up to sign, for linear eigenvalue problems on a unit sphere, but its validity for nonlinear eigenvalue problems is not established. Numerical experiments did not reveal non-uniqueness, but this does not constitute a proof.