In this paper, we consider minimization formulas which arise typically in optimal control and weak KAM theory for Hamilton-Jacobi equations. Given a minimization formula, we define a uniqueness set for the formula, which replaces the original region of minimization without changing its values. Our goal is to provide a necessary and sufficient condition that a given set be a uniqueness set. We also provide a characterization of the existence of a minimal uniqueness set with respect to set inclusion.
Yasuhiro Fujita. Hitosh Ishii. "Uniqueness sets for minimization formulas." Differential Integral Equations 25 (5/6) 579 - 588, May/June 2012. https://doi.org/10.57262/die/1356012681