“…In particular, Minty's lemma [2,6,7,10,14,15,18] has been shown to be an important tool in the variational field including variational inequality problems, obstacle problems, confined plasmas, free-boundary problems, elasticity problems and stochastic optimal control problems when the operator is monotone and the domain is convex. The classical Minty's inequality and Minty's lemma offered the regularity results of the solution for a generalized nonhomogeneous boundary value problem [2] and, when the operator is a gradient, also a minimum principle for convex optimization problems [6].…”