Existence of scalar minimizers for simple convex integrals with autonomous Lagrangian measurable on the state variable

The minimizer y(⋅) constructed here is bi-monotone, i.e. it increases or decreases outside of a subinterval where it is constant, with y′∉{0}∪(α(y),β(y)) a.e.. (Here (α(s),0),(0,β(s)) are intervals, with one extremity at zero, where L(s,⋅) is affine.)