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

The existence of a minimizer is proved here for the integral View the MathML source∫abL(x,x′)dt, among the AC functions with x(a)=Ax(a)=A and x(b)=Bx(b)=B. The Lagrangian L:R×R→[0,+∞]L:R×R→[0,+∞] may have View the MathML sourceL(⋅,ξ)non-lsc measurability sufficing for ξ≠0ξ≠0 provided e.g. L(⋅)L(⋅) is lsc at View the MathML source(s,0)∀s; while L(s,⋅)L(s,⋅) is assumed convex lsc and superlinear. Under such basic hypotheses no known weak sequential lower semicontinuity results are applicable. The minimizer y(⋅)y(⋅) constructed here is bi-monotone, i.e. it increases or decreases outside of a subinterval where it is constant, with View the MathML sourcey′∉{0}∪(α(y),β(y)) a.e.. (Here (α(s),0),(0,β(s))(α(s),0),(0,β(s)) are intervals, with one extremity at zero, where L(s,⋅)L(s,⋅) is affine.)