Mathematics for Economists

Calculus of variations

Theorem

If the kernel function F



t,x,x



is convex in



x,x



then every solution of

the EulerñLagrange equation is a global minimum.

Letxbe a solution of the equation. LetΨ(y)be the value of the

functional aty.For everyythe variation

ψ(λ) = Ψ(x+λ(yx))=Ψ(( 1 λ)x+λy)=

=

Zb

a

L



t,( 1 λ)x+λy,( 1 λ)x



+λy





 ( 1 λ)

Zb

a

L



t,x,x



+λ

Zb

a

L



t,y,y



=

= ( 1 λ)ψ( 0 )+λψ( 1 ).