Consider the following minimization problem. Given the Lagrange function which is sufficiently regular (e.g. ∈ C2) and the function , find function such that
Definition 1. A function is called a candidate for a solution of (A.1) if it is defined on Ω, obeys the boundary condition, and is piece-wise differentiable.
A necessary condition for a candidate , to be a solution of (A.1) is that the following system of differential equations applies:
This fundamental theorem was proven by Leonhard Euler in 1744. With this proof, he created ...