Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point

Theorem
Let $y, F$ be real functions.

Let $ y$ be continuously differentiable for $x \in \left[{{ a}\,.\,.\,{c}} \right) \cap \left({{ c}\,.\,.\,{ b}} \right]$ and satisfy


 * $ y \left({ a} \right)=A,\quad y \left({ b} \right)=B$

Let $J[ y]$ be a functional of the form


 * $ \displaystyle J[ y]= \int_a^b F \left({ x, y, y'} \right) \mathrm d x$

Then the functional $J$ has a weak extremum if $ y$ satisfies the following system of equations:

$\begin{cases} & \displaystyle F_y- \frac{ \mathrm d }{ \mathrm d x }F_{ y'}=0 \\ & \displaystyle F_{ y'} \big \rvert_{x=c-0} =  F_{ y'} \big \rvert_{ x=c+0} \\ & \displaystyle \left({ F - y' F_{ y'} } \right) \big \rvert_{ x= c-0} =  \left({ F - y' F_{ y'} } \right) \big \rvert_{ x= c+0} \end{cases}$

where, by the use of limit from the left and from the right, the following abbreviations are denoted as follows:


 * $\displaystyle y \left({ x} \right) \rvert_{ x= c+0  }= \lim_{ x \to c^+} y\left({ x } \right),\quad y \left({ x} \right) \rvert_{ x \to  x= c-0  }= \lim_{ x \to c^-} y\left({ x} \right)$

The last two equations are known as the Weierstrass-Erdmann corner conditions.

Proof
Rewrite $J[ y]$ as a sum of two functionals:

Recall that end points $x = a, x = b$ are fixed.

The function $y(x)$ has to be $C^0$ at $x=c$, but otherwise this point can move freely.

From general variation of functional, and noting that $y=y\left({ x}\right)$ is an extremal, write down variations for $J_1[ y]$ and $J_2[ y]$ separately:

$\displaystyle \delta J_1 = F_{ y'}  \rvert_{ x \to c-0 } \delta y_1+ \left[   F- y' F_{y'} \right] \big \rvert_{ x \to c-0 } \delta x_1$

$\displaystyle \delta J_2 = - F_{ y'} \rvert_{ x \to c+0} \delta y_1- \left[ F- y' F_{y'} \right] \big \rvert_{ x \to c+0} \delta x_1$

Note that $ \delta J_1$ and $ \delta J_2$ involve the same increments $ \delta x_1$ and $ \delta y_1$.

Since $y = y \left({x}\right)$ is an extremum of $J$, we have:

Since $ \delta x_1$ and $ \delta y_1$ are arbitrary, both collections of terms have to vanish independently.