Sufficient Conditions for Weak Extremum

Theorem
Let $J$ be a functional such that:


 * $\displaystyle J\sqbrk y=\int_a^b \map F {x,y,y'}\rd x,\quad \map y a=A,\quad \map y b=B$

Let $y=\map y x$ be an extremum.

Let the strengthened Legendre's condition hold.

Let the strengthened Jacobi's necessary condition hold.

Then the functional $J$ has a weak minimum for $y=\map y x$.

Proof
By the continuity of function $\map P x$ and the solution of Jacobi's equation:


 * $\displaystyle\exists\epsilon>0:\paren {\forall x\in\closedint a {b+\epsilon}:\map P x>0}\land \paren { \tilde a \notin \closedint a {b+\epsilon} }$

Consider the quadratic functional


 * $\displaystyle\int_a^b\paren{Ph'^2+Qh^2}\rd x-\alpha^2\int_a^b h'^2\rd x$

together with Euler's equation


 * $-\dfrac \rd {\rd x} \sqbrk {\paren {P-\alpha^2}h'}+Qh=0$

The Euler's equation is continuous $\alpha$.

Thus the solution of the Euler's equation is continuous $\alpha $.

Since


 * $\displaystyle\forall x\in\closedint a {b+\epsilon}:\map P x>0$

$\map P x$ has a positive lower bound in $\closedint a {b+\epsilon}$.

Consider the solution with $\map h a=0$, $\map {h'} 0= 1$.

Then


 * $\displaystyle\exists\alpha\in\R:\forall x\in\sqbrk a b:\map P x-\alpha^2>0$

Also


 * $\forall x\in\hointl a b\quad\map h x\ne 0$

By Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite:


 * $\displaystyle\int_a^b\paren{\paren {P-\alpha^2}h'^2+Qh^2}\rd x>0$

In other words, if $c=\alpha^2$, then:


 * $\exists c>0:\displaystyle\int_a^b\paren{Ph'^2+Qh^2}\rd x>c\int_a^b h'^2\rd x\quad\paren {\star}$

Let $y=\map y x$ be an extremal.

Let $y=\map y x+\map h x$ be a curve, sufficiently close to $y=\map y x$.

By expansion of $\Delta J\sqbrk{y;h}$ from lemma of Legendre's Condition:


 * $\displaystyle J\sqbrk{y+h}-J\sqbrk y=\int_a^b\paren{Ph'^2+Qh^2}\rd x+\int_a^b\paren{\xi h'^2+\eta h^2}\rd x$

where


 * $\displaystyle\forall x\in\closedint a b\quad\lim_{\size {h}_1\to 0}\lbrace{\xi,\eta}\rbrace=\lbrace{0,0}\rbrace$

and the limit is uniform.

By Schwarz inequality:

Notice, that the integral on the right does not depend on $ x $.

Integrate the inequality $ x $:

Let $ \epsilon > 0 $ be a constant such that:


 * $ \displaystyle \left \vert \xi \right \vert \le \epsilon$, $ \left \vert \eta \right \vert \le \epsilon $

Then:

Thus, by $ \left ( { \star } \right ) $


 * $ \displaystyle \int_a^b \left ( { Ph'^2 +Qh^2 } \right ) \rd x >0 $

while by $ \left ( { \ast } \right ) $


 * $ \displaystyle \int_a^b \left ( { \xi h'^2 + \eta h^2 } \right ) \rd x $

can be made arbitrarily small.

Thus, for all sufficiently small $ \left \vert h \right \vert_1 $, which implies sufficiently small $ \left \vert \xi \right \vert $ and $ \left \vert \eta \right \vert$, and, consequently, sufficiently small $ \epsilon $:

Therefore, in some small neighbourhood $ y = y \left ( { x } \right ) $ is a weak minimum of the functional.