Sufficient Conditions for Weak Extremum

Theorem
Let $ J $ be a functional such that:


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

Let $ y = y \left ( { x } \right ) $ 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 = y \left ( { x } \right ) $.

Proof
By the continuity of function $ P \left ( { x } \right ) $ and the solution of Jacobi's equation:


 * $ \displaystyle \exists \epsilon > 0 : \left ( { \forall x \in \left [ { a \,.\,.\, b + \epsilon } \right ] : P \left ( { x } \right ) > 0 } \right ) \land \left ( { \tilde a \notin \left [ { a \,.\,.\, b + \epsilon } \right ]  } \right )   $

Consider the quadratic functional


 * $ \displaystyle \int_a^b \left ( { Ph'^2 + Qh^2 } \right ) \mathrm d x - \alpha^2 \int_a^b h'^2 \mathrm d x$

together with Euler's equation


 * $ \displaystyle - \frac{ \mathrm d }{ \mathrm d x } \left [ { (P - \alpha^2 )h' } \right ] + 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 \left [ { a \,.\,.\, b + \epsilon } \right ] : P \left ( { x } \right ) > 0 $

$ P \left ( { x } \right ) $ has a positive lower bound in $ \left [ { a \,.\,.\, b + \epsilon } \right ] $.

Consider the solution with $ h \left ( { a } \right ) = 0 $, $ h' \left ( { 0 } \right ) = 1$.

Then


 * $ \displaystyle \exists \alpha \in \R : \forall x \in \left [ { a \,. \,. \, b } \right ] : P \left ( { x } \right ) - \alpha^2 > 0 $

Also


 * $ \forall x \in \left ( { a \,. \,. \, b } \right ] \quad h \left ( { x } \right ) \ne 0 $

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


 * $ \displaystyle \int_a^b \left ( { \left ( { P - \alpha^2 } \right ) h'^2 + Qh^2  } \right ) \mathrm d x > 0 $

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


 * $ \exists c > 0 : \displaystyle \int_a^b \left ( { Ph'^2 + Qh^2 } \right ) \mathrm d x > c \int_a^b h'^2 \mathrm d x \quad \left ( { \star } \right ) $

Let $ y = y \left ( { x } \right ) $ be an extremal.

Let $ y = y \left ( { x } \right ) + h \left ( { x } \right ) $ be a curve, sufficiently close to $ y = y \left ( { x } \right ) $.

By expansion of $ \Delta J \left [ { y; h } \right ] $ from lemma of Legendre's Condition:


 * $ \displaystyle J \left [ { y + h } \right ] - J \left [ { y } \right ] = \int_a^b \left ( { Ph'^2 + Qh^2 } \right ) \mathrm d x+\int_a^b \left ( {  \xi h'^2 + \eta h^2 } \right ) \mathrm d x $

where


 * $ \displaystyle \forall x \in \left [ { a \,. \,. \, b } \right ] \quad \lim_{ \left \vert h \right \vert_1 \to 0 } \left \{ { \xi, \eta } \right \} = \left \{ { 0, 0 } \right \} $

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 ) \mathrm d x >0 $

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


 * $ \displaystyle \int_a^b \left ( { \xi h'^2 + \eta h^2 } \right ) \mathrm d 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.