Fundamental theorem of the calculus of variations

This article deals with the fundamental theorem of the calculus of variations, not to be confused with the fundamental lemma of the calculus of variations

The fundamental theorem of calculus of variations ( English Fundamental Theorem of the Calculus of Variations is) a basic set of mathematical sub-region of the calculus of variations and closely related to the Weierstrass set from minimum . It deals with the central question in the calculus of variations, under which conditions real-valued functionals assume a minimum .

Formulation of the fundamental theorem

The fundamental theorem of the calculus of variations can be formulated as follows:

Let be a reflexive Banach space over and be a non-empty , weakly closed and at the same time limited subset in it . ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle M\subset X}$

Continue to be a weakly sub-continuous functional. ${\displaystyle \phi \colon M\to \mathbb {R} }$

Then the functional takes on at a minimum. ${\displaystyle \phi }$ ${\displaystyle M}$

In other words:

There is an element with ${\displaystyle x_{0}\in M}$

${\displaystyle \phi (x_{0})=\inf _{x\in M}{\phi (x)}=\min _{x\in M}{\phi (x)}}$.

proof

Following the presentation by Fučík , Nečas and Souček , the proof can be given as follows:

According to Eberlein – Šmulian's theorem , the reflexivity of the Banach space implies that every bounded sequence in it has a weakly convergent subsequence . ${\displaystyle X}$

So there are under the mentioned conditions in a sequence of elements which on the one hand in the limit value ${\displaystyle M}$ ${\displaystyle x_{n}\in M\;(n\in \mathbb {N} )}$ ${\displaystyle \mathbb {R} }$

${\displaystyle \lim _{n\rightarrow \infty }\phi (x_{n})=\inf _{x\in M}{\phi (x)}}$

and which on the other hand converges weakly to an element . ${\displaystyle M}$ ${\displaystyle x_{0}\in M}$

This element is the searched minimum position for . ${\displaystyle x_{0}}$ ${\displaystyle \phi }$

Because in connection with the semi-continuity of the following chain of inequalities results : ${\displaystyle \phi }$

${\displaystyle \inf _{x\in M}{\phi (x)}\leq \phi (x_{0})\leq \liminf _{n\to \infty }{\phi (x_{n})}=\lim _{n\rightarrow \infty }\phi (x_{n})=\inf _{x\in M}{\phi (x)}}$

However, that means

${\displaystyle \phi (x_{0})=\inf _{x\in M}{\phi (x)}=\min _{x\in M}{\phi (x)}}$

and the theorem is proven.

Consequences from the fundamental theorem

Two direct conclusions can be attached to the fundamental theorem:

(I)

(a) The conditions of the fundamental theorem are fulfilled if there a non-empty, closed , bounded and convex subset of the reflexive -Banach space and the functional is continuous and convex . ${\displaystyle M}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle \phi }$

That means: in this case has a minimum position . ${\displaystyle \phi }$ ${\displaystyle x_{0}\in M}$

(b) If, in addition , it is strictly convex , the minimum point is even clearly determined. ${\displaystyle \phi }$ ${\displaystyle x_{0}\in M}$

(II)

(a) If the reflexive -Banach space is a weakly sub-continuous and at the same time coercive functional , the assertion of the fundamental theorem also applies. ${\displaystyle \phi \colon X\to \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$

That means:

It is then

${\displaystyle \inf _{x\in M}{\phi (x)}>{-\infty }}$

such as

${\displaystyle \phi (x_{0})=\inf _{x\in M}{\phi (x)}=\min _{x\in M}{\phi (x)}}$

for at least one ${\displaystyle x_{0}\in X}$

(b) In the case that is coercive, continuous and convex or strictly convex, corollary (I) is correspondingly valid. ${\displaystyle \phi \colon X\to \mathbb {R} }$

Note to prove the inferences

1. Because of the weak closure of , the functional is weakly subcontinuous if and only if for every real number the archetype of the associated interval is weakly closed. ${\displaystyle M}$ ${\displaystyle \phi }$ ${\displaystyle a}$ ${\displaystyle {\phi }^{-1}(]\infty ,a])}$ ${\displaystyle (-\infty ,a]}$
2. A continuous and convex functional on a convex subset of a Banach space is always weakly sub-continuous.

Different version of the Fundamental Theorem

A slightly different, but related version of the Fundamental Theorem is the following:

Be a non-empty Hausdorff room and be further ${\displaystyle M}$

${\displaystyle \phi \colon M\to \mathbb {R} \cup \{+\infty \}}$ a sub-continuous functional.

Furthermore there is a real number with: ${\displaystyle r}$

(i) ${\displaystyle M_{\phi \;,\;r}\colon =\{x\in M\mid \phi (x)\leq r\}\neq \emptyset }$

(ii) is consequential compact . ${\displaystyle M_{\phi \;,\;r}}$

Then:

There is an element with ${\displaystyle x_{0}\in M}$

${\displaystyle \phi (x_{0})=\inf _{x\in M}{\phi (x)}=\min _{x\in M}{\phi (x)}}$.

See also

• Bauer's maximum principle
• Minimal solution
• Theorem of minimum and maximum

literature

• Philippe Blanchard , Erwin Brüning : Direct methods of the calculus of variations . A textbook. Springer Verlag , Vienna, New York 1982, ISBN 3-211-81692-5 ( MR0687073 ).
• Philippe Blanchard, Erwin Brüning: Variational Methods in Mathematical Physics . A unified approach. Translated from the German by Gillian M. Hayes. (=  Texts and Monographs in Physics ). Publisher = Springer Verlag, Berlin 1992 ( MR1230382 ).
• Philippe G. Ciarlet : Linear and Nonlinear Functional Analysis with Applications . Society for Industrial and Applied Mathematics , Philadelphia, PA 2013, ISBN 978-1-61197-258-0 ( MR3136903 ).
• Svatopluk Fučík, Jindřich Nečas, Vladimír Souček: Introduction to the calculus of variations . Extended edition of the lecture notes Úvod do variačního počtu (=  Teubner texts on mathematics ). Teubner Verlagsgesellschaft , Leipzig 1977.

Individual evidence

1. ↑ ^{a b c d e f} Svatopluk Fučík, Jindřich Nečas, Vladimír Souček: Introduction to the calculus of variations. 1977, pp. 16-19.
2. ^ Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations: A textbook. 1982, p. 1 ff.
3. ^ Philippe Blanchard, Erwin Brüning: Variational Methods in Mathematical Physics. 1992, p. 1 ff.
4. ^ Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations: A textbook. 1982, p. 16 ff.