Fundamental lemma of the calculus of variations

In the calculus of variations , the so-called plays fundamental lemma of variational calculus or Hauptlemma the calculus of variations ( English fundamental lemma of calculus of variations or Dubois-Reymond lemma ) a central role. It is sometimes also named with the fundamental theorem of the calculus of variations , but does not coincide with it. It is an important lemma which is attributed to the German mathematician Paul Dubois-Reymond .

In its simplest version, the fundamental lemma makes the following statement:

Let be a compact real interval and be a continuous function . ${\displaystyle I=[a,b]\subset \mathbb {R} }$ ${\displaystyle g\colon I\to \mathbb {R} }$

Assume that for every continuously differentiable function with : ${\displaystyle h\colon I\to \mathbb {R} }$ ${\displaystyle h(a)=h(b)=0}$

${\displaystyle \int _{a}^{b}{g(t)\;h(t)}\;\mathrm {d} t=0}$

Then is the null function . ${\displaystyle g}$

Another, but somewhat broader version of the fundamental lemma , which also includes multi-dimensional integration , is as follows:

Let be an open subset of and be a locally integrable function . ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{N}\;(N\in \mathbb {N} )}$ ${\displaystyle g\colon \Omega \to \mathbb {R} }$

The following applies to every infinitely differentiable function with a compact carrier : ${\displaystyle h\colon \Omega \to \mathbb {R} }$

${\displaystyle \int _{\Omega }{g(x)\;h(x)}\;\mathrm {d} x=0}$

Then applies almost everywhere . ${\displaystyle g=0}$

For direct application, note that a locally integrable function is given by the formula ${\displaystyle g\colon \Omega \to \mathbb {R} }$

${\displaystyle T_{g}(h):=\int _{\Omega }{g(x)\;h(x)}\;\mathrm {d} x}$

a distribution on defined. According to the above lemma, two such distributions and are equal if and only if and almost everywhere agree (for proof, consider ). ${\displaystyle T_{g}}$ ${\displaystyle \Omega }$ ${\displaystyle T_{g_{1}}}$ ${\displaystyle T_{g_{2}}}$ ${\displaystyle g_{1}}$ ${\displaystyle g_{2}}$ ${\displaystyle g_{1}-g_{2}}$

Individual evidence

1. ^ ^{A b} Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations: A textbook. 1982, p. 78 ff.
2. ↑ George Leitmann: The Calculus of Variations and Optimal Control: An Introduction. Plenum Press, New York (et al.) 1981, p. 14 ff.
3. ↑ Dubois-Reymond, Explanatory Notes on the Beginnings of the Calculus of Variations, Mathematische Annalen, Volume 15, 1879, pp. 283-314, here pp. 297, 300
4. ^ Oskar Bolza, lectures on the calculus of variations, Teubner 1909, p. 26. According to Bolza, the oldest proof comes from Friedrich Stegmann , textbook of the calculus of variations, Kassel 1854, but there are more restrictive assumptions.
5. ^ Philippe G. Ciarlet: Linear and Nonlinear Functional Analysis with Applications. 2013, p. 314.
6. ↑ The corresponding article Fundamental lemma of calculus of variations in the English language Wikipedia provides information on other versions .