# Isoperimetric problem

One of the fundamental problems in the classical calculus of variations. The isoperimetric problem consists in minimizing a functional

$$ J _ {0} ( y) = \ \int\limits _ { x _ {1} } ^ { {x _ 2 } } f _ {0} ( x, y, y ^ \prime ) dx $$

under constraints of the form

$$ J _ {i} ( y) = \ \int\limits _ { x _ {1} } ^ { {x _ 2 } } f _ {i} ( x, y, y ^ \prime ) \ dx = c _ {i} ; $$

$$ f _ {i} : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R ,\ i = 1 \dots m, $$

and certain boundary conditions.

The isoperimetric problem reduces to the Lagrange problem when new variables $ z _ {i} $ are introduced satisfying the differential equations

$$ \dot{z} _ {i} = f _ {i} ( x, y, y ^ \prime ),\ \ i = 1, \dots, m, $$

with boundary conditions

$$ z _ {i} ( x _ {1} ) = 0,\ \ z _ {i} ( x _ {2} ) = c _ {i} ,\ \ i = 1, \dots, m. $$

Necessary conditions for optimality in the isoperimetric problem have the same form as do the simplest problems in the calculus of variations related to the Lagrange function

$$ L ( x, y, y ^ \prime , \lambda _ {0}, \dots, \lambda _ {m} ) = \ \sum _ {i = 0 } ^ { m } \lambda _ {i} f _ {i} ( x, y, y ^ \prime ). $$

The name "isoperimetric problem" goes back to the following classical question: Among all the curves with given perimeter in the plane, find the one that bounds the largest area.

#### References

[1] | G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947) MR0017881 Zbl 0036.34401 |

[2] | L.Ya. Tslaf, "Calculus of variations and integral equations" , Moscow (1970) (In Russian) |

[3] | M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian) |

#### Comments

As stated above, the original isoperimetric problem is the problem of finding the geometric figure with maximal area and given perimeter. I.e., the problem is to find functions $ y _ {1} ( x) $, $ y _ {2} ( x) $ such that

$$ \int\limits _ { x _ {1} } ^ { {x _ 2 } } y _ {1} y _ {2} ^ \prime d x $$

is minimized, subject to

$$ \int\limits _ { x _ {1} } ^ { {x _ 2 } } \sqrt {( y _ {1} ^ \prime ) ^ {2} + ( y _ {2} ^ \prime ) ^ {2} } \ d x = l , $$

where $ l $ is a given constant.

#### References

[a1] | L.E. [L.E. El'sgol'ts] Elsgolc, "Calculus of variations" , Pergamon (1961) (Translated from Russian) MR0344552 MR0279361 MR0209534 MR1532560 MR0133032 MR0098996 MR0051448 Zbl 0101.32001 |

**How to Cite This Entry:**

Isoperimetric problem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Isoperimetric_problem&oldid=52024