# 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.

How to Cite This Entry:
Isoperimetric problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isoperimetric_problem&oldid=47444
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article