Isoperimetric problem
One of the fundamental problems in the classical calculus of variations. The isoperimetric problem consists in minimizing a functional
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under constraints of the form
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and certain boundary conditions.
The isoperimetric problem reduces to the Lagrange problem when new variables are introduced satisfying the differential equations
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with boundary conditions
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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
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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 ,
such that
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is minimized, subject to
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where 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 |
Isoperimetric problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isoperimetric_problem&oldid=47444