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Results tagged with calculus-of-variations
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user 5371
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
17
votes
Accepted
Surface equivalent of catenary curve
A model equation for an inextensible, flexible, heavy surface in a gravitational field was deduced by Poisson Lagrange and later the problem was also studied by Poisson (see the references in the link …
- 22.3k
6
votes
2
answers
656
views
Minimal surface which divides a convex body into two regions of equal volume
Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume?
Background/motivation.
A 2D version of the …
- 22.3k
26
votes
Accepted
What was Weierstrass's counterexample to the Dirichlet Principle?
Weierstrass simply observed that not every problem in the calculus of variations would have a solution. He considered the example
$$D[y]=\int_{-1}^{1}x^2\left(\frac{d y}{dx}\right)^2dx\to \min,$$
wher …
- 22.3k
16
votes
Accepted
Smallest area shape that covers all unit length curve
Whereas I don't know of any recent progress in this problem, let me mention one result for
closed curves.
Theorem. A closed plane curve of length $L$ and curvature bounded by $K$ can be contained …
- 22.3k