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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.
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
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
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