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Results tagged with calculus-of-variations
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user 20302
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
1
vote
Accepted
strong form of variational problem in control theory
Note that the second integral is the integral over the triangle $\newcommand{\bR}{\mathbb{R}}$
$$ \Delta_T=\big\{ (s,t)\in\bR^2;\;\;0\leq s\leq t\leq T\big\}. $$
Fubini's theorem shows that for any i …
- 33.5k
3
votes
Reference for a dual isoperimetric problem and solution
I think that a straight variational approach has a good chance of yielding the desired conclusion. Fix two points $z_0,z_1$ on the boundary of the unit disk. Denote by $C$ the positively oriented …
- 33.5k
7
votes
Good book on Calculus of Variations
The book of Gelfand and Fomin is a good place to start. It worked for me. I would like to include another nice and short source namely Chapter 19, vol. II of Feynman's Lectures on Physics.
If …
- 33.5k
3
votes
Good reference for globally formulated calculus of variations on Riemannian manifolds?
Under certain nondegeneracy conditions, a Lagrangian $L$ on a manifold $M$, i.e., a function on the total space $TM$ of the tangent bundle of $M$ defines a diffeomorphism
$$\Psi_L: TM\to T^*M$$
k …
- 33.5k
2
votes
Accepted
Symmetry Properties of Minimizers - Calculus of Variations
Suppose that $\newcommand{\eF}{\mathscr{F}}$ $\newcommand{\bR}{\mathbb{R}}$ $\eF: C\to \bR$ is a convex functional defined on a closed convex subset $C$ of a say real Banach space $U$. (You can all …
- 33.5k
2
votes
Coercivity for functional and complete orthonormal system
$J$ is not coercive in $W^{1,2}$ For that to happen you need to show that $\Vert u_n\Vert_{1,2}\to \infty$ implies $J(u_n)\to \infty$. Take for example the function $u_n$ which is identically $0$ …
- 33.5k
8
votes
Accepted
Results about existence/uniqueness of solution to Euler-Lagrange equations?
The so called direct method of the calculus of variations provides one such existence and uniqueness result.
Here is the gist of it. Suppose that $X$ is a reflexive Banach space, e.g. a Hilb …
- 33.5k