Questions tagged [calculus-of-variations]
This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.
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Solution to a constrained optimal control problem — is $v(t) = \frac{S}{T}$ optimal?
Given distance $S$, time $T > 0$ and bounds on the velocity $v_{\min} < v_{\max}$,
$$ \begin{array}{ll} \underset{v : [0, T] \to {\Bbb R}}{\text{minimize}} & \displaystyle\int_0^T f (v(t),t) ...
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Maximize the area enclosed by a fixed-length arc and a circle (endpoints of the arc may slide along the circle)
Let the coastline be the open arc of the unit circle between polar angles $0$ and $\phi$ (so its length is $\phi$).
For a fixed free-arc length $s>0$, and for each pair of endpoints $A,B$ on this ...
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Minimise $I(u)=\int_{0}^{1}{u}dx$ subject to $\int_{0}^{1}{\sqrt{1+(u’)^2}}dx=\frac{\pi}{3}$ [duplicate]
Starting with ($u(0)=u(1)=0$), find the stationary values of: $$I(u)=\int_{0}^{1}udx$$ subject to $$\int_{0}^{1}{\sqrt{1+(u’)^2}}=\frac{\pi}{3}$$
i.e find a curve that minimises area but has a fixed ...
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Gradient bound for $C^1$ function in unit disk [duplicate]
For a function $f : D^n \to \mathbb{R}$, $f \in C^1(D^n) \bigcap C(\overline{D^n})$, suppose $||f||_{L^\infty} \le 1$, we want to show $\inf | Df | \le c$, for all such $f$.
A known result https://www....
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Maximizing $\int_0^1 f(x) \, {\rm d} x$ given $\int_0^1 x f(x) \, {\rm d} x = 0$
Let $f : [0,1] \to [-1,1]$ be an integrable function such that
$$\displaystyle\int_{0}^{1} x f(x) \, {\rm d} x = 0$$ What is the maximum possible value of $\displaystyle\int_{0}^{1} f(x) \, {\rm d} x$?...
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Minimise $\int_{-1}^{1}{\left (x^2(u’)^2+x(u’)^3+(u’)^6\right)}dx$ with $u(-1)=u(1)=0$
Starting with $$ I(u) := \int_{-1}^{1}{\left(x^2(u’)^2+x(u’)^3+(u’)^6\right)} {\rm d} x,$$ with boundary conditions $u(-1)=u(1)=0$ and using the Euler-Lagrange equation:
$$\begin{align}\frac{d}{dx}\...
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A calculation of an integral under restricted variations
Recently I read a paper (https://link.springer.com/article/10.1007/BF02921575)
and got stuck on a calculation of an integral under restricted variations where the paper does not provide any details.
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Topological degree to show sets link
In Chapter II.8 of Variational Methods by Struwe, there is the definition of linking sets in a Banach space:
Definition Let $V$ be a Banach space, $S \subset V$ a closed subset and $Q$ a submanifold ...
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Existence of a convergence sequence in weak-star topology
In a book concerning calculus of variations written by Giusti, I read the following
" Let $Q$ be a cube. For every $\lambda\in [0,1]$ there exists a sequence $\chi_h$ of characteristic functions ...
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The jacobian of projection $r_s$ on $C^{1,1}$ surface converges uniformly to 1, when manifold has nonpositive sectional curvature
Let $M$ be a $3$--dimensional Cartan--Hadamard manifold (complete, simply connected, nonpositive sectional curvature). Suppose $S\subset M$ is a $C^{1,1}$ surface which encloses a domain $E$. Let $D_0$...
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Holomorphic analogue of Fundamental Lemma of Calculus of Variations
The fundamental lemma of calculus of variations essentially states that given a "smooth enough" ($C^1$ or $C^{\infty}$ or whatever) on an open domain $\Omega$, if we know that $\int_{\Omega} ...
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Uniqueness for periodic OT on $\mathbb T^d$ with small Dirac pins
Let $M=\mathbb T^d$ with $d\ge 2$, let time be the circle $S^1=\mathbb R/\mathbb Z$, and fix four phases $\theta_0=0$, $\theta_1=\tfrac14$, $\theta_2=\tfrac12$, $\theta_3=\tfrac34$.
Fix an anchor ...
user1675092
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Poisson's equation in a domain with a hole
Let $N > 2$ and let $\omega, \Omega \subset \mathbb{R}^N$ be open and bounded sets with smooth boundary. Assume both sets contain the origin. For $\sigma > 0$, consider the boundary value ...
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Are there any solutions with cusps for the problem of extrema of the functional $v(y(x))=\int_0^{x_1}y'^3dx$
I am solving through problems in the Calculus of Variations book by L. D. Elsgolc and I am stuck with the following one:
Are there any solutions with cusps for the problem of extrema of the ...
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Symmetrization of functional derivatives
I'm dealing with a mathematical problem stemming from quantum field theory (QFT). However, at the moment, I'm not concerned with the physics aspect of it and, hence, I wish to view it in purely formal ...
user724377