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.
3,197 questions
2
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Solution to a Constrained Optimal Control Problem: Is $v(t) = S/T$ Optimal
The problem is stated as:
$\min_{v}\int^T_0f(v(t),t)dt$
Subject to the following constraints:
$s'(t)=v(t)$,
$s(0)=0$,
$s(T)=S$,
$v_{\min}\le v(t),S/T\le v_{\max}$
where:
$T$ and $S$ are given ...
- 23
2
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0
answers
107
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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 ...
- 6,087
5
votes
1
answer
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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 ...
- 1,127
1
vote
0
answers
81
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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....
- 142
34
votes
10
answers
2k
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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 $\displaystyle\int_{0}^{1} f(x) \, {\rm d} x$?...
- 459
3
votes
1
answer
103
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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}\...
- 1,127
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0
answers
21
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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.
...
- 69
1
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0
answers
56
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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 ...
- 43
0
votes
1
answer
34
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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 ...
- 928
2
votes
0
answers
62
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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$...
- 663
3
votes
1
answer
278
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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} ...
- 43
1
vote
0
answers
41
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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
2
votes
0
answers
81
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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 ...
- 1,198
4
votes
1
answer
48
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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 ...
- 330
2
votes
0
answers
80
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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
2
votes
0
answers
112
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P-Laplacian weak solution.
I am working on this PDE exercise using variational calculus. This is what I have done so far, and I have some doubts. If someone could help me with a guide or correct anything that might be wrong, I ...
- 85
7
votes
1
answer
147
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Does the functional have a minimizer?
Given the following functional with fixed parameters $a>0$ and $b>1$,
$$F[x;u,u']: = \int_{1}^b\left(x^3-\frac{ax}{u^2}\right)(u')^2~\mathrm{d}x$$
Suppose the space of functions satisfy $u(1) = ...
- 4,163
2
votes
1
answer
56
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If the composition $G(u)$ is zero almost everywhere, and $u=0$ on $\partial U$ then how to conclude $G(0)=0$ (Evans 8.4.1)
I am looking at Theorem 2 from Evans PDE 2nd Edition chapter 8.4.1 about Lagrange Multipliers. Here he is trying to prove there is a real number $\lambda$ such that
$$\int_U Du\cdot Dv\;dx=\lambda\...
0
votes
0
answers
72
views
Variation of equator with keep area unchange on round sphere
As an exercise, I attempted to calculate the second-order variation of the equatorial length on a round sphere to verify some geometrically clear facts.
Since the equator is the shortest bisector of ...
- 6,956
2
votes
1
answer
201
views
Computing the first variation of a variational integral
Having started to read Giaquinta/Hildebrandt "Calculus of Variations I", 2 ed. 2004, in order to get a better mathematical foundation for my understanding of the Hamilton principle of least ...
- 545
1
vote
0
answers
76
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Tautochrone is a Brachistochrone
As both the Tautochrone and Brachistochrone have the same solutions (ie. they lie along a cycloid), I was wondering how one might show that the Tautochrone is perhaps a Brachistochrone (ie. that the ...
0
votes
0
answers
25
views
Static Potentials as Mean-Field Limits
Let $X$ be a Polish space. Consider two dynamics:
A mean-field dynamic on the space of probability measures $\mathcal{P}(X)$, governed by an interaction potential $F(x, \mu)$.
A gradient flow on $X$ ...
0
votes
1
answer
87
views
Optimal Control w/ Control Equality Constraints
Let's say I have an optimal control problem of the following for:
\begin{align*}
&\max_{\vec{u}}\int_a^bf(\vec{x}(t),\vec{u}(t))\,dt\\[10pt]
\text{subject to:}&\qquad \frac{d\vec{x}}{dt}(t) = \...
- 10.1k
-1
votes
1
answer
71
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Is the notation of $\nabla_k(\delta\Gamma^k_{ij})$ in the variation of the Einstein-Hilbert action mathematically valid?
I'm trying to understand the variation of the Ricci tensor in the derivation of the Einstein field equations, and I have encountered a confusion regarding the notation and interpretation of the term $\...
- 905
3
votes
1
answer
80
views
Integral equality for determinants in $W^{1,4}$
I am trying to show the following:
Let $\Omega=B_1(0)\subset \mathbb{R}^2, g\in W^{1,4}(\Omega)$ and $A=\{u \in W^{1,4}(\Omega,\mathbb{R}^2)\mid \operatorname{tr}(u)=\operatorname{tr}(g)\}$. Then:
$\...
- 307
1
vote
1
answer
62
views
Gamma limit of constant family
Let $(X,d)$ be a metric space, and given a function $F_0:X\to [0,+\infty]$ let $\{F_\varepsilon\}_{\varepsilon>0}$ a family of function such that
$$ F_\varepsilon = F_0 \quad \forall \varepsilon>...
- 101
1
vote
1
answer
86
views
Existence of solution of variational problema
Given the function $f(\xi)=\sqrt{1+\|\xi\|^2}$ with $\xi \in \mathbb{R}^n$, define de functional
$$
E(u)=\int_\Omega f(\nabla u(x))\,dx
$$
with $\Omega \subset \mathbb{R}^n$ open, bounded and having ...
- 101
2
votes
0
answers
59
views
Failure of $\Gamma$-compactness in non-separable metric spaces
For metric spaces, the notion of $\Gamma$-convergence can be defined as follows.
Let $(X,d)$ be a metric space, let $(F_j)_{j \in \mathbb N}$, $F_j \colon X \to \overline{\mathbb R}$, be a sequence ...
- 51
1
vote
0
answers
43
views
Notation for field variations in the variational bicomplex
While reading Compère's Advanced Lectures on
General Relativity (https://arxiv.org/abs/1801.07064), I stumble upon an issue that I encountered before, but that got even more pressing after ...
- 2,044
4
votes
0
answers
50
views
Is weak lower semicontinuity of an integral functional preserved under convexification of the integrand?
Let $\Omega \subset \mathbb{R}^n$ be a bounded open set, and consider the integral functional $\mathcal{F} \colon W^{1,p}(\Omega; \mathbb{R}^m) \to \mathbb{R} \cup \{+\infty\}$ defined by:
$$
\mathcal{...
- 41
4
votes
0
answers
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views
Functional maximizations on manifolds under constraints defined on sub-manifolds
I was trying to find the explicit form of a probability distrubution through entropy maximization when I realized I don't know how to perform the needed calculation.
The task is to maximize a ...
- 41
0
votes
1
answer
83
views
How to find the shortest optical path through an annular medium with uniform refractive index?
I'm studying a 2D geometric optics problem and would like help determining the shortest optical path between two points in the presence of a circular annular medium with a uniform refractive index.
...
- 3
4
votes
1
answer
105
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Are minimal perimeter sets locally analytic?
I deeply apologize for this question , since i have asked another one today , but i am a little bit confused.
Is this fact true ?
"If we take a set $E \subset \mathbb{R}^n$of minimal perimeter , ...
- 61
2
votes
1
answer
136
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Measure of topological boundary of a minimal set
Good morning to everyone.
I am doing a lot of confusion with these concepts and despite having read a lot I cannot go into the details in the remaining time.
The question is:
"If I have a ...
- 61
1
vote
1
answer
44
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Weaker coercivity condition for Tonelli's existence theorem - bounding the gradient of a minimizing sequence
I would like to show to show that Tonelli's existence theorem still holds for a variational problem
$\inf_{u \in A} F(u)$
with
$F(u)=\int_{\Omega}f(x,u(x),Du(x))dx$
and
$A=\{u \in W^{1,p}(\Omega)| u=...
- 307
0
votes
0
answers
49
views
Example of Martingale where almost all trajectories are RCLL with finite variation
It is clear that the only Martingale where a.a. trajectories are RCLL and of finite variation is the 0 Martingale.
But I have read that if we replace continuous with RCLL then there are others too.
...
- 11
1
vote
0
answers
103
views
Local minimizers of norm of a linear operator
I'm struggling to understand something about what I saw referred to as the lower norm function and its possible relation to singular values.
Let $A\in \mathbb{R}_{m\times n}$ be a matrix with $m\geq n$...
- 8,566
12
votes
2
answers
998
views
Proof of shortest path avoiding ball
I have read in a number of places that the shortest path between two points $a,b\in \mathbb{R}^2$ that avoids a disk $D$ between them (by "between"
I mean the disk intersects the line $a-b$) ...
1
vote
1
answer
178
views
How to show $\alpha$ is a circle?
$\alpha(s):[0,2\pi] \rightarrow \mathbb R^2$ is smooth closed curve. And $s$ is arc length parameter, namely $|\alpha_s|=1$. Denote the geodesic curvature of $\alpha(s)$ as $k_g(s)$.
Besies, we define ...
- 6,956
1
vote
0
answers
46
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Explanation of the qualitative description of the solution to Goddard problem.
Background: Goddard problem asks what is the optimal thrust profile of a vertically ascending rocket with a given amount of fuel, taking into acount gravity and atmospheric drag. It asks what is the ...
- 1,416
1
vote
0
answers
92
views
Existence of solution to PDE via functional
I want to show that the problem
\begin{equation}
\left\{ \begin{aligned}
-\Delta u &= u^p & \text{in } \mathbb{B},\\
u &= 0 & \text{in } \partial\mathbb{B}, \\
...
0
votes
0
answers
56
views
Understanding the Mountain Pass Level in the Mountain Pass Theorem
The usual Mountain Pass Theorem is:
Let $X$ be a Banach space and $I \in C^1(X, \mathbb{R})$ such that $I(0) = 0$. Also suppose
$G_1)$ There exists $\rho, \alpha > 0$ such that $I(u) \geq \alpha$ ...
- 1,753
1
vote
0
answers
87
views
Minimization with implicit dependency over function spaces
Technically, I am working on an optimal control problem. However, through some trickery, I managed to eliminate the dynamics. What I am left with is the following minimization problem:
$$ \min_{g \in ...
- 66
0
votes
1
answer
98
views
Do endpoints of a path of local maxima have to be local maxima?
I'm currently working on my PHD paper and I got stuck with this question.
(It is a paper on game theory, but the problem is purely mathematical)
Let
$$
S = [0,1]^8,
$$
the set of 8-tuples in the unit ...
0
votes
0
answers
56
views
Bounding High Order Variations of Functionals
In Strang 1964, there is a claim (Eq. 15) concerning bounds on the $L^2$ norm of higher-order variations of a linear functional. The paper offers only a brief, one-sentence justification for this ...
- 101
1
vote
1
answer
61
views
Weierstrass-Erdmann conditions
Consider the functional
$$
J[y] = \int_{0}^{1} \left[ (y'(x))^2 + (y'(x))^3 \right] dx, \quad \text{subject to} \quad y(0) = 1 \quad \text{and} \quad y(1) = 2.
$$ (minimising problem)
Determine which ...
- 6,580
0
votes
0
answers
74
views
Approximating the extremizer of $J[y]=\int_{0}^{1} \left[(y')^2+y^3 \right]\,dx$ using Rayleigh-Ritz method. The boundary data is$y(0)=4$ and $y(1)=1$
So, I want to approximate the extremizer to the functional $J[y]=\int_{0}^{1} \left[(y')^2+y^3\right] \,dx$
the boundary data is $\hspace{0.2cm}$$y(0)=4$,$\hspace{0.2cm}$ $y(1)=1$
According to ...
- 690
1
vote
0
answers
84
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Numerical optimization of discretized energy functional with spatial fields and derivative constraints
I am solving a variational optimization problem where the objective functional depends on two unknown spatially-varying functions, E(u) and G(u), as well as their first and second derivatives. The ...
- 11
1
vote
0
answers
99
views
Relations among various weak solutions of $-\text{div}(|\nabla u|^{p-2}\nabla u) = |u|^{q-2}u$.
Consider the following nonlinear PDE-
$$-\Delta_{p} u = |u|^{q-2}u\,\,\, \text{in $\Omega$,}$$
where $\Delta_{p} u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $u\in W^{1,p}_{0}(\Omega)$, $\Omega \subset \...
- 189
2
votes
0
answers
43
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Comparing trace of two functions who agree a.e. except on a small set
Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ ...
- 1,737