Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.

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Convergence in the distributional sense (mean field games dynamics)

I am trying to go through the papers by Gueant, Lions and Lasry on Mean field games. One of their examples is the Mexican wave (which happens in football stadia). Straight to the point the Lagrangian ...
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25 views

Partial Differentiation with respect to a functional

Suppose that there are two independent variables $x, y \in \mathbb{R}$. Define two functionals $f_i : \left(\mathbb{R} \times\mathbb{R}\right) \rightarrow \mathbb{R}, i=1,2$ \begin{align} ...
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1answer
21 views

The shortest line problem extremum extinction

Let's consider a functional $J(y)=\int_{0}^{1}{ \sqrt{1+\frac{dy}{dx} ^{2}} dx}$, $y(x_{1})=y_{1}, y(x_{2})=y_{2}$. Using Euler-Lagrange equations we can get that the set of functions which attains ...
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1answer
38 views

Integration of partial derivative $\frac{dL}{dq}$ with respect to $t$ where $q$ is implicitly a function of $t$

Is $\int_{t1}^{t2} \frac{\partial L}{\partial q}\delta{q} dt$ equal to $\left[\frac{\partial L}{\partial \dot{q}}\delta{q}\right]_{t1}^{t2} $ if $q$ implicitly depends on $t$ ? If not I ...
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59 views

Is there any situation in which a geodesic maximize the path length between two points?

Some people (even in here) claim that geodesics are, in general, stationary curves. Locally speaking, geodesics always minimize arc length (see Manfredo, for example). But I can't visualize a surface ...
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18 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
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12 views

Variational calculus on integrals and derivatives

I am studying mechanics but am a novice in variational calculus. While reading a book on Lagrangian mechanics, I blocked when the author states that by calculating the variation of the following ...
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12 views

Stable geodesics

Consider a function defined on some space of smooth curves in a manifold (think of the "action functional"). I understand what a "critical point" of such a function is, but what is understood by a ...
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1answer
29 views

Solving a functional problem with break points

I'm totally out with solving functional problems when we need to find broken extremals, can someone show me how to solve the following problem: Can the following problem have break (corner) ...
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1answer
60 views

Find maximum $\int^{1}_{0}\{f(x)\}^3dx$

I would appreciate if somebody could help me with the following problem: Question: Find maximum $\int^{1}_{0}\{f(x)\}^3dx$ when (1). $f(x) : \text{conti-and} \int^{1}_{0}f(x)dx=0$ (2). $-1\leq ...
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34 views

Understanding derivation in Calculus of Variations book

I'm reading about Calculus of Variations and about the general variation of a functional. I bumped into few obstacles in my book I can't get over with. I have scanned the pages where I have my ...
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1answer
33 views

A little confusion about calculating variance?

given $a = 10% $, $b = 50%$, $c = 25%$ and $d = 15% $ calculate the variance? ( this is a made up question to make thinks clear for me) what i have done is 1st eliminate percentages? then i ...
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1answer
23 views

Harmonic solutions

Assume that $\Omega\subset R^2$ is an open bounded set with a smooth boundary, $g:\partial\Omega\to R$ is a continuous map and $\{b_i \ | \ i=1,2,\ldots,d\}$ is a finite subset of $\Omega$. ...
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0answers
10 views

Minimal distance in expectation problem

Two random variables $\tilde{X},\tilde{Y}$ have the joint distribution $F:X\times Y\to [0,1] $ The given function is $g^*=(g_1^*,g_2^*):X\times Y\to R^2$ The control function is ...
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61 views

Second variation of the domain functionals.

I am reading a paper which deals with the second variation of the domain functionals and applications . The following problem is solved . The eigen value $\lambda(t)$ which is characterised by the ...
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29 views

solving euler-lagrange equation in constrained functional optimization

The problem to solve is the minimization of a functional of two functions, $F(y,z) = \int_a^b f(y,z)dx$ , subject to a constraint $g(y,z,y',z') = 0$. The augmented functional is then $L(y,z,y',z') = ...
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1answer
21 views

Finding function that maximizes ratio of area to length

I'm new to variational analysis, so I need someone to check, if I'm going in the right direction. Let's say I need to find a curve $y(x)$ with $y(0) = 1$ and $y(1) = 0$ that maximizes ratio of area ...
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18 views

Quantity of order $\Delta x$

I have the following from Gelfland's Calculus of Variations book: $$\frac{\partial J}{\partial y_k} = F_y\left(x_k,y_k\frac{y_{k+1}-y_k}{\Delta x}\right)\Delta x + ...
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1answer
54 views

Is this functional well defined?

I came across the following formulation of the problem. Minimize the functional $L[u]$ given by $L[u] = \int^b_a \sqrt{(1+(u'(x))^2}$ over $U = \{u\in C([a,b])\cap C^1((a,b)):u(a)=\alpha, u(b) = ...
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16 views

Checking Weierstrass's condition in a functional problem

My problem is to check for the Weierstrass's (necessary) condition in the following functional problem in my Calculus of Variations course: $$\min J[y]=\int_0^1xy'^3\;dx,\;\;\;\;y(0)=0,\;y(1)=1$$ ...
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Determine whether the extremals of a functional are weak or strong?

Is there any thumb rule for determining whether the extremals of a functionals (which are determined using Euler Lagrange equations) are maximum or minimum, weak or strong. I read about Weierstrass ...
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Numerical optimization in function space

I'm new to calculus of variations. I'm curious about how to apply simple numerical optimization techniques in function space. Consider the classical problem: finding the shortest path between two ...
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27 views

Prove a formula after change of variable?

If I have a change of variable $(x,u)\to (X,U)$ given by $$X=x+\epsilon u,U=u-\epsilon u.$$ How to prove the formula $$\frac{\partial U(X,0)}{\partial \epsilon}=\phi(X,u(X))-u'(X)\xi(X,u(X)),$$ where ...
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A change of variable problem?

How to calculate (a)? The definition of infinitesimal generators is as follows, It is first a change of variables, then calculate the derivative: I am so confused by the notations. Can anyone ...
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1answer
43 views

The minimizing problem over a sequence of shrinking balls

Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define $$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$ where $1<q<5$. Hence we know that each ...
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38 views

First variation — A differentiation problem.

My question: This is just differentiation and I did it. I got $$T'(\epsilon)[L(T(\epsilon),x(T(\epsilon);\epsilon),\dot x(T(\epsilon);\epsilon))]+\int_0^{T(\epsilon)}L_x(t,x(t,\epsilon),\dot ...
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0answers
17 views

How to determine convexity of functional

I am a rookie to functional and calculus of variation. I want to know whether there exists any sufficient condition for convexity of a functional besides the definition. Actually for function, one ...
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1answer
32 views

Find the extremals of a functional of the form $\int^{x_1}_{x_0}F(y',z')dx$

I was working on Problem 3 in Ch. 2 of Gelfand & Fomin's Calculus of Variations, which reads: Find the extremals of a functional of the form $$\int^{x_1}_{x_0}F(y',z')dx$$ given that ...
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1answer
21 views

Calculation of the second variation of the functional $I(y)=\int_{-1}^1 [x^2(y')^2+x(y')^3]\,dx$

My question: I don't understand the last equation about second variation. According to definition, shouldn't it be $\int_{-1}^1 [2x^2+6xy'] (\eta)^2$? Can anyone help me with this? Where am I ...
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Calculus of variation: question about Lagrange multiplier?

In the example of Newton's problem with volume constraint. When using the method of Lagrange multipliers,the new Lagrangian is $L(x,h,h')=\frac{x}{1+h'^2}+\lambda xh $. I don't understand why we ...
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1answer
26 views

Differentiation under the integral sign problem

I have to take the derivative of the following function with respect to $\varepsilon$: $$\phi(\varepsilon)=\int_{a}^{b+\varepsilon C}F(x, y(x)+\varepsilon\eta(x), y'(x)+\varepsilon\eta'(x))\;dx$$ My ...
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1answer
23 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
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Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
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26 views

Mountain pass theorem

Let $I$ be a real functional over a Hilbert space $H$, satisfying all the conditions in the Mountain pass (M-P) theorem. My question is, can the assumption in the M-P theorem that $I[v]\leq 0$ for a ...
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How to minimize $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $

I'm new in optimizations and i am trying to understand how to obtain $ v $ that minimizes $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $ where $\rho(x)$ - continuous ...
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2answers
29 views

Understanding part of a theorem of Calculus of Variations

I have trouble understanding the following statement (From Gelfland's Calculus of Variations book): If $\phi[h]$ is a linear functional and if ...
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1answer
42 views

Why Dirichlet's energy uses a **squared** norm?

$E = \int_{\Omega}\left \| \nabla u(x)\right \|^2 dx$ So, Dirichlet's energy measures the integral of the squared norm of the gradient. Why squared norm? What would we get if we use just a norm? It's ...
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Problem about deformation theorem

I'm reading Evans PDE, on chapter 8.5 the proof of deformation theorem about the calculus of variation. On page 504 Evans wrote on the top: "we verify that the map $u\to dist(u,A)+dist(u,B)$ is ...
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Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
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0answers
26 views

Check whether the extremal has weak minima or weak maxima

The functional $$\int_0^1(y'^2 + x^3)dx,$$ given $y(1)=1,$ achieves its weak maximum on all its extremals weak minimum on all its extremals weak maximum on some, but not on all of its extremals weak ...
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1answer
38 views

What does the term “perturb” mean?

I've been studying Calculus of Variations and I came a cross with the term "perturb" in my study material, but the term was not defined. The sentence where I read it from was: "Rigid extremals are ...
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1answer
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Functional derivative - understanding some basics

I have the following functional $$ L[u] = \int_0^l dx [-\frac{\lambda}{2}u^2 + \frac{1}{4}u^4] = \int dx J[u]$$ Now, I need to calculate $$ \frac{\delta L}{\delta u} $$ As I understand, since I can ...
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1answer
36 views

Functionals' Taylors Theorem

Consider functional $F:B\to \mathbb{R}$, where B is a Banach space eg. $B=H^{1}(\mathbb{R}^{d},\mathbb{C})$. Then Taylor's theorem for functionals is: Suppose that the line segment between u ∈ ...
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2answers
87 views

Calculus of Variations: Understanding functional derivative

I am trying to understand the basics of the Calculus of Variations and the first thing to understand is the functional derivative. I failed to find a good introductory material, so I am trying to make ...
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1answer
79 views

Checking: finding extremals for a functional

I'm trying to find the extremals of the functional $$J[y] = \int_0^1 (y')^2 + y^2 + 4ye^x \, {\rm d}x,$$ imposed that $y(0) = 0$ and $y(1) = 1 $. I got that there can't be extremals, and that's weird ...
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2answers
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Infimum of $\frac{||u'||^p_{L^p}}{||u||^p_{L^p}}$ for $u \in W^{1,p}_0((0,1))$

Good afternoon everyone! It is very easy to show that the infimum mentioned in the title is strictly positive, but it seems much more difficult to show that it is attained within the Sobolev space of ...
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1answer
49 views

Integration by parts problem

If $\textbf{x}\in \Omega \subset\mathbb{R}^n,$ where $\Omega$ is a bounded open set, $u:\Omega\rightarrow\mathbb{R}, \;\eta:\Omega\rightarrow\mathbb{R},\;u'=\nabla u = ...
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How does integration by parts work with multivariable functions

How does integration by parts work with multivariable function? Lets say I have the functions $f(\textbf{x})$ and $g(\textbf{x})$, where $\textbf{x}\in\mathbb{R}^n$. How would integration by parts be ...
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1answer
50 views

Problems with Calculus of Variations lecture material

I'm having trouble understanding the derivation in my Calculus of Variations course material and I was hoping if someone could clarify this out. Here is my reference (as I have rewritten it, the ...
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1answer
51 views

What does $f(u)=\min!$ mean in calculus of variations?

I have a very simple notation related question. There are notes to calculus of variations [specifically: Zeidler's book "Nonlinear Functional Analysis and its Applications II/B" page 506] which states ...