Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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147 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
5
votes
2answers
412 views

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernece to this paper [Olivier ...
2
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0answers
47 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
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0answers
1k views

Geodesics of a cone

To find Geodesics on a cone I used the cylindial coordinates $x=rcos\theta$ $y=rsin\theta$ $z=z$ Is this parameterization correct.How can I know how to parameterize? Then arc length $ds^2=r^2 ...
1
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1answer
138 views

minimal surface of revolution integrating Euler Lagrange result

In finding minimal surface of revolution after applying to euler lagrange equation $d u \over dx \sqrt{1+(u')^2}$$=0$. Then $ u \over \sqrt{1+(u')^2}$$=constant$. Then solving $\int{ c \over ...
2
votes
1answer
98 views

Proof of Euler Lagrange equation in calculus of variation

I am learning calculus of variations from the article page 9 There it says $J[u]=\int_a^bL(x,u,u')\, dx$ and $u'$ is represented by $p$. $$\langle\nabla J[u],v\rangle={dJ[u+tv] \over dt} \text{ ...
2
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2answers
99 views

Doubt on calculus of variation

In wiki http://en.wikipedia.org/wiki/Calculus_of_variations#Example, the first example of calculus of variation is the minimize distance between 2 points. In my understanding, value of functional $J$ ...
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1answer
24 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
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1answer
113 views

Variational Methods, why KL divergence is the difference between true distribution and approximating distribution.

Likelihood = $L(\textbf{w}) = P(V\mid \textbf{w})$. $$\ln P(V\mid \textbf{w}) = \ln \sum_H P(H,V\mid \textbf{w})$$ $$= \ln \sum_H Q(H\mid V)\frac{P(H,V\mid \textbf{w})}{Q(H\mid V)}$$ $$\geq ...
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0answers
382 views

Integration of the Euler-Lagrange equation with explicit dependence on x

In the variational calculus, if the Lagrangian $L[y(x),y'(x)]$ is not an explicit function of $x$, the Euler-Lagrange equation takes on the following form: $$\frac{d}{dx}\left(L-y'\frac{\partial ...
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1answer
146 views

Integrating the Euler-Lagrange equation

Let us have a Lagrangian $L(y,y') = f(y)\sqrt{1+y'^2}$, where $y=y(x)$. The corresponding Euler-Lagrange equation is $$\frac{f'}{\sqrt{1+y'^2}} - f\frac{y''}{(1+y'^2)^{3/2}}=0$$ This expression should ...
0
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1answer
111 views

Minimizing cost function (Eikonal)

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $S \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=S$ and ...
2
votes
2answers
138 views

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ ...
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0answers
31 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
0
votes
2answers
34 views

Quadratic Minimization

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If ...
2
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2answers
64 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
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vote
0answers
107 views

Brachistochrone Problem to find out the path by which a bead travels in least time

The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time. So I proceeded ...
3
votes
1answer
45 views

how to use variational principle to find the best value for parameter $\lambda$?

I need to minimize the following integral by varying parameter $\lambda$: $$\int_0^\infty(f(x)-g(x,\lambda))^2dx$$ The functions $f(x)$ and $g(x,\lambda)$ are known and they satisfy ...
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0answers
20 views

Double standards on recognizing expression as functional

The Calculus of Variations starts with a definition of functional Such an expression, the argument of which is a function, is called a functional. Particularly, they say that $J = ...
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0answers
22 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
4
votes
1answer
155 views

Shortest distance between two points calculus of variation

Ok this problem might be trivial but when solving it using calculus of variations it's not so stupid. Suppose we have a fixed boundary condition $f(a) = f(b) = 0$ and we want to find the shortest ...
2
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0answers
107 views

Calculus of Variation - Question with Integral constraint

I'm stuck on this question, Let $V$ be an open ball in $\mathbb R^3$ in $ { x,y,z \text{ st }x^2 +y^2 +z^2<1}$ I need to minimise the integral $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
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89 views

Maximize an integral

I have the following integral to maximize but I don't know what to do with this $f(b,y)$ in the first integrand: $$ F=\intop_{a}^{b}[f(b,y)+\intop_{a}^{b}f(x,y)dx]dy $$ a and b are constants. Do I ...
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0answers
34 views

Finding extremals over y in two ways

We can find the extremal of $$\int_0^1(\frac{1}{2}y'^2+yy'+y'+y)dx$$ amongst all y with $y(0)=1$ by imposing the natural boundary condition $\frac{\partial F}{\partial x}=y'+y+1=0$ at $x=1$.Solving ...
4
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1answer
334 views

Difference between Variation of Calculus problems and Control Theory problems?

Variation of Calculus seems to have problems without the control with variables such as state and time. Then again Control Theory problems seems to have problems with one extra variable that is ...
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0answers
50 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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0answers
50 views

How are Hamilton function and Hamilton-Jacobian-Bellman function related to each other?

Can someone here knowing Finnish help me with this question in mathematical physics? I am trying to understand the solution to the problem 2: [here] ...
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0answers
53 views

Euler Lagrange equations

I need to minimise $$\int\limits_\Omega|\nabla H_\epsilon(\phi)|\,dx\,dy$$ with respect to $\phi$. Where $H_\epsilon$ is the regularised Heaviside function, so that it is differentiable. This can be ...
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0answers
40 views

Non-convexity of an energy functional

How would I go about showing that the following Mumford Shah functional is not convex? $$E_{MS}(u,C)= \int_{\Omega} |u_{0}(x,y) -u(x,y)|^{2}\ dx\ dy + \mu \int_{\Omega \backslash C}|\nabla ...
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0answers
60 views

Non-standard applications of Noether's theorem

The particular "Noether's theorem" that I'm referring to is the one that appears in the calculus of variations: if the Lagrangian in a variational problem is invariant under a one parameter group of ...
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1answer
61 views

Using the Euler-Lagrange equation to evaluate an integral

I think this is a very basic question. I'm just not sure how to use $\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\partial F}{\partial y'}\right)=\frac{\partial F}{\partial y}$ to find the general ...
1
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1answer
23 views

minimizing a function involving exponential term

Let $w\ge e$ . I want the following $$ \min_{r\geq0} r(e^r-w) $$ Is there any way to find it. Thanks.
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3answers
93 views

Finding Survival Function given hazard rate

Let X be a random variable defined for 0 < x < 4 with hazard rate $$lambda(t)=1/(4-t)$$ for 0 <= t <= 4. find the survival function, S(x) = P(X>x). Using the formula S(x) e^-integral from ...
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1answer
41 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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vote
2answers
359 views

Isoperimetric inequality on the sphere via calculus of variations

The isoperimetric inequality on the sphere of radius 1 asserts that for any closed curve on the sphere, $$L^2 \geq A(4\pi - A)$$ where $L$ is the length of the curve and $A$ is the area it encloses. ...
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2answers
141 views

Are all critical points of energy geodesics?

Let $\gamma$ be a smooth curve in a Riemannian manifold and consider the arclength functional $L(\gamma) = \int_a^b |\gamma'(t)|\, dt$ and the energy functional $E(\gamma) = \frac{1}{2}\int_a^b ...
2
votes
1answer
678 views

DuBois-Reymond Lemma

I know thats the following statement is true. $f,g$ are continuous function $[a,b]$.Suppose $\int\limits_a^bf(t)h(t)+g(t)h'(t) \, dt=0$ for every $h$ belonging to $C_0^{\infty}[a,b]$, then $g$ is ...
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0answers
44 views

Integrating a Functional

Reading the quote the so called "Feynmann path integral", which, as far as I understand, means "integrating" a functional (action) on some infinite-dimentional space of configurations (fields) of ...
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157 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
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1answer
91 views

An application of the mountain pass lemma

I am trying to show the existence of classical solution for the following problem using the mountain pass theorem : $$ \left\{ \begin{array}{ccccccc} u^{''} + \lambda u + u³ = 0 (0<t<\pi)\\ ...
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0answers
90 views

Semilinear Poisson Equation Using Direct Method of Calculus of Variations

The following problem comes from: http://people.physics.anu.edu.au/~gvn105/analyticMethPDE.pdf 12.9 Exercises 12.3: Let $\Omega$ be a bounded domain in the plane with smooth boundary. Let $f$ be ...
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31 views

Case C: Euler's equation in Simmon's textbook

Working through Simmons' Differential Equations with Applications and Historical Notes and we're stuck in Case C, page 360. Case C: If x is missing from the function $f(x,y,y')$, then Euler's ...
2
votes
1answer
86 views

bilinear continuous, coercive form

Let $k\in \mathbb{R}, k\neq 1$, consider the space $$ V = \{u\in H^1(0,1): u(0) = ku(1)\}$$ Let $$a(u,v) = \int_0^1 (u'v'+ uv)\; dx - \left(\int_0^1 u\; dx\right) \left(\int_0^1 v\; ...
2
votes
1answer
71 views

Differential of Lagrangian

My professor wrote this $\frac{\partial L}{\partial q}\dot{q}=\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})$. Due to the fact that I am very very very very bad at Math, could you explain me about ...
3
votes
2answers
80 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
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70 views

Image histogram equalization using variational calculus

In an image processing course at Coursera.org, on the section on PDE and calculus of variations, the professor gave the following as the functional to be optimized for image histogram modification: ...
11
votes
1answer
606 views

A variation of the isoperimetric problem in the plane

The isoperimetric problem in the plane: « The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed ...
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1answer
40 views

about lower semicontinuous functional

Let $X$ a topological space.My book define : A functional $\varphi: X \rightarrow R$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a, + \infty)$ is open in $X$ for any $a \in R.$ (1) And the book ...
3
votes
1answer
307 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
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1answer
140 views

Question about variational principles involving light rays.

In question 9 (see this link: http://view.samurajdata.se/psview.php?id=28b2e4b5&page=1 ), I've shown the light rays are follow a parabolic paths using the Euler-Lagrange equation and Fermat's ...