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

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Solving a PDE Using Variational Calculus

I am currently trying to understand a proof of the following Claim Let $(\mathcal{M},g)$ be a compact, oriented Riemannian manifold and let $f:\mathcal{M} \to \mathbb{R}$ be a function. Then $$ \...
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1answer
33 views

How to handle a diffusion equation with a variable diffusion coefficient with finite elements?

Based on the notes here I created a finite elements solver for the stationary heat equation (Poission's equation) $$-u''(x) = f(x)$$ However I would like to solve the stationary heat equation that ...
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26 views

variational system and non autonomous Hamiltonian

Let $H$ be the Hamiltonian vector field of the hamiltonian function $h(q,p)$, in local coordinates $(q,p)$, $$ H=\frac{\partial h}{\...
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23 views

Show that the functional $F(u)=\int ((u_x^2-1)^2+u_y^4)$ has zero infimum on a Sobolev space

Let $\Omega=(-1,1)\times (-1,1)$. We consider the functionnal $$F(u)=\int_{\Omega}\left[\left(\left(\frac{\partial u}{\partial x}\right)^2-1\right)^2+\left(\frac{\partial u}{\partial y}\right)^4\right]...
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calculus of variation and second derivative

Starting from the Hamiltonian system $$ \dot{z}(t) = H(t,z(t)) $$ we deduce the variational system $$ \dot{\delta z} = dH(t,z(t)).\delta z $$ $\delta z$ means we consider curves close to $z(t)$ ...
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Time-dependent inequalities in optimal controller

I need to build the optimal controller, i.e. one that maximizes: $J = \int_{0}^{t_f} f(u) \mathrm{d}t$ For the following time-dependent system: $\dot{x} = g(x, u, t)$, $x(t) \geq l(t)\; \forall t \...
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37 views

What is the definition of sequential precompactness?

I know that topological space $X$ is called precompact if any sequence in $X$ has a subsequence convergent in X. In my book of calculus of variation I have encountered the word sequential ...
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2answers
37 views

Derivative with respect to function

I am looking to calculate the derivative of a functional $\phi(\rho)$ with respect to $\rho$, that looks like $$\phi[\rho](x)=\rho(x)\int_0^1\log|x-y|\rho(y)dy.$$ I have read that the Gateaux ...
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50 views

Commuting Covariant Derivatives in Derivation of First Variation Formula

I'm following the book "A Course in Minimal Surfaces" by Colding and Minicozzi. I'm stuck on section 1.3, The first variation formula. We are given a Riemannian manifold $M$ with metric $g$ and ...
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1answer
14 views

Confusing on lower semi continuous and its application in minimize problem

I have a problem about the definition of lower semi-continuous (lsc) function and minimize problem: Is a function defined on a compact set which is lsc will has FINITE lower bound? In many books, ...
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24 views

Connect two points in $\mathbb R^3$ using integral lines of some vector fields

Suppose we are in $\mathbb R^3$ and let $\nu\in S^2$ be fixed. We have a regular function $u:\mathbb R^3\to \mathbb R$ with the following property: $u$ is decreasing along the integral lines of the ...
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17 views

conclude the local mass balance form from the global one

I am engineer and not so deep in mathematics and i need your help please. Let $\textbf{x}=[x,y,z]^T$ denotes a position vector of a point in a domain $\Omega_t$ say for example $\Omega_t = $ a$~$...
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41 views

Lower semi-continuous functional

Define the following functional on $W^{2,p}_0(\Omega)$ \begin{equation} E(w)=\frac{1}{p}\int_{\Omega}|D_{ij}^2w|^pdx \end{equation} where $|D_{ij}^2w|=|D_{xx}^2w|+|D_{yy}^2w|+|D_{xy}^2w|$. Is this ...
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1answer
38 views

Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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1answer
36 views

Sequence of lowersemmi continuous functions

Given a sequence of lowersemi continuous functions $f_n : (0,1)\rightarrow [0,\infty)$ with $$\sup_n Varf_n<\infty,\ \ \ \int_0^1f_n=1.$$ Here, $Varf_n$ is the point-wise variation of $f_n$, that ...
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46 views

Curve with finite length

Let $u: [a,b]\rightarrow \mathbb{R}^d$ is a curve. We define $$Var_{[a,b]}u=\sup\{\sum_{i=1}^n|u(x_i)-u(x_{i-1})|: a\le x_0<x_1<\ldots<x_n\le b, n\in\mathbb{N}\}$$ I am trying to find a ...
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219 views

Calculus of variation proof confusion.

So I was reading the proof on the shortest distance between two points being a line (https://en.wikipedia.org/wiki/Calculus_of_variations#Example) but one line of the proof is baffling me. The ...
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2answers
88 views

Euler-Lagrange Equation has no solution?

I've been asked to compute the Euler-Lagrange equation and second variation of the functional $$I[y]=\int_{a}^{b}(y'^2+y^4)dx$$ with boundary conditions $y(a)=\alpha$, $y(b)=\beta$. It's easy to see ...
2
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1answer
38 views

Is the first variation of a Jacobian determinant always zero?

I'm trying to find the Euler-Lagrange equations for the functional $$F[\mathbf{u}]=\iint \det{(D\mathbf{u})} \, dx \, dy$$ Where $\mathbf{u}:\mathbb{R}^{2} \to \mathbb{R}^{2}$ and $\det{(D\mathbf{u})}$...
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Check whether a functional has an extremal or NOT

Find the extremal of the functional $$J(y)=\int_a^b F(x,y,y')\,dx$$where , $F(x,y,y')=y'+y$ , for admissible functions $y$. From Euler-Lagrange equation , $\displaystyle \frac{d}{dx}\left(\frac{\...
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25 views

Isomorphism induced by $\Delta$

I know the laplacian induces an isomorphism between $E$, the space of the $H^1_0$ radially symmetric functions, and $H^{-1}$. I would need to see a proof of that fact. Do you know where I can find it?
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Why should the map $-\Delta^{-1}$ continuous?

I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$ in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
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1answer
57 views

The distribution with the lowest possible variance.

Let $\phi$ be the family consisting of all random variables $X$ such that $P(X\in [0,1])=1$, $EX=\frac{1}{5}$, $P(X\leq\frac{3}{10})=\frac{1}{2}$. Calculate $\inf \{\rm{Var}(X):X \in \phi\}.$ Could ...
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42 views

Consider the functional $J(y)=\int_{a}^{b}F(x,y,y')dx$.

Where , $$F(x,y,y')=y'+y$$ for admissible function $y$. Then what is the extremal. I did the solution using the special case of Euler-Lagrange equation where the $x$ is missing and arrived at the ...
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62 views

infimum of a functional in $W^{1,p}((0,1))$

Consider the functional $$\mathcal{F}(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ u\in W^{1,p}((0,1)),$$ where $\alpha\ge 0$ and $1<p<\infty$. Given $a<b$, find the value of $$\inf\{\mathcal{F}...
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32 views

Find the infimum value of a functional.

Consider the functional $$F(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ \ u\in W^{1,p}(0,1)$$ where $\alpha\ge 0$ and $1<p<\infty$. Given $a<b$, find the value of $$\inf\{F(u): u\in W^{1,p}(...
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40 views

General Brachistochrone problem

Let $\gamma$ is a continuous curves that connect $(0,0), (x,y), x>0,y<0$ in $\mathbb{R}^2$ and $u : [0,L(\gamma)]\rightarrow\mathbb{R}\times(-\infty,0]$ is the arc length parameterrization of $\...
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243 views

Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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1answer
104 views

How to take a partial derivative of an integral containing a fourier series?

(Chapter 2, p. 68, Problem 24) From Goldstein Classical Mechanics. Problem: The one-dimensional harmonic oscillator has the Lagrangian $L=m\dot{x}^2/2-kx^2/2.$ Suppose you did not know the ...
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1answer
34 views

Why does the Lagrange equation have to be zero?

I know it's a pretty basic question, but I still don't get it since starting Lagranian mechanics this year. I tried to read Stone and Goldbart's "Mathematics for Physics" and they said: Suppose ...
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1answer
160 views

Functional optimization: maximize a double integral where the functional appears twice

Please help me solve the following optimization problem. Suppose that you have to choose a function $U: [0,1]\mapsto [0,1],$ which must be nondecreasing ($U'\geq 0$) to maximize the following integral:...
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66 views

Short introduction script to calculus of variations

Does anybody know of a short lecture script/website/small book that gives an introduction to the calculus of variations? It should be suitable for total beginners, with steps explained in detail. At ...
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1answer
52 views

Stuck with a problem of calculus of variation in the proof that a minimizing curve is a geodesic

I'm reading the proof of the proposition that states that every minimizing curve is a geodesics when it is given an unit speed parametrization. In the proof appears the following quantity : $$ \...
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89 views

Functional derivative or chain rule?

Just a quick question... I have two functions – $V(a,b,c)$ and $F(a,b,c)$ – and I wish to calculate the derivative of one with respect to another ($\frac{\partial V}{\partial F}$). Am I right in ...
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1answer
36 views

Linearization of divergence of a vector field?

Let's $X$ is a fixed smooth vector field on semi-Riemannian manifold $(M,g)$. For a symmetric 2-tensor field $s$, and for sufficiently small values of $t$, $\tilde{g}=g+ts$ is a semi_Riemannian metric ...
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29 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
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1answer
37 views

Variational calculus - inequality

Let $u \in L^1_{loc}(\Omega)$, and suppose that $\int _{\Omega} u(x)\eta (x)\; dx \geq 0$, $\forall \eta \in C ^{\infty}_{0} (\Omega)$ . Then $u(x) \geq 0$ , a. e. $x \in\Omega$
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Characterize $|\nabla f|$ as minimal function which satisfies an upper gradient inequality

Let $f \in C^1( \mathbb R^n, \mathbb R) .$ Then one by chain rule has $$ (*)\qquad |f(g(1))-f(g(0))| \leq \int_0^1 |\nabla f|(g_t)|g'(t) |\ dt, \quad \forall g \in C^1([0,1],\mathbb R^n). $$ I have ...
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2answers
23 views

Proof for special concave form of functions

If $f(x)$ is a non decreasing concave function between $0$ and $1$ for $x\ge 0$, then for $a>1$ I am confident that $af(x)>f(ax)$, but I am not quite sure how to prove it. Any help would be ...
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Sufficient conditions for an optimization problem

Given an optimization problem \begin{equation} \max{F(x)} \text{ subjected to }T(x)=u \end{equation} Where $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is concave function and $T:\mathbb{R}^n \...
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108 views

Calculation of a Frechet derivative

Say I have an infinite sequence $X=(x_i)$, $i=1,2,3,\ldots$ such that it's in $\ell^2$ space, i.e. $\sum_{i=1}^\infty|x_i|^2<\infty$. Now, this function that takes this infinite sequence to a real ...
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How can I find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1} \sqrt{1+y'^2+z'^2}$$ such that :$$y^2+z^2=1$$ and $$y(0)=z(1)=1$$ $$y(1)=z(0)=0$$
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Variational function versus variational solution

I want to minimize the functional $F[f(x)]$ and I'm going to try this in two different ways: First I am going to numerically minimize the functional $F[f(x)]$, leading to the "true solution" $f(x)$. ...
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33 views

Fastest path with limited acceleration

An object on point $A$ with the initial velocity of $\dot{\bf{x}} (t)$ have a maximal acceleration of $a$. What is the fastest path for the object to get to point $B$? I thought this should be all ...
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1answer
38 views

Partial derivative of an integral transform

I have a Lagrangian of the form $\mathcal{L}(x,f)=[s(x)-\int_a^b A(x,x')f(x')dx']g(x)$, where $a,b$ are constants, and $g(x),s(x)$ and the kernel $A(x,x')$ are given . I am interested in computing ...
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Are triangles the strongest shape?

They are according to the buzz on the Internet (and most stable too), despite competition from circles. Mythbasters even proclaim that "triangles are the strongest shape because any added force is ...
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45 views

Does Euler-lagrange Equation hold with discontinuous integrand?

Suppose I have a function $Heaviside(x-1) \sqrt{1+f'(x)^2}$ or some other discontinuous function, how do I find the minima with E-L Eqn or in other way?
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53 views

Why would integrating acceleration give the following solution?

Suppose I have a mass with equation of motion described by: $x^{''}(t) = F(t) - 1$, $0<t<T$, all initial conditions equal to zero $F(t)$ is some unknown force My text claims that the equation ...
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32 views

Analytic version of Hilbert's XIX problem

The famous Hilbert's nineteenth problem, initially stated in the $C^\omega$ category, was reduced by Bernstein and Petrowsky to the analogous statement in the $C^\infty$ category (and, after ...
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309 views

Poincaré hyperbolic geodesics in half-plane and disc models

The objective of this post is to state that 1) the Poincaré hyperbolic metric results in a solution of complete geodesic circles in both half-plane and disk models. 2) the choice of one or other ...