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

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55 views

Energy Method for Regularizing Effect of Heat Equation

I am trying to show the following: Let u solve the homogeneous heat equation in the cylinder $\Omega$ x $(0, \infty)$ with vanishing dirichlet data and initial condition g. Multiple the PDE by $tu_t$ ...
2
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1answer
26 views

Understanding a necessary step in a solution in variational calculus

I'm reviewing calculus of variations using a pdf that I found online (link) and in the example about the minimal surface of revolution, the writer simplified an equation tagged $(3.16)$ as follows: $$...
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58 views

Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = \sup_{\...
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1answer
40 views

Using Euler-Lagrange equations to differentiate a Laplacian

Assume I am given a functional of the form: $$ I_0[u]:= - \int \nabla u \cdot \nabla u dx $$ then, I know that by the Euler-Lagrange equations, I have: $$ \frac{\delta I_0}{\delta u }= 2\Delta u $$ ...
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2answers
42 views

Dirichlet's Energy on $X:=\{u\in C^1([0,1]) |\ u(0)=0, u'(1)=1\}$?

Let $X:=\{u\in C^1([0,1]) |\ u(0)=0, u'(1)=1\}$. I want to show that $J(u):= \int_0^1 (u'(x))^2dx$ doesn't have an infimum on $X$. Hi, this looks an awful lot like an application for Dirichlet's ...
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33 views

Displacement of differential equation

In [Forni and Sepulchre, arXiv:1305.3456] the authors state that given the differential equation \begin{equation} \dot{x}=f(x,u),\quad (1) \end{equation} where $f:\mathbb{R}^{n\times m}\to\mathbb{R}^...
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69 views

What's the other way without the Euclidean approximations to prove that a geodesic lies on the great circle?

The geodesics are the shortest curves that can be drawn between two points in a space. If the surface is a spherical one on which we are trying to get the geodesic between two points then it is said ...
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67 views

Non-integral constraints in calculus of variations

This is from Daniel Liberzon's book on Optimal Control, see section 2.5.2 in http://liberzon.csl.illinois.edu/teaching/cvoc.pdf. Consider a basic calculus of variations problem: $$ J(y) = \int_a^b L(...
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90 views

Extremal of a functional $I=\int\limits_0^{x_1} y^2(y')^2dx$

The extremal of the function $$I=\int\limits_0^{x_1} y^2(y')^2dx$$ that passes through $(0,0)$ and $(x_1,y_1)$ is a constant function a linear function of x part of a parabola part of an ...
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25 views

dimension of a optimal control problem with one state and two control variables

I have a optimal control problem where I have a two control and one state variable. (The field is economics but my question is purely on mathematics) The maximization program is ; $$max\int_{0}^{\...
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173 views

Geodesic on the surface of a cone (calculus of variations)

I have been trying to solve an exercice I found on a book. It is about a geodesic on the surface of a cone. The answer is plainly provided at the end of the book without any hint or detail and my ...
4
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1answer
37 views

Surjectivity of Derivatives in infinite dimensional spaces

I have a trouble about an exercise in Techniques of Variational Analysis, Borwein, J.M., Zhu, Q.J (Ex. 2.1.2): Let $X$ be a Banach space and let $f: X \to \mathbb{R}$ be a Fréchet differentiable ...
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1answer
27 views

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
35 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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30 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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1answer
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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21 views

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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26 views

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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1answer
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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1answer
53 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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1answer
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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44 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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47 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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220 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 ...
2
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2answers
91 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
42 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 $\det{(D\mathbf{u})}$...
4
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70 views

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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0answers
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?
2
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1answer
59 views

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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1answer
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 ...
3
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1answer
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 $\...
3
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2answers
246 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
105 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
35 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
164 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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67 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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92 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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1answer
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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19 views

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 ...