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

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A homogeneous double integral equation

I've happened to stumble upon an interesting double integral equation: $$ 0=\int_0^\ell\int_0^\ell f(s,t)\mu(t)\mu^\prime(s)\,ds\,dt $$ Here $f$ and $\mu$ are at least continuous (if you want higher ...
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65 views

Minimizing distance between two curves. Can the Calculus of Variations be used?

Given two curves, one might want to find the minimum distance between two points. It is fairly straightforward to find minimums of the function $$(x_1-x_2)^2+(y_1-y_2)^2$$ which corresponds to the ...
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1answer
47 views

Conceptual problems when minimizing a simple functional

I have a problem with what seems a very simple functional maximization. Let's define: $$ J[z]=\int \left( u(z)-\frac{\dot z^2}{2} \right) dt $$ Where $u(z)=-z^2+5$. The problem is to find $$ ...
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1answer
45 views

variational calculus with probabilistic boundaries

I'm interested to find the solution to the following variational problem: $$ J[y]=\int_{T=0}^{\infty}\int_{t=0}^{T}L(t,y(t),y'(t))p(T)dtdT $$ where $p(T)$ is a probability distribution function ...
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23 views

Find the variation of this integral functional

Find the variation of $A(Y) = 2\pi\int \limits _0 ^1 |(Y(x)|\sqrt{1+Y'(x)^2} \Bbb dx$ with respect to $Y$. I have no idea how to solve this problem.
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33 views

$\text{min }\frac{1}{2} \int (f')^2$ in $C^1([0,1])$ given both D and N boundary conditions.

Does there exist a minimizer in $C^1([0,1])$ (or $H^1([0,1])$) for $$\frac{1}{2}\int (f')^2 dx, \text{ given the boundary conditions: } f(0)=0, f(1) = a, f'(1) = b?$$ When $a=b$, we have the ...
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32 views

Sturm-Liouville Variational Problem

I'm entirely clueless with this problem. No formal training in variational methods. Show that for function $\phi\left ( x \right )$ with $$\phi\left ( a \right )=\phi\left ( b \right )=0$$ and ...
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1answer
36 views

Differentiation under an integral with respect to a function

Consider the functional $F$ defined via the integral $$ F(\mu)=\int_0^\ell\int_0^\ell f(s,t)\mu(s)\mu(t)\,ds\,dt. $$ How would I differentiate this with respect to $\mu$? I realize that this has ...
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1answer
26 views

Are these critical points minima to the variational problem?

Let $\Omega\equiv (0, 1)\times(0, 1)\subset\mathbb{R}^2$ and consider the variational integral \begin{equation*} I[u]\equiv\int_{\Omega}\frac{1}{2}|Du|^2\ \mathrm{d}x-\frac{5\pi^2}{2}|u|^2\ ...
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2answers
41 views

Variation Problem with Euler-Lagrange Differential Equation

I'm just trying to understand that type of equations, and I can't solve this, kind of a simple minimization problem. Maybe someone can help me ? Here is my equation $$\eqalign{ & A(y(x)) = ...
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1answer
66 views

Energy functional and Euler Lagrange equation

We know that for potential energy functional, its derivative is called the Euler Lagrange equation and physically, it means that at the given point there is a force balance. Now if the energy ...
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38 views

Time it takes a particle of $2x^2+3y^2+4z^2=9$ to reach the xy plane using derivatives.

Consider in $\mathbb{R}^3$ the surface $2x^2+3y^2+4z^2=9$. Suppose a particle leaves the point $(1, 1, 1)$ located in the surface along the normal at that point to the $xy$ plane at a speed of one ...
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9 views

Rewriting the reparametrization of a function as a sum

Say I have some function $$\mu:[0,\ell]\to\mathbb{R}^+,$$ and a bijective function $$\phi_\varepsilon:[0,\ell]\to[0,\ell].$$ Let $s$ represent the variable in $[0,\ell]$ and say that ...
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1answer
21 views

Find the differential (if exist) of the function $h(\vec{x}) = \frac{f^3(\vec{x})+f(\vec{x})g^2(\vec{x})}{f^2(\vec{x})+g(\vec{x})}.$

Let $A\subset\mathbb{R}^n$ a noempty open set and $\vec{x}_0\in A$. Let $f,g:A\to\mathbb{R}$ two differentiable function in $\vec{x}_0$ so that, $g(\vec{x})>0$, $\forall \vec{x}\in A$. Consider ...
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1answer
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 $tu_t$ ...
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1answer
25 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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57 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) = ...
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1answer
38 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
41 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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1answer
31 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 ...
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1answer
67 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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62 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 ...
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1answer
81 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 ...
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1answer
24 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 ; ...
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153 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 ...
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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
26 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
32 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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23 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 ...
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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 ...
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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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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
36 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
35 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
45 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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1answer
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 = $ ...
2
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0answers
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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1answer
44 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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2answers
218 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
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 ...
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1answer
35 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 ...
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2answers
65 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 ...
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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
56 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
32 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 ...