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

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Difficulty in solving calculus of variations problem.

I am solving a problem on calculus of variation in which $F(x,y,y')$ is given as $F(x,y,y')=e^yy'^2$ After solving Euler equation I got this $2y'' +2y'-y'^2=0$. I don't know how to proceed further. ...
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38 views

textbook on calculus of variation which focuses on the following topics

I need a textbook (or set of online lecture notes) on calculus of variation which focuses on the following topics "Variation of a functional, Euler-Lagrange equation, Necessary and sufficient ...
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27 views

Calculate $\Delta J$ for a functional

$J(y)=\int_{0}^{1} (x^2-y^2+(y')^2)dx$ $y(x)=x, h(x)=x^2$ I need to calculate $\Delta J$ and I am given this from the answer key: $\Delta J = J(y + \epsilon h)-J(y) = J(x + \epsilon x^2) - J(x)$ ...
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35 views

When do minimizers exist?

I'm trying to solve a problem set for my functional analysis course and I'm stuck at the following problem: Decide if the following problem has a minimizer Let $g\in C^0([0,1])$. Minimize ...
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28 views

Is this correct use of Lagrange multipliers?

Given the standard isoperimetric problem: Minimize the functional $$ A[y]=\int_a^bF(y,y',x)dx $$ subject to the functional-constraint $$ B[y]=\int_a^bG(y,y',x)dx = c=\text{constant}$$ (where $ ...
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2answers
20 views

Continuous representative for functions in $W^{1,2}(\mathbb{R})$

I want to prove that $K(x,y) = \frac{1}{2}e^{-|x-y|}$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$ and as a hint I have given that for $f\in W^{1,2}(\mathbb{R})$ I should use its continuous ...
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28 views

Variational calculus with inequality constraints

I'm interested in finding the solution to the following variational problem: $$ J[y]=\int_{t=0}^{T}L(t,y(t),y'(t))\,dt $$ subject to the constraint: $$ y'(t) \ge 0 $$ My question is what are the ...
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83 views

Feynman problem on action

It is very weird for me that a newbie can ask a new (may be silly, sorry...) question but must have 50 reputation to comment. When I see a good question like this but have no answer what I have to ...
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26 views

Weak derivative of $\int_0^x g(t)\,dt$ is $g$ on $(0,1)$

I'm working on a functional-analysis problem set, and the question is: Let $g\in L^1(0,1)$, and define $f(x) = \int_0^x g(t)\,dt$. Show that $f\in W^{1,1}(0,1)$ and that the weak derivative of $f$ ...
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77 views

Minimization of Expected Value

I'd like to know how I can minimize, with respect to $\hat{y}(x)$, $$ \DeclareMathOperator{\Tr}{Tr} \mathbb{E}_{p(x,y)}[(\hat{y}(x)-y)^2 + (\hat{y}(x)-y)\Tr(\nabla^2_x\hat{y}(x)) + ...
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15 views

Total variation of a vector valued measure

If I have understood correctly, a vector valued measure $\mu$ is simply a vector of measures, that is $\mu=(\mu_1,\dots,\mu_n)$, where $\mu_i$ is a possibly signed measure on the measure space ...
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51 views

Examining $\inf \int_0^1(x')^{2}(t)\mathrm dt$

I need to examine $$\inf \int_0^1(x')^{2}(t)\mathrm dt$$ with such conditions: \begin{cases} x(0)=0, x(1)=1 & \\ \int_0^1x(t)dt=0 & \end{cases} I started with writing Euler-Lagrange ...
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86 views

Maximization of a complex function with real outputs

Let $a$ and $b$ be functions from $\mathbb{C}$ to $\mathbb{R}$. Suppose that for some real $\alpha$ and $\beta$, there exist complex $z$ and $w$ such that $a(z) = \alpha$ and $b(w) = \beta$. Given ...
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38 views

How would you solve this surface integral?

Suppose you had the surface integral $\iint \limits_{A} = x^{3}(1-x^{4}-y^{4})dx \ dy$ where $A$ is the region defined by $x \geq 0, \; y \geq 0, \; x^{4}+y^{4} \leq 1$. How would you solve this ...
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15 views

What does it mean “Distance between $k$-planes induced by the identification plane-projection matrix”?

I'm reading some parts of Functions of bounded variation and free discontinuity problems by Ambrosio, Fusco, Pallara. At the very beginning of page 82 there's written "Let $G_k$ be the complete ...
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22 views

Can't understand where the second term in this derivative comes from

Given an integral $$S=\int_{\theta_1}^{\theta_2}q(p,t,\theta)f(\theta)\:d\theta $$ My understanding is that I can partially differentiate this w.r.t., say, t, in the following way: $$\frac{\partial ...
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16 views

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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44 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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42 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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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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21 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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$\text{min }\frac{1}{2} \int (f')^2$ 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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29 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
31 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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21 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
38 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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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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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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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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50 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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23 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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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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30 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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39 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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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
57 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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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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Geodesics in a hyperbolic plane like space

For $|\rho| < 1$ and $\sigma >0$ consider the Riemannian metric \begin{equation} g:= \begin{pmatrix} \frac{1}{\left( 1-\rho^2 \right)y^2} & \frac{-\rho}{\sigma\left( 1-\rho^2 \right)y^2} \\ ...
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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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17 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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101 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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36 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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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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21 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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16 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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22 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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25 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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31 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 ...