Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.

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Local minima: Sufficient conditions. Comparison of Calculus verses Calculus of Variations

My lecturer has written: Let $y=x^*+\epsilon \eta$ where $x^*,\eta,y\in \mathbb{R}^2$ $0\leq f(y) - f(x^*) = \epsilon V_1 + \epsilon^2 V_2 + O(\epsilon^3)$ $V_1 = \nabla f(x^*)\eta$ $V_2 = ...
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Constrained optimization minima and maxima and non-degeneracy answer check

Find the critical points of $$\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\f{f(\1,\2,\3)}\def\l{\lambda}$$ $$\f=\1\2+\2\3+\3\1$$ subject to constraint $\1+\2+\3=1$ First I will construct the Lagrangian: $$L ...
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1answer
48 views

How to calculate this functional derivative?

How can I calculate the functional derivative of this functional? $$F[x](t) = \int_{0}^{t}x(t_1)a(t_1)\left \{ \int_{0}^{t_1}x(t_2)b(t_2) \,dt_2\right \} dt_1 .$$ Where $a(t)$ and $b(t)$ are real ...
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27 views

Confusion with Euler-Lagrange Derivation

This is mostly a re-hash of this thread, but it did not receive an adequate answer. In the derivation that I am reading, there is one step that is not justified. Perhaps obvious, but it is not clear ...
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35 views

isoperimetric problem:how to solve the given question

Determine $y(x)$ for which $\int_{0}^{1} x^{2} + y^{'2}dx$ is stationary, subject to $\int_{0}^{1}y^2=2$, $y(0) = 0$, $ y(1) = 0$. how to solve it? I tried it: $f=x^{2} + y^{'2}$ and $g=y^2$ then ...
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extremal problem-how to check istrong minima,maxima condition

The functional $I[y(x)]=\int_{0}^{2}(xy^{'}+y^{'2})dx$,y(0)=1,y(2)=0 possess a.strong minima b.strong maxima c.strong maxima but not weak minima d.weak maxima but not strong minima How do we show ...
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199 views

Rayleigh-Ritz-method-how to solve the given problem

how to solve this: An approximate solution of the problem $y^"-y^{'}+4x\epsilon^x =0$, $y^{'}(0)-y(0)=1$,$y^{'}(1)+y(1)=-\epsilon$ is: here we have to calculate the value of y(x)? what i did is: ...
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32 views

how to find the optimal function with lagged cost? (calculus of variations)

I need to find the function $b( )$ that maximizes this guy ($c()$ and $\beta()$ are functions too, and $c()$ is convex): $$\int_{0}^{T} \! e^{-\delta v}\beta(v) \left[\int_{0}^{v} b(s) \; ds - ...
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Open problems in variational analysis/PDEs

I wasn't sure whether this question was more appropriate for StackExchange or Overflow, but in any case I would really appreciate it if any active researchers in the field responded. I'm a PhD ...
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49 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
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29 views

green function, functional derivative

I am trying to find ${\delta F}/{\delta u}$ for the functional: $F[u]=\int u(x)\int G(x,y)u(y)dy dx $ G is green function for laplace operator. is there Euler-Lagrange version for double intrgral? ...
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1answer
25 views

Maximization of ratio of two functionals

I am trying to find a function prescribed in polar coordinates $r = f(\theta)$ that maximizes the following quantity $$\frac{\int_0^{2\pi}r^3\cos\theta\, d\theta}{\int_0^{2\pi}r^4\, d\theta}$$ ...
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1answer
24 views

Least surface of volume with constraints

We know that in 2D/3D the shape with the least surface of a certain volume is a circle/sphere (e.g. soap bubbles). Now Imagine we have a flat surface (tabletop) that can be used as part of the surface ...
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32 views

Minimum travel time of a fuel-less train: brachistochrone problem

It is suggested that a rail network should include a frictionless tunnel where fuel-less trains run under gravity. The trains are released from rest at the point of departure and run freely until ...
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1answer
35 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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1answer
128 views

Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
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1answer
41 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
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23 views

Energy functional in Geodesic Active Contours

I have read some papers about Geometric active contours of the author C.Gout and Le Guyader [1] Segmentation under geometrical conditions using geodesic active contours and interpolation using level ...
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104 views

Minimize Energy in Image processing - Geodesic active contours

I've read some papers in Geodesic active contours (Image processing), which use the minimization of an Energy, consist of Internal Energy and External energy, for example, in the paper of Kass (Snake: ...
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Minimum surface attaching two parallel (non-planar) circles.

While studying for a comprehensive exam, I found this old problem: Consider two parallel coaxial wire circles, not in the same plane, to be connected by a surface of minimum area that is ...
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1answer
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Prove that the maximizing point configuration on the unit circle for a Vandermonde like functional is a picket fence

For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | \lambda_j - \lambda_k | $. I want to show that $H$ is globally maximized by ...
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58 views

Calculus of variations: big-O notation?

I have a formula in my text-book $$y(x+C) = y(x) + \frac{dy}{dx}C + O(C^2)$$ Can someone explain this formula?
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80 views

Shallow tent like soap film

A soap film circle in $x-y$ plane with center at origin can be carefully pricked with a blunt soapy pin at center and drawn out a little bit on $z$-axis forming a surface of revolution somewhat like a ...
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2answers
108 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
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A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
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2answers
162 views

The relationship between two definitions of star-shaped domain

There are two definitions of star-shaped domain. One is given in wikipedia as follows. Def1: A set $S$ in the Euclidean space $\mathbb{R}^n$ is called a star domain (or star-convex set, star-shaped ...
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Missing explanation in this paper of Masmoudi.

In this paper, on page 4, beginning in the line above 3.8, the authors begin a discussion of a given variational problem. I follow their argument until they begin the line of reasoning that begins ...
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175 views

Erroneous calculus of variations reference in V. I. Arnold's Mathematical Methods of Classical Mechanics?

The beginning of section 12, Calculus of variations (chapter 3, Variational principles) in V. I. Arnold's Mathematical Methods of Classical Mechanics (2nd edition, p. 55) reads: For what follows, ...
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38 views

Continuity of a functional with respect to two different norms

Let $J$ be a functional defined on $E = C^1[a,b]$ by $$J(y) = \int_a^b \sqrt{1 + (y^{\prime}(x))^2} \, dx.$$ Define the following two norms on $E$: $$\|y\|_{\infty} = \max_{a\leq x\leq b} |y(x)|$$ ...
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90 views

Maximizing a particular integral / functional

I have a (probably simple) question whose answer seems obvious but I cannot prove it. It relates to the calculus of variations. Let scalar $A = \Re[\int_a^bB(x)C(x)dx$], where $B$ and $C$ map ...
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1answer
75 views

Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
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Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
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Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
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How to prove $\gamma$ is continuous?

In the paper A remark on least energy solutions in $\mathbb{R}^N$, page 2407, it said, if $u_0\in H^1(\mathbb{R}^2)$, set $\gamma(t)=t^{-1/4}u_0(x/t)$. Then $\gamma(t)$ is a continuous path in ...
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Newton's method for the brachistochrone

Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are ...
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93 views

Variational Principles: Lagrange Multipliers

I am trying to minimize the functional $$I[\textbf{x}] = \int ||\dot{\textbf{x}}||^2 dt$$ subject to the constraint $\textbf{x}(t) \in \{\textbf{s} \in \mathbb{R}^3 : ||\textbf{s}|| = 1 \}$. The ...
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Does calculus of variations have a close connection to Feynman's ''differentiation under the integral sign''?

Most of the calculus I've studied seems separate math problems in to "derivative" or differential applications and integral applications. The one exception seems to be "calculus of variations," which ...
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Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
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Functional Extremum

Let a functional $H[\phi]$ of a map $\phi\in\mathbb{R}^{\mathbb{R}^4}$ be given by: $$ H(x^0) = \int_{\mathbb{R}^3} ...
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Finding maximum rate of change of total derivatives

consider $PV =nRT , P,V,T =$ pressure , volume , temperature respectively. $nR =$ constant let $n=R=1$ differentiate with respect to $t$ (time) $dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * ...
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Perturbation of the boundary of a strictly stable minimal surface

Let $\Sigma \subseteq \mathbb{R}^3$ be a minimal surface with boundary $\Gamma$. Now let us assume that $\Sigma$ is strictly stable, that is, $\lambda_1(\Sigma,L) >0$, where $L$ is the stability ...
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Why $\widetilde m = argmin_{m \in \mathcal P(X)} E[m]$ implies $\widetilde m (\arg \min \frac{\delta E}{\delta m}[\widetilde m]) =1$?

Consider $E: \mathcal P (X) \rightarrow \mathbb R \cup \{ \infty \}$ a functional (with a convex and dense domain, $E< +\infty$) over $\mathcal P(X)$ the set of probability measures of a metric ...
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Variation of the determinant of a Jacobian

I am following a derivation in a Calculus of Variation problem. After introducing a one-parameter family of one-to-one mappings from $R^{2}$ to itself, $$z({x},\epsilon)$$, $x = (x_1,x_2)$, such that ...
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Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$ Prove that ...
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43 views

The curve of shortest length bounding given area

Is formula #7 in this MIT OCW course incorrect? I think it should be $$f(x)=(\sqrt{1-(mx-c)^2}+d)/m$$ Also, presumably this answer to a very similar problem is also wrong. Because this is not the ...
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Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
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Spherical rearrangement

Let $u\colon\Omega\subset\mathbb{R}^N\to\mathbb{R}$ be a non negative measurable function, and $\Omega$ open and bounded. Consider $u^*$ the spherical rearrangement $$ u^*(x)=\sup\{t\geq0 : \mu\{x: ...
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Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
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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 ...
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2answers
262 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 refernce to this paper [Olivier Chapelle, ...