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

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Proof of Fundamental Lemma of Calculus of Variations

Let me preface this question by saying I'm actually a physicist, not a mathematician, so a lot of the language I see you guys using here is over my head, so if you can keep it simple, that would be ...
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2answers
110 views

To find an extremal of a given functional

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
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1answer
67 views

How to transform $ \int_a^b f(x^2+y(x)^2)\sqrt{1+y'(x)^2}\;dx$ into polar coordinates

I have the following homework problem (from Calculus of Variations course) : Show that if in $$ \min \int_a^b f(x^2+y(x)^2)\sqrt{1+y'(x)^2}\;dx$$ polar coordinates are used, then the ...
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1answer
53 views

Why stronger norm defines weak local minimizer? [closed]

Why the stronger norm defines weak local minimizer, while the weaker norm defines strong local minimizer? For example, when minimizing a functional on $C^1[a,b]$, one can also consider the weaker ...
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1answer
36 views

Special integrands in the calculus of variations

Most techniques in the calculus of variations that I know of, deal with integrands of the form $W(x, \phi(x), \nabla \phi(x)): \Omega \times \mathbb{R}^n \times \mathbb{R}^{n \times n} \to ...
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1answer
45 views

Solving a differential equation $F-y'F_{y'}=C$, with $F(y,y')= \frac{1+2y'^2}{3y^3\sqrt{1+y'^2}}$

If $$F= F(y,y')= \frac{1+2y'^2}{3y^3\sqrt{1+y'^2}},$$ where $y=y(x)$ and $y'= y'(x)=\frac{dy}{dx}$, then how to solve the differential equation: $$F-y'F_{y'}=C, $$ that is: ...
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1answer
36 views

Solving the functional $\min \int_0^1y^2y'^2\;dx,\;y(0)=0,\;y(1)=1$

I'm trying to solve the following problem: Determine smooth extremums in $$\min \int_0^1y^2y'^2\;dx,\;y(0)=0,\;y(1)=1$$ by (a) using the fact that the functional does not contain ...
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1answer
62 views

Taylor series expansion in calculus of variations

I am reading a book on calculus of variations, so I stumbled upon this integral, which the author expands by taylor series expansion, where $y$ and $y'$ are functions of $x$ and $\tilde{y}(x) = y(x) ...
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113 views

Find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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51 views

Complex solution to Euler-Lagrange equation?

I'm currently working on Calculus of Variations and I came across an integral which I had to minimize. The integral I have to minimize is $$\int_0^1(1+y'^2)^2\,dx$$ After getting the Euler-Lagrange ...
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46 views

Solving the functional $\min \int_0^1xy^2+x^3y\;dx$

I'm having a course on Calculus of Variations and I'm doing my first homework problems. One of them is the following: Determine the smooth extremum which satisfies the boundary conditions for: ...
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1answer
65 views

Question about integral of the product of two continuous functions.

I'm having a hard time understanding why the following lemma is true: If a $f(x)$ is continuous on $[a,b]$, and if $$\int_a^b f(x)g(x) \,dx = 0 $$ for every function $g(x)$ continuous on ...
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1answer
41 views

A question of variational inequality on H. Brezis' functional analysis book.

On page 134, H. Brezis gives an example of the connection of minimization problem and variational inequality. Here I quote: Suppose $F$ : $\mathbb R \to \mathbb R$ is a differentiable function and ...
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56 views

Is $(-\Delta)^{s}$ c0incident with $(-\Delta)^{s/2}$?

We already know the following facts: $$\displaystyle (-\Delta)^su(x):=c_{n,s}\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy, $$ where $s\in (0,1)$. $$\int_{\mathbb{R}^N} ...
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1answer
42 views

How to find upper bound Bessel's zero by Rayleigh quotient?

I'm trying to find an upper bound of the first zero (not including $0$) of Bessel's function of orden zero, $J_0$. The method proposed is using the Rayleigh quotient evaluated at a simple function. ...
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55 views

The constraint subset of $H_0^1(\Omega)$ is a $C^1$-submanifold.

This problem comes from the constraint problem in CoV. (the lagrange-multiplier case) Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. We define the sub-manifold $$ M:=\{u\in ...
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39 views

The best constant of Poincare inequality can be determined by eigenvalue of Laplace operator

Given $\Omega\subset \mathbb R^N$ be open bounded and smooth boundary. Then for $u\in H_0^1(\Omega)$ we have well-known Poincare inequality $$ \int_\Omega \lvert u\lvert^2dx\leq C\int_\Omega ...
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1answer
29 views

Some calculation details in elliptic PDE operator.

Define function $f$: $\mathbb R^N\to \mathbb R$ by $$f(\xi):=A\xi\cdot\xi $$ where $A$ is $N \times N$ uniformly elliptic matrix, i.e., $A\xi\cdot\xi\geq \theta\lvert\xi\lvert^2$ for some ...
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1answer
48 views

Using Direct method to prove Rayleigh Quotient Theorem

Define the elliptic PDE operator $Lu:=-\partial_j(a_{ij}\partial_iu)+cu$ where $A=(a_{ij})$ is and uniformly elliptic matrix and $c\geq 0$, i.e., $A\xi\cdot\xi\geq \theta \lvert\xi\lvert^2$ for ...
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38 views

The $p$-Laplacian is strongly monotone

I am studying the solution of $p$-Laplacian by finding the minimizer of the following energy, among the space $W_0^{1,p}(\Omega)$, $p\geq 2$, $$ E[u]=\frac{1}{p}\int_\Omega \lvert\nabla ...
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69 views

Regularity energy minimizing harmonic maps

I am using the book "Geometric Measure Theory- An introduction" by Fanghua and Xiaoping. I'm studying the proof of the following Lemma (Lemma 2.1.8 page 38). This chapter is dealing with the theory ...
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13 views

Existence and interchange of integrals

Let $F: [a,b] \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ written as $F(t,u,p)$ and $F$ is continuous w.r.t $t$ and $C^1$ w.r.t. $u$ and $p$. Let $u\in AC([a,b],\mathbb{R}^d)$ Then the ...
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52 views

Maximum moment of inertia of arc

Is there is variational calculus solution to the problem of maximum moment of inertia of a wire of uniform density per unit length $s$ between two fixed endpoints, about z-axis in 3-Space? ...
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58 views

Is there a way to graphically show that a solution is the minimum or stationary solution to a functional?

I'm looking for the functional analogue to the visual representations of function optimization you most commonly see. To illustrate, if we have some function: $$ f(x) = (x-1)^2+1 $$ We can look at ...
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2answers
76 views

Dirichlet Principle in Sobolev Spaces

According to Zeidler, 1995, in his book "Applied Functional Analysis: Application to Mathematical Physics". Dirichlet problem is a problem to minimize $$F(u)=\frac{1}{2}\int_G(u')^2\ dx-\int_G fu\ ...
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1answer
20 views

Extremizing a functional with a non-elementary solution to the Euler-Lagrange equation

Someone recently asked this question, but deleted it before I could post an answer. I figured I might as well share it for future internet travellers. The question is problem $10$ here. It states ...
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Calculus of Variations with discontinuous Lagrangian

Consider the classical problem of extremizing a functional of the form $$S[x] = \int_a^b L\left(t,x,\dot{x}\right)\ dt.$$ In almost all cases of consideration, the integrand $L$ is considered to be a ...
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1answer
86 views

Solve Integral equation with convolution

I have to solve the following integral equation \begin{align*} \int_{-\infty}^\infty e^{-y^2} \log \left( \int_{-\infty}^\infty e^{-(y-x-t)^2} f(t) dt\right) dy=-cx^2 \end{align*} where $c$ is some ...
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1answer
40 views

Finding the unique weak solution of Non-linear boundary problem

We are given the equation \begin{cases} -\Delta u=0&x\in \Omega\\ \partial_\nu u+\beta(u)=0&x\in\partial\Omega \end{cases} where $\Omega$ is bounded bounded smooth boundary and $0<a\leq ...
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Reference request: calculus of variations

I am searching for a good book to self-study calculus of variations. It should be fairly complete; build up gradually from the very basics; offer detailed explanations; have some emphasis on ...
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31 views

Calculus of Variations-First and Second Order Deviations

I'm new to Calculus of Variations and the Method of Least Action (L=T-V) What I'm unsure about is how first and second order deviations are used in finding the least action. I know it's used to find ...
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2answers
80 views

Calculus of Variations. Lagrangian Hamiltonian Mechanics Mathpages.

Over at http://www.mathpages.com/home/kmath523/kmath523.htm is an article about Lagrangian and Hamiltonian Mechanics with a derivation of the Euler-Lagrange equations of motion. Mid-way through is ...
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1answer
112 views

Functional Derivative (Gateaux variation) of functional with convolution

I have the following functional \begin{align*} F[f]=\int f(x) \log(g(x)) dx$ \end{align*} where $g(x)$ is given by convolution $g(x)=y(x) * f(x)=\int y(\tau) f(x-\tau) d\tau$, so \begin{align*} ...
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Independence of function and its derivative in calculus of variations

It's common to see in calculus of variation that the integrand $f$ of functional $F[y]=\int f(y,y',x)dx$ is a function of $y,y'$ and $x$. Why do we regard the derivative $y'$ as an independent ...
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A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
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22 views

Fixed vs. variable functions in functional variation

Given the functional $$F[y,y'] = \int L[y(x),y'(x),y_0(x),y_0'(x)]\ dx\ ,$$ I want to find the function $y(x)$ that extremizes $F$ for known functions $y_0(x)$ and $y_0'(x)$. Since both $y_0(x)$ and ...
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34 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
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1answer
118 views

Minimization of Variational - Total Variation (TV) Deblurring

Under the Linear Blurring Model - $ f = H \ast u $. I'm trying to calculate the Euler Lagrange of with respect to $ u $ of the functional: $$ E \left( u \right) = {\left\| f - H \ast u ...
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52 views

How to find the curve extremizing a given functional?

Given a functional $$I(y)=\int_1^2 {\frac {\sqrt{1+(y'(x))^2}}{x}}dx ,$$ with $y(1)=0$ and $y(2)=1$. How to find the curve extremizing this functional?
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Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation

For learning purposes, I'm trying to prove that the shortest function passing through the two points $(x_1, y_1)$, $(x_2, y_2)$ is a straight line, without using the Euler-Lagrange equation. My ...
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1answer
25 views

Smooth and Lipschitz domains

We know that an open ball $B_{r}\subseteq R^{n}$ is a smooth domain. It follows that this is a Lipschitz domain. How can I show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is ...
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1answer
40 views

Solving for f(t) in presence of f'(t)

Here's the situation: I have a function $$e(t) = \frac{a~d(t)}{b + d(t)}$$ with first derivative $$e'(t) = \frac{a~b~d'(t)}{[b+d(t)]^2}$$ where $a$ and $b$ are constants. For a given constant $K$ I ...
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1answer
68 views

What does this sentence regarding the Riemannian metric mean?

I am slowly working through a text on ordinary differential equations and I don't understand what this particular exercise is even asking of me. The exercise says to determine the geodesics in ...
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83 views

Solving differential equation by weak formulation and minimizing a functional

I want to give a weak formulation of the boundary value problem \begin{align*} -(c(x)(u'(x)-1))' & = 0 \textrm{ on } \Omega = (-1,1) \\ u(-1) = u(1) & = 0 \end{align*} where $c(x)$ is ...
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29 views

Minimum of functional

Find for which $w:\mathbb{R}^2\rightarrow \mathbb{R}$ attains following functional a minimum $$ F[w] = \int_0^1 \frac1{p(x)} \left(\int_0^1 w(x,y)f(x,y)\,dy\right)^2 \,dx + \int_0^1 ...
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0answers
31 views

Ritz Approximation

I have the following problem: $1/r * d/dr(r*d \theta/dr) - N_1(\theta-\theta_a)^2 -N_2(\theta^4-\theta_a~^4)=0$ where $N_1, N_2,$ and $\theta_a$ are constants Subject to $r=1$ : $\theta=1$ $r=L$ ...
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29 views

Euler Lagrange of a Curve

Let $C(s) = (x (s), y(s))$ be a closed curve inside a plane where $s$ is the parametric arc length parameter. What is Euler Lagrange equation for the following functional $$-\int_0^L \nabla C ds$$ ...
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25 views

second variation of a modified Dirichlet energy

Given: $E(h)=\int (\frac{1}{2}h_x^2-F(h))dx$ $f(h)=F'(h)$ and $f'(h) $is integrable $\int \phi dx=0$ (may not need this) Trying to show: $E(h+\epsilon\phi)=E(h)+\epsilon \int h_x \phi _x \, ...
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34 views

Euler Lagrange Equations for the following minimization problem

Say I have a curve C in the plane Using Euler-Lagrange I am trying to solve the following minimization problem: $$\int \int (E - R) \nabla C dx dy$$
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Euler Lagrange of the following equation.

I've worked the Euler -Lagrange for the following equation $$E(u(x,y),v(x,y)) = \int \int (I_x u + I_y v + I_t) + \alpha ((u_x^2 + u_y^2) + (v_x^2 + v_y^2))dx dy$$ and got the following equations: $$0 ...