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

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0
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1answer
19 views

Regularity of limit measure and prove that $|\mu_h|\stackrel{*}{\rightharpoonup}|\mu|$

I have some questions. First of all, let $\mu_h$ a sequence of Radon measures and suppose that $\mu_h$ weakly-converge to another measure $\mu$. Now, this limit measure $\mu$ is still Borel? Is it ...
0
votes
2answers
60 views

How to Invert the Euler Lagrange Equations?

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations: ...
0
votes
2answers
30 views

Minimum curve for the distance between two points at the plane

The problem is to determine the curve y=y(x) in the plane, the lenght of which is given by the functional: \begin{equation} I(y)=\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx=\int_{x_1}^{x_2}F(x,y,y')dx ...
0
votes
1answer
24 views

Calculus of Variation: Euler-Lagrange Equation in 1D

I am currently trying to get into calculus of variation for a course of Image processing. In the lecture we learned that a smooth function u[a,b]->R that minimises: $$\int_a^b F(x,u,u') dx $$ ...
0
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1answer
25 views

Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...
0
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0answers
46 views

The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
2
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0answers
31 views

How to use find the Lagrange Multipliers in multidimensional Calculus of Variations

Suppose I wish to minimise the integral $$I = \int_{s_0}^{s_1}\int_{t_0}^{t_1}F\, dt ds$$ Where $F$ is a function of the six variables $x(s,t)$, $y(s,t)$, and their four partial derivatives, ie $$F ...
0
votes
1answer
53 views

Euler-Lagrange equation with constraints outside the integral

So I've been studying Euler-Lagrange equations, and on an assignment I have the problem to find them for $J(y)=\int_a^bF(x,y,y')dx-By(b)+Ay(a)$ Where $y(a)$ and $y(b)$ are free, $A$ and $B$ are ...
2
votes
1answer
84 views

Surface of constant mean curvature

From PDE Evans, 2nd edition: Chapter 8, Exercise 12: Assume $u$ is a smooth minimizer of the area integral $$I[w]=\int_U (1+|Dw|^2)^{1/2} \, dx,$$ subject to given boundary conditions $w=g$ on ...
0
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0answers
15 views

Definition of the Second Variational Derivative In terms of The first

I know that for functional $F$ the first variational derivative at $f$ with increment $h$ is defined as \begin{align*} \delta F[f,h]= \lim_{\alpha \to 0 } \frac{F[f+\alpha h]-F[f] ]}{\alpha }. ...
3
votes
2answers
41 views

dirichlet principle: why $u-g\in W_0^{1,2}(\Omega)$?

Let $D\subset\mathbb R^n$ be open and bounded. Consider $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial\Omega$. Let $g\in W^{1,2}(D)$ and $f\in L^\infty(D)$. Then the minimizer of $$ I(u)=\int_\Omega ...
1
vote
0answers
25 views

Calculus of variations - unilateral constraints [duplicate]

Question about Evans states, chapter 8.4.2! We have $I[w] := \int_U \frac{1}{2}|Dw|^2 - fw\, dx$, among all functions $w$ belonging to the set $$\mathcal{A} : = \{w \in H_0^1(U) : w \geq h \, ...
1
vote
1answer
58 views

The functional $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ is weakly lower semicontinuous

I am studying calculus of variation, and I need to prove that $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ with $f \in L^2(U)$ is weakly lower semicontinuous on $H_0^1(U)$. In classes, I only ...
0
votes
1answer
15 views

Euler equation for the functional has the form: $f_y-f_xy'-\frac{fy''}{1+y'^2}=0$

I want to show that the Euler equation for the functional $J(y)= \int_a^b f(x,y) \sqrt{1+y'^2}dx$ has the form: $$f_y-f_xy'-\frac{fy''}{1+y'^2}=0$$ $$L(x,y,y')= f(x,y) \sqrt{1+y'^2} dx$$ ...
0
votes
1answer
46 views

optical flow Euler-Langrange equation

I have a problem understanding how optical flow functional is plugged into Euler-Lagrange equation. The functional is: $\iint[(I_xu+I_yv+I_t)^2+\alpha^2(||\nabla u||^2 +||\nabla v ||^2)]dxdy$ ...
1
vote
0answers
31 views

Geodesic equation for surface of sphere

One of the standard problems of calculus of variations is showing that geodesics on the surface of the sphere are great circles. But I don't understand the equation. The equation for great circle ...
0
votes
1answer
37 views

Dirichlet energy

From PDE Evans, 2nd edition: Chapter 8, Exercise 17: Let $u,\hat{u} \in H_0^1(U)$ both be positive minimizers of the Dirichlet energy $$I[w] := \int_U |Dw|^2 \, dx,$$ subject to the constraint ...
10
votes
2answers
262 views

Elliptic regularization of the heat equation

This is from PDE Evans, 2nd edition: Chapter 8, Exercise 3: The elliptic regularization of the heat equation is the PDE $$ u_t - \Delta u -\epsilon u_{tt}=0 \quad \text{in }U_T, \tag{$*$}$$ where ...
1
vote
0answers
61 views

Derivation of Von Karman Equations

I'm reading Howell's Applied Solid Mechanics to gain background for a research project. I'm struggling with the following derivation in the text that the authors refer to as a "lengthy exercise." The ...
0
votes
1answer
102 views

Euler-Lagrange equation [duplicate]

This is PDE Evans, 2nd edition: Chapter 8, Exercise 2: Find $L=L(p,z,x)$ so that the PDE $$-\Delta u + D\phi \cdot Du = f \quad \text{in }U$$ is the Euler-Lagrange equation corresponding to the ...
0
votes
1answer
49 views

Calculus of Variations - Function dependent of $y'$ only

I completely understand the proof for the Euler-Lagrange equation for a general function $F(x,y,y')$. However, when I try to use the same proof technique on a function $F(y')$, I run into a curiosity ...
4
votes
2answers
241 views

Maximum area of a fenced playpen on the side of a house.

Here's an interesting problem: you just got a really cute puppy, and you want it to have a large rectangular playpen to run around in. What's more, your neighbor just happened to have 100 feet of ...
0
votes
0answers
16 views

Calculus of variational in 2 dimension with constraints

Let $S$ be a 2D region and its boundary is $\partial S$. $u(x,y)$ is defined in S. The functional is of the following type: $J[u] = \int_S F(x,y,u,u_{x},u_{y}) \mathrm{d}s + \int_{\partial S} ...
7
votes
1answer
137 views

What function maximizes area for a constant arc length?

Suppose I have a continuous function $f$, such that $f(0) = f(1) = 0$. Given the length $l$ of the curve between $0$ and $1$, which function maximizes the area under the curve? I know that if $l \leq ...
1
vote
0answers
67 views

How to prove $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$

It was shown in the book "Finite Elements Methods for Navier-Stokes Equations" by Girault and Raviart that $$H_0^1(\Omega)=H_0(\operatorname{div};\Omega)\cap H_0(\operatorname{curl};\Omega).$$ The ...
2
votes
0answers
45 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
0
votes
1answer
49 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...
0
votes
1answer
14 views

Minimizing constrained functions on $l^p$

Suppose we have some functionals $H,G:l^p(\mathbb{N}^+)\to\mathbb{R}$, and we want to find some $p \in l^p(\mathbb{N^+})$ which minimize $H$, subject to the constraint that $G(p)=0$ is constant. As ...
0
votes
0answers
32 views

An MCQ involving Rayleigh - Ritz method for the functional $I(y) = \int_{0}^{1}(\frac{1}{2}(y^{'})^2 - y)dx$

An MCQ involving Rayleigh - Ritz method for the functional $$I(y) = \int_{0}^{1}(\frac{1}{2}(y^{'})^2 - y)dx$$ Let $y_\text{app}$ be polynomial approximation, involving only one coordinate ...
1
vote
1answer
19 views

An MCQ for finding the extremal of the functional $J = \int_{a}^{b} F(x, y, y^{'})$

Consider a functional $$J = \int_{a}^{b} F(x, y, y^{'}),$$ where $F(x, y, y^{'}) = \frac{1 + y^{2}}{(y^{'})^2}$ for admissible function $y(x).$ Which of the following are extremals for $J$? $y(x) = ...
4
votes
1answer
57 views

Intuition of weak star convergence.

Given $\Omega=(0,1)$, consider the following sequence $$ v_j(x)\colon=\begin{cases} \;a &\text{if }jx-\lfloor jx \rfloor\le\theta\\ \;b &\text{otherwise} \end{cases} $$ where ...
2
votes
1answer
82 views

Functional Maximization

So how do we solve a problem like this: Find the function $s(x)$ such that $s(x)$ maximizes $$\int_0^{s^{-1}(k)} s(x) dx $$ where $x\in[0,10]$, $s(x)\in[0,1]$, and $k\in[0,1]$ ($k$ is a constant). ...
0
votes
0answers
47 views

Ideal shape for underwater habitat

Is there an analytic solution to this problem or do I need to compute a discrete approximation using a relaxation procedure - or something similar? I want to find the shape of a roughly spherical ...
1
vote
1answer
46 views

Calculate the (variational) derivative of the following equation;

Consider $ E[u]= \int^1_0 \big(u'(x)\big)^2+\big(u(x)\big)^2-2f(x)u(x) dx.$ Calculate the variational derivation for a function $v$; in other words, calculate $\frac{d}{d\epsilon}E[u+\epsilon v]$ at ...
0
votes
0answers
14 views

An MCQ to determine extremal of a functional $J(y) = y^2(1) + \int_{0}^{1}(y^{'})^2(x)dx,$ [duplicate]

The following problem occurs in an exam: Consider a functional $J(y) = y^2(1) + \int_{0}^{1}(y^{'})^2(x)dx,$ $y(0) = 1, $ where $y \in C^2 ([0, 1]).$ If $y$ extremizes $J$, then $y(x) = 1 ...
3
votes
2answers
72 views

Advice on second order non-linear ordinary differential equation

I'm currently working on some problems concerning the calculus of variations and I have come up with the following differential equation that I now want to solve: $$1 + y'(x)^2 - y''(x)(y(x)-\lambda) ...
2
votes
1answer
55 views

Divergence identity

From PDE Evans (2nd edition), page 515, we are given $$\sum_{i=1}^n \left(\left(Du \cdot x + \frac{n-p}p u \right)p|Du|^{p-2}u_{x_i}-|Du|^px_i \right)_{x_i}=0. \tag{10}$$ Then the author goes ...
2
votes
2answers
54 views

Lagrangians independent of $x$

In PDE Evans, 2nd edition, the following formula is printed as equation $\text{(9)}$ in §8.6 (on page 514): $$\sum_{k=1}^n (L_{p_i}u_{x_k}-L\delta_{ik})_{x_i}=0 \quad (k=1,\ldots,n) \tag{9}$$ ...
2
votes
2answers
85 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the ...
1
vote
3answers
43 views

Proving an identity

Given $a,b\in\mathbb{R}$ with $a < b$ and defining $F(z):=\int_0^z f(s) \, ds$ with $z \in \mathbb{R}$, how can one establish that $$F(a+b)=F(a)+f(a)b+ b^2\int_0^1 (1-s)f'(a+sb) \, ds,$$ which is ...
4
votes
1answer
126 views

solution of $y^{\prime \prime} + y^n = 1$ [closed]

I am not able to figure out the solution for the differential solution $$y^{\prime \prime} + y^n = 1$$ I want to specifically find an answer for $$y^{\prime \prime} + y^2= 1$$and $$y^{\prime \prime} + ...
0
votes
1answer
39 views

Potential energy of a hanging string of a prescribed length

Consider a homogeneous, flexible string of a prescribed length hanging in a vertical plane where its ends are fixed at two points P and Q. Determine the equilibrium configuration of the string by ...
3
votes
1answer
60 views

What is a functional? And how is it defined for the length?

Im reading about Calculus of varations and there is a lot of references to "the functional" i.e we want to find the minimum of the functional etc. From what i have read, "the functional" is simply the ...
2
votes
1answer
21 views

Does $\log$ minimize this functional for its Abel equation?

Suppose that we have the functional equations ("Abel equation", it is called) for a function $F: [1, \infty) \rightarrow \mathbb{R}$ given by $$F(1) = 0$$ $$F(ex) = F(x) + 1$$ where $e$ is the ...
4
votes
1answer
70 views

Calculus of variations question with two variables

If $u(x)$ and $v(x)$ satisfy $u(0)=1$, $v(0)=-1$, $u(\pi/2) =0$, $v(π/2) =0$ on extremals of functional $$ \int_0^{\pi/2}\left[\big({\frac{du}{dx}\big)^2 +\big(\frac{dv}{dx}\big)^2 +2 \,u v ...
4
votes
1answer
101 views

Calculus of variations: two integrals

I would like to find the extrema of the following integral with respect to $u\left(s\right)$: ...
1
vote
0answers
47 views

What's the maximum speed of snake so that the frog can escape?

Suppose there's a round pond, a frog which can swim as 1 meter / second, and a snake that moves along the pond ridge but cannot swim. If the frog can reach any point on the ridge of the pond before ...
0
votes
1answer
46 views

A weird Calculus of Variations problem

I became stuck with the following Calculus of Variations problem. The problem is related with something called as the "Nadaraya-Watson" model in statistics. We have $N$ inputs ${x_n}$ and each of ...
0
votes
0answers
47 views

What's the “real” definition of variational derivative

This seems a notational question. Given a functional $S[f]=\int L(f,f';x)\ dx$, I want to derive $\delta S[f]$. There are quite a lot of literature interchanging integrate and variation. That is, ...
0
votes
0answers
38 views

How to take derivative of integral of square matrix function

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+ \int |A^TG(x)-C^TJ(x)|^2 (1-H(x)) \, dx+\lambda_1 A^2+\lambda_2 B^2+\lambda_2 C^2$$ where $A^T$ is transpose of vector $A$. $A$ ...