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

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1answer
22 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
0
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0answers
22 views

Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
1
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0answers
20 views

Mountain pass theorem

Let $I$ be a real functional over a Hilbert space $H$, satisfying all the conditions in the Mountain pass (M-P) theorem. My question is, can the assumption in the M-P theorem that $I[v]\leq 0$ for a ...
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0answers
10 views

How to minimize $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $

I'm new in optimizations and i am trying to understand how to obtain $ v $ that minimizes $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $ where $\rho(x)$ - continuous ...
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2answers
28 views

Understanding part of a theorem of Calculus of Variations

I have trouble understanding the following statement (From Gelfland's Calculus of Variations book): If $\phi[h]$ is a linear functional and if ...
0
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1answer
36 views

Why Dirichlet's energy uses a **squared** norm?

$E = \int_{\Omega}\left \| \nabla u(x)\right \|^2 dx$ So, Dirichlet's energy measures the integral of the squared norm of the gradient. Why squared norm? What would we get if we use just a norm? It's ...
0
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1answer
43 views

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 ...
4
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1answer
315 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
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0answers
10 views

Problem about deformation theorem

I'm reading Evans PDE, on chapter 8.5 the proof of deformation theorem about the calculus of variation. On page 504 Evans wrote on the top: "we verify that the map $u\to dist(u,A)+dist(u,B)$ is ...
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0answers
11 views

Gateaux but not Frechet differentiable functional [duplicate]

For functional between Banach spaces X,Y: By Gateaux differentiable at $u\in X$ I mean that there exists bounded linear operator $dF(u)$ s.t. $F(u+t\xi)-F(u)=dF(u)\xi+o(t)$ for all $\xi\in X$. For ...
0
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1answer
26 views

Functionals' Taylors Theorem

Consider functional $F:B\to \mathbb{R}$, where B is a Banach space eg. $B=H^{1}(\mathbb{R}^{d},\mathbb{C})$. Then Taylor's theorem for functionals is: Suppose that the line segment between u ∈ ...
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8answers
6k views

Why Circle encloses largest Area?

In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
2
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0answers
20 views

Check whether the extremal has weak minima or weak maxima

The functional $$\int_0^1(y'^2 + x^3)dx,$$ given $y(1)=1,$ achieves its weak maximum on all its extremals weak minimum on all its extremals weak maximum on some, but not on all of its extremals weak ...
2
votes
2answers
79 views

Calculus of Variations: Understanding functional derivative

I am trying to understand the basics of the Calculus of Variations and the first thing to understand is the functional derivative. I failed to find a good introductory material, so I am trying to make ...
0
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1answer
35 views

What does the term “perturb” mean?

I've been studying Calculus of Variations and I came a cross with the term "perturb" in my study material, but the term was not defined. The sentence where I read it from was: "Rigid extremals are ...
2
votes
1answer
113 views

Null Lagrangians and “Local Degree”

Let $u: U\subset\mathbb R^n \rightarrow \mathbb R^n$ be a smooth function, $U$ bounded. Let $x_0$ and $r$ be such that $B_r(x_0)$ is disjoint from $\partial U$. Let $\eta$ be a smooth bump function ...
0
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1answer
18 views

Functional derivative - understanding some basics

I have the following functional $$ L[u] = \int_0^l dx [-\frac{\lambda}{2}u^2 + \frac{1}{4}u^4] = \int dx J[u]$$ Now, I need to calculate $$ \frac{\delta L}{\delta u} $$ As I understand, since I can ...
0
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0answers
22 views

solve $J(y)=y^{2}(1)+\int_{0}^{1}y^{'}{^2}(x)dx$? [closed]

Consider the following linear functional: $$J(y)=y^{2}(1)+\int_{0}^{1}y'{^2}(x)dx,$$ where $y(0)=1$ and $y \in C^{2}[0,1]$. If $y$ extremizes $J$ then, 1. $y(x)=1-\frac{1}{2}(x^{2})$ 2. ...
0
votes
1answer
67 views

Checking: finding extremals for a functional

I'm trying to find the extremals of the functional $$J[y] = \int_0^1 (y')^2 + y^2 + 4ye^x \, {\rm d}x,$$ imposed that $y(0) = 0$ and $y(1) = 1 $. I got that there can't be extremals, and that's weird ...
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2answers
33 views

Infimum of $\frac{||u'||^p_{L^p}}{||u||^p_{L^p}}$ for $u \in W^{1,p}_0((0,1))$

Good afternoon everyone! It is very easy to show that the infimum mentioned in the title is strictly positive, but it seems much more difficult to show that it is attained within the Sobolev space of ...
0
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1answer
45 views

Integration by parts problem

If $\textbf{x}\in \Omega \subset\mathbb{R}^n,$ where $\Omega$ is a bounded open set, $u:\Omega\rightarrow\mathbb{R}, \;\eta:\Omega\rightarrow\mathbb{R},\;u'=\nabla u = ...
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0answers
34 views

How does integration by parts work with multivariable functions

How does integration by parts work with multivariable function? Lets say I have the functions $f(\textbf{x})$ and $g(\textbf{x})$, where $\textbf{x}\in\mathbb{R}^n$. How would integration by parts be ...
1
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1answer
48 views

Problems with Calculus of Variations lecture material

I'm having trouble understanding the derivation in my Calculus of Variations course material and I was hoping if someone could clarify this out. Here is my reference (as I have rewritten it, the ...
6
votes
3answers
774 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
0
votes
1answer
47 views

What does $f(u)=\min!$ mean in calculus of variations?

I have a very simple notation related question. There are notes to calculus of variations [specifically: Zeidler's book "Nonlinear Functional Analysis and its Applications II/B" page 506] which states ...
2
votes
0answers
100 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 : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
0
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0answers
29 views

Constrained Trajectory Optimization

Context: Let $X(\mathbb{R}^3)$ be space of paths connecting point $A$ and point $B$. Then the minimization of a functional $S:X(\mathbb{R}^3)\to \mathbb{R}$ given by ...
0
votes
2answers
33 views

Is it possible to solve for $y(x)$ from $\min \int _a^bf(x^2+y^2)\sqrt{1+y'^2}\;dx$

I have the following problem: 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 problem will be converted into one that ...
3
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1answer
26 views

Solving a functional problem $\min\int_0^1(ay^2+2byy'+cy'^2)\;dx,\;y(0)=0, y(1)=1$

I have the following problem in my Calculus of Variations course: Find all smooth extremums if $a,b$ and $c$ are positive numbers $$\min\int_0^1(ay^2+2byy'+cy'^2)\;dx,\;y(0)=0, y(1)=1$$ I ...
1
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1answer
34 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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0answers
9 views

Calculus of variations with double integral, inner integral's upper limit is outer variable of integration

I want to find the function $\gamma(t)$ that makes the following stationary. $S(t) = \int_0^t \gamma(t'') \Big(\beta\int_0^{t''}(1-\gamma(t'))dt' + e_0\Big) dt''$ Constraints: $\beta\gt0$, ...
1
vote
1answer
44 views

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 ...
2
votes
2answers
31 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 ...
1
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1answer
53 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 ...
1
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1answer
27 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?
11
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2answers
349 views

Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
2
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1answer
39 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: ...
1
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0answers
64 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 ...
1
vote
1answer
32 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 ...
0
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1answer
41 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) ...
3
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0answers
27 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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0answers
45 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: ...
1
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1answer
50 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 ...
1
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1answer
36 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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0answers
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} ...
1
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1answer
24 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. ...
3
votes
3answers
417 views

Procedure for Gâteaux Derivative with functionals

Not after an answer, just the method/procedure as I'm stumped... We have the functionals: $$ T[y] = \int_2^3 \left( 3\left| \frac{dy}{dx}\right|^2 - 8y \right)dx $$ $$ S[y] = \cosh(T[y]) $$ Now, to ...
4
votes
1answer
53 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 ...
0
votes
1answer
28 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 ...
0
votes
0answers
21 views

Solution of a Volerra type equation

If $y(t)=1+\int_0^{t} y(v)e^{-(t+v)}dv$ then $y(1)=$ (a) 0 (b) 1 (c) 2 (d) 3 It is a Volterra equation. To solve it we apply successive approximation method or Resolvent kernel method. But we ...