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

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1answer
39 views

Find the extremals of a functional of the form $\int^{x_1}_{x_0}F(y',z')dx$

I was working on Problem 3 in Ch. 2 of Gelfand & Fomin's Calculus of Variations, which reads: Find the extremals of a functional of the form $$\int^{x_1}_{x_0}F(y',z')dx$$ given that ...
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1answer
32 views

Calculation of the second variation of the functional $I(y)=\int_{-1}^1 [x^2(y')^2+x(y')^3]\,dx$

My question: I don't understand the last equation about second variation. According to definition, shouldn't it be $\int_{-1}^1 [2x^2+6xy'] (\eta)^2$? Can anyone help me with this? Where am I ...
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0answers
29 views

Calculus of variation: question about Lagrange multiplier?

In the example of Newton's problem with volume constraint. When using the method of Lagrange multipliers,the new Lagrangian is $L(x,h,h')=\frac{x}{1+h'^2}+\lambda xh $. I don't understand why we ...
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1answer
55 views

Differentiation under the integral sign problem

I have to take the derivative of the following function with respect to $\varepsilon$: $$\phi(\varepsilon)=\int_{a}^{b+\varepsilon C}F(x, y(x)+\varepsilon\eta(x), y'(x)+\varepsilon\eta'(x))\;dx$$ My ...
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1answer
26 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 ...
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0answers
89 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]\\ ...
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0answers
36 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
13 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
30 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
81 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 ...
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0answers
21 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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10answers
2k views

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
2
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1answer
144 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 ...
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1answer
47 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 ...
0
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1answer
22 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 ...
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1answer
56 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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2answers
156 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 ...
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1answer
93 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
51 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 ...
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1answer
59 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
56 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 ...
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1answer
52 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 ...
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1answer
69 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 ...
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0answers
33 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 ...
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2answers
36 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
30 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 ...
2
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0answers
46 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: $\beta\gt0$, ...
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1answer
300 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 ...
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2answers
223 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
76 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 ...
2
votes
1answer
84 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 ...
1
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1answer
40 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 ...
2
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1answer
50 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
39 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
97 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) ...
2
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0answers
122 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 : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
3
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0answers
87 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
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
72 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
50 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} ...
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1answer
59 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. ...
4
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1answer
60 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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0answers
78 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
30 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 ...
0
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1answer
72 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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0answers
49 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 ...
2
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0answers
76 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 ...
0
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0answers
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 ...
1
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1answer
98 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? ...