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
508 views

Determine the minimum and maximum values of an integral subject to end conditions

The question: determine the minimum and maximum values of the integral $$\int_0^1 yy'dx$$ subject to the conditions $y(0)=0$ and $y(1)=1$. There is no explicit y dependence, so our Euler-Lagrange ...
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0answers
16 views

Variational Calculus with Discrete Objective

I'm trying to infer a smooth, non-negative function from some given data ($\vec{m},\vec{\alpha},\vec{\beta}$). That is, I want to solve (I think) $$ \mathop{\arg\!\min}_{g \in ...
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0answers
40 views

Show that the graph of the function $z(x,y) = x\tan(y)$ is a minimal surface

Show that the graph of the function $z(x,y) = x\tan(y)$ is a minimal surface I'm really lost on how to do this question. I know we have to use the Euler equation to show this, but other than that ...
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15 views

Question about trace operator

From the general trace theorem we know for instance that if $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$, then there exists a function $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega}=f$. But is it ...
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1answer
560 views

Sufficient conditions for functional extrema

In the calculus of variations, we can develop a sufficient condition for a functional $J: S \to \mathbb{R}$, $$J(y) = \int_a^b f(x,y,y') \, dx$$ to have a local maximum or minimum, where $S \subseteq ...
2
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1answer
86 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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0answers
11 views

Prove Ladyzhenskaya- Babuska-Brezzi condition for Poisson problem with homogenoeus Dirichlet boundary condition

I'm considering the problem: \begin{equation} \label{eq:PM} \begin{cases} \mathbf{u} -\nabla p=0\quad \text{ in } \Omega\\ \mathrm{div} \mathbf{u}=-f \quad \text{ in } \Omega\\ p=0\quad \text{ in } ...
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1answer
30 views

Solving a 2-variable Second Order Linear Partial Differential Equation

Part 1: Initial Problem I am trying to solve the following partial differential equation. $$py + q= a\frac{\partial f}{\partial x} + by\frac{\partial f}{\partial y} + c\frac{\partial^2f}{\partial ...
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3answers
770 views

Find the extremal to the functional $J(y) = \int_{0}^{1} ((y')^2 -y)dx$ and discuss whether they provide a max/min

I am having a hard time getting my head around Functionals and Calculus of Variations, My question is: Given a functional and using the Euler-Lagrange equation to find an extremal, how do we show ...
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0answers
41 views

circle tangent path [closed]

Here is my question. What is the shortest path which starts at the center of a circle and visits all of the tangents at some point. Let the path be given by $[x(t),y(t)]$, for any given tangent ...
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0answers
69 views

Green's first identity and the calculus of variations

UPDATE: I was able to solve this problem using iterative integration by parts. However, I still cannot find how Green's first identity would apply here. Suppose I had a multiple integral over ...
3
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0answers
88 views

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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0answers
42 views

Calculation of Variational Derivative

Following is from Olver's book on Lie groups and Differential Equations: Define the variational problem: \begin{eqnarray*} \mathcal{L}= \int_\Omega L(x,u^{(n)}) \end{eqnarray*} where $u^{(n)}$ ...
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0answers
22 views

Can be the derivative the marginal variation of a function?

I know that derivative means in other words slope of the function. $$f(x) = x^2-4x+4$$ $$\frac{d}{dx} = 2x-4$$ So the slop of function at the point 3 is: $2*(3)-4=2$ Can be the derivative the ...
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0answers
15 views

Simply Supported Beam Deflection

I would appreciate confirmation on my method and answers for part a), also if anyone knows how to solve this problem that would be appreciated.
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0answers
8 views

Factoring Weierstrass Excess function

I have a Lagrangian $$ L(x,\dot{x}) = \dot{x}^2(1+\dot{x})^2 $$ For this I have the Weierstrass Excess function as $$ E(x,\dot{x},x') = x'^2(1+x')^2 - \dot{x}^2(1+\dot{x})^2 - ...
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0answers
39 views

Explanation of this integral identity in the proof of Wirtinger's inequality from Hardy-Littlewood-Polya

I report the following excerpt from the book "Inequalities" by Hardy-Littlewood-Polya, page 184, where Wirtinger inequality is proven using variational methods. I'm trying to understand what is the ...
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2answers
20 views

Equilibria of the system Calculus of Variations

The harmonic oscillator is described by the action functional $$J[x] = \int_{t_0}^{t_1}(mx'^2 −\frac{1}{2}kx^2) dt$$ where m is the mass and k is the spring constant. a. Show that the equation of ...
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0answers
19 views

calculus of variation

Among all curves joining a given point (0, b) on the y-axis to a point on the x-axis and enclosing a given area S together with the x-axis, find the curve which generates the least area when rotated ...
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0answers
16 views

Time independent vs. time dependent lagrange multiplier

What are the differences between these two in applications? For example: $$max\sum_{t=0}^{\infty} \beta^t u(c_t)$$$$s.t.f(c_t,c_{t+1},x_t,x_{t+1})=0$$ What are the differences between: ...
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1answer
658 views

DuBois-Reymond Lemma

I know thats the following statement is true. $f,g$ are continuous function $[a,b]$.Suppose $\int\limits_a^bf(t)h(t)+g(t)h'(t) \, dt=0$ for every $h$ belonging to $C_0^{\infty}[a,b]$, then $g$ is ...
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0answers
30 views

The definition of the First Variation - Calculus of Variation

I have the following definition of the functional derivative $ \frac{\delta S}{\delta\gamma}$, where $S$ is my functional and $\gamma$ is a curve: $$\tag{1} \int^B_A \frac{\delta S}{\delta\gamma} ...
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0answers
28 views

One dimensional obstacle problem - how to determine coincidence set

I was wondering whether someone could comment on my line of reasoning and, if possible, point me to some relevant literature etc. In general any help will be much appreciated! Suppose $\Omega = ...
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1answer
481 views

Proof of fundamental lemma of calculus of variation.

Suppose $\Omega$ is an open subset of $\mathbb{R}^n$ and let $L^1_\text{Loc}\Omega$ denote all locally integrable functions on $\Omega$ and $C^{\infty}_0\Omega$ for smooth functions whose support lie ...
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1answer
25 views

$J[y]=\int_a^bF(x,y,y')dx$ with constraint and free boundary

Suppose the variation problem $$J[y]=\int_a^bF(x,y,y')dx$$ with free boundary and constraint $\int_a^bG(x,y,y')=l$, how can formulate the corresponding Euler-Lagrange equation? For fixed boundary ...
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1answer
34 views

Why the need of Sobolev spaces in this proof of isoperimetric inequality?

I was reading the chapter about isoperimetric inequalities in DaCorogna's book "Introduction to The Calculus of Variations". The isoperimetric inequality is proved to be equivalent to Wirtinger ...
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1answer
41 views

Is This Linear Functional Bounded?

In the space of $L^1[0,1]$, is the following linear functional $I:L^1[0,1]\rightarrow \mathbb{R}$: $$ I(f) = \int_0^1 x^3f(x)dx $$ bounded on this section of the norm ball $\{f:\|f\|_1 = 1 \text{ ...
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0answers
31 views

Where does Jacobi's accessory equation come from?

I'm reading Charles Fox's An Introduction to the Calculus of Variations and in section 2.4 he just suddenly introduces Jacobi's accessory equation and I don't understand where it's coming from. ...
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1answer
12 views

Quasiconvexity (in the sense of Morrey) implies Rank-One convexity

I am trying to understand why Quasiconvexity implies Rank-One convexity. In a standard proof of this fact a sequence of functions is constructed, which converges weakly to zero in $W^{1,p}.$ in ...
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0answers
23 views

Making some mistake in finding the solution to the catenary

I seem to be getting a little bit different solution to the catenary problem. Maybe I'm just not seeing how its equivalent to the regular solution. Here's what I've got Starting with $$U=\lambda g ...
0
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1answer
28 views

isoperimetric problem in variational methods

I have a problem that requires to find the smallest perimeter of a curve that goes trough points a,b and encloses an area A between the curve and x axis. I used the calculus of variations approach. ...
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0answers
20 views

Weierstrass-Erdmann corner conditions well explained

I am for the moment taking a course in calculus of variations and I find it very hard to understand the concept of broken extremals and Weierstrass-Erdmann condition if someone would give a nice and ...
2
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2answers
149 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
2
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1answer
68 views

Why is finding a closed form of a geodesic so difficult?

What is it about geodesics (does this term apply in $R^n$ or just $R^3$?) that makes finding a closed form of the curve so difficult? My thoughts: The minimal curve may not be unique which ...
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2answers
35 views

Minimizing a functional of two functions with three boundary conditions

What are the natural boundary conditions for the following calculus of variations problem: Minimize: $$J[y] = \int_0^b (1+(y_1')^2 + (y_2')^2)) \,dx$$ subject to the boundary conditions $$y_1(0) = 0 = ...
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1answer
51 views

Isoperimetric problem with constraint

Solve the isoperimentric problem to minimize $$\int_0^1 y(x)\,{\rm d}x$$ subject to the constraint $$\int_0^1 \sqrt{1 + y'(x)^2}\,{\rm d}x = \frac{\pi}{2}$$ with $y(0) = 0$ and $y(1) = 0$. My Work: ...
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0answers
16 views

Exact Galerkin solution

Let's consider the following linear variational problem (a < b) : and its finite element Galerkin discretization, based on a trial and test space on some uniform mesh M of [a, b]. the space ...
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1answer
27 views

From Euler-Lagrange equation to time-dependent problem

I am reading this pdf which is about an image denoise model. Essentially, we want to find a function $u$ such that $u$ minimize the following functional: $$F(u) = \lambda \int_\Omega |f-u|^2 \,dx\,dy ...
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0answers
15 views

Find the smooth extremal for variational problem

Given the Lagrangian function: $L(t,x,x')=e^t(1+x'^2)$. where $x'$ is the time derivative of x. Find the smooth extremal for the Lagrangian function and determine whether any two points in the ...
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1answer
23 views

calculus of variations - connecting functionals in different regions.

This is a seemingly small but annoying problem I can't seem to find an answer to. I have a problem of the form $$J(y) = \int^{x_1}_{0} F(x,y,y') \,ds + \int^{x_2}_{x_1} G(x,y,y') \,ds$$ where the ...
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0answers
21 views

Lifting the Einstein-Hilbert action into the frame bundle

If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some ...
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1answer
28 views

Proving a trajectory is critical for an action functional

Suppose a particle with mass $m$ moves freely in $\mathbb{R}^{n}$ under the influence of a conservative force field with potential $\Phi\in C^{2}([t_{1},t_{2}]\times\mathbb{R}^{n})$; that is, its ...
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0answers
51 views

Example of an oscillation Young measure

I'm taking a course in which Young measures are introduced for oscillation and concentration. I have understood the examples the lecturer has given us for concentration Young measures, but cannot get ...
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0answers
20 views

Why can every twist map be realized as the time-1 map of a time-dependent Hamiltonian?

if have problems getting my head around the following claim made by Moser in "Monotone twist mappings and the calculus of variations" and Gole in "Symplectic twist maps". Setting: Let $F : ...
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0answers
20 views

Extremals of a Functional with two functions

Find the extremals of the functional $$J[y, z] = \int_0^\frac{π}{2} ((y')^2 + (z')^2 + 2yz) \,dx$$ subject to the boundary conditions $y(0) = 0, y(\frac{π}{2})= 1, z(0) = 0, z(\frac{π}{2}) = 1$ Do ...
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0answers
22 views

Can I use calculus of variations in this dynamical system

I have a dynamical system of the form $$\dot{\underline{x}} = \underline{f}(\underline{x})$$ where $\underline{x} = \underline{x}(t) \in R^4$ and $\underline{f}$ is smooth. With one of the systems ...
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0answers
35 views

Variational inequalities and their relation to weak formulations of PDEs

this topic is continuously boggling me as I go through material of a class on nonlinear PDEs. I will illustrate the broader issue with the following example. Consider the reaction diffusion equation ...
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30 views

Functional analysis with KKT conditions

I want to solve an optimization problem min $F(x_{ik}) $ subject to $x \in X$. $F$ here is a function or functional that I wish to determine. I want my optimal ...
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0answers
58 views

A calculus of variation problem with Non-homogeneous Boundary Conditions

Find function $u(x,y,z)$ so that the function $I(u)$ to be optimized $$I(u)=\iiint_{\Omega}(u_x^2+u_y^2+u_z^2-xyzu)\,\mathrm dx\,\mathrm dy\,\mathrm dz $$ that $$\Omega=\{(x,y,z):0\le x,y,z\le1\}$$ ...
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1answer
59 views

A calculus of variation problem with an obligation: minimizing $I(x,y,z)=\int_0^1 \sqrt{\dot x^2 +\dot y^2 +\dot z^2} dt$

I have to minimize this : $$I(x,y,z)=\int_{0}^{1} \sqrt{\dot x^2 +\dot y^2 +\dot z^2} dt$$ with this obligation : $$z=x^2+y^2$$ And : $$x(0)=y(0)=z(0)=0$$ $$x(1)=y(1)=z(1)/2=1$$ I tried some ...