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

learn more… | top users | synonyms

9
votes
1answer
79 views

Proof Nehari manifold of semilineal subcritical $-\Delta u = f(u)$ in $\Omega$ is not empty.

Given the problem $$ \left\{ \begin{array}{rll} -\Delta u& = f(u) & \text{in }\Omega \\ u & = 0 & \text{in } \partial\Omega \end{array} \right. $$ In a bounded domain $\Omega\subset ...
-1
votes
1answer
33 views

Domain and Range of a Functional

Define the functional $$F[y] = \int_0^1 f(x,y)p(x) \, dx$$ 1)What is the domain and range of the functional if $p(x) = 1$? 2)How does it change for more general functions $ρ$? My work: 1) The domain ...
0
votes
1answer
30 views

Extremal of functionals

By a functional we mean a mapping from some space of curves to $\mathbb{R}$. A functional can have infinitely many extremals or none. If $F$ is the length of curve connecting the north and south pole ...
2
votes
0answers
39 views

Fundamental lemma of calculus of variation, about hypothesis

We can find on the web several forms of the fundamental lemma of calculus of variation, the simplest one I could find (French wikipedia ) is: for $f\in C^1([a, b])$ $$ \int_a^b f(x) g(x) dx = 0, ...
0
votes
0answers
23 views

Deduce Euler-Lagrange equation

I'm trying to get the Euler-Lagrange equation: $$\frac{d}{dx}\Big(\frac{\partial\mathfrak{L}}{\partial \dot f}\Big)-\frac{\partial \mathfrak{L}}{\partial f}=0$$ But I can't see this: Let $p$ and $q$ ...
0
votes
1answer
33 views

Euler-Lagrange equation and unknown coefficients

I want to show that the nonlinear functional $$ J(u) = \int_0^1 (u'(x))^2 + b(x)u^2(x) + f(x) u(x) \,\textrm{d}x $$ attains its minimum in exactly one point of the Sobolev space $W_0^{1,2}(0,1)$. ...
1
vote
0answers
33 views

Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
2
votes
2answers
64 views

Solving Euler-Lagrange Equation with delta function

I am trying to understand a physical system and have arrived at the following equation: $$\mathcal{S} = \int_{z = -\infty}^{z = \infty} dz \left\lbrace f_\rho[\rho] + \dfrac{m}{2} \mid ...
0
votes
1answer
34 views

How to compute a geodesic (homework question)

I am interested to compute the a geodesic in the Euclidean space $\mathbb{R}^1$. I am not familiar with differential geometry but I know that a geodesic is the solution of the following optimization ...
0
votes
0answers
56 views

Norm in finding local extrema for functional

In "The Calculus of Variations" by Bruce van Brunt, he says: Let $J:C^2[x_0,x_1]\to\mathbb{R}$ be a functional of the form $$J(y)=\int_{x_0}^{x_1}f(x,y,y^\prime)dx,$$ where $f$ is a function ...
0
votes
0answers
20 views

An example of convergence to Young measures

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\lam}{\lambda}$ I am trying to prove the following claim: Let $\{u:[0,1]\to \mathbb{R} \mid u \, \, \text{ is differentiable a.e}, u(0)=u(1)=0 \}^{*} $. ...
0
votes
2answers
28 views

Chain rule for variational derivatives and differentiation of an integral?

Assume that I have the following functional: $$ F[u_1,u_2,...u_N,\nabla u_1,...,\nabla u_N,...;t]=\int_{\Omega} f( u_1(x,t),u_2(x,t),...,u_N(x,t),\nabla u_1 , \nabla u_2,... ) dV $$ where $\Omega ...
0
votes
0answers
14 views

Using the Fourier Series in Variational Optimization Problems

Say I have a functional $L(f)$ which takes as input the function $f:\mathbb{R}\to\mathbb{R}$, and I want to find the function that optimizes $L$. Unfortunately, there's no way to define a functional ...
0
votes
1answer
21 views

How to calculate the variation of a matrix?

Suppose we have two diagonal matrices $$ A_{\mu \nu}=\left(\begin{array}{cccc} \rho(t) & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0 ...
1
vote
2answers
69 views

Geodesics in Poincare Disk

I would like to find the geodesics in the Poincare disk. I know that the metric is $$\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$$ so $$s=\int \frac{\sqrt{1+y'^2}}{1-x^2-y^2}\, dx$$ Then I try to find y(x) using ...
0
votes
0answers
18 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
0
votes
1answer
25 views

Chain rule in partial derivatives

I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow To be more specific, it's the diferentiation of (6.9) right at the ...
0
votes
0answers
43 views

Ordinary differential equations as variational problems

Considering an ordinary differential equation of first order in the implicit form $$ F(q(t),\dot q (t))=\alpha,\,\,\, q(0)=q_0 $$ with $\alpha,\, q_0$ constants, what is the relation of the solution ...
0
votes
0answers
26 views

For a convex function, why does lower semicontinuity imply weak lower semicontinuity?

I have come across the statement that, for a convex function, all notions of lower semicontinuity are equivalent. That is: weak lower, sequential lower, and weak sequential lower semicontinuity are ...
2
votes
1answer
25 views

A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
1
vote
1answer
22 views

Calculus of variation for geodesic

I need to minimize $$J[v]=\int\sqrt{P(x)+R(x)(v')^2}dx$$ By Euler equation, I get $$\frac{d}{dx}\frac{Rv'}{\sqrt{P+Rv^{'2}}}=0$$ Then I need to solve a complex ODE, but I don't know how to deal ...
1
vote
1answer
28 views

calculus of variations or optimize over function form

I have a question about optimizing the following quantity over function form . Given unknown function $f(\theta)$ such that $f(\theta)\geqslant 0$ and $\int f(\theta)d\theta\leq \infty$. And ...
0
votes
0answers
22 views

Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
0
votes
0answers
38 views

How to obtain the closed form solution to this nonlinear system of ODEs

I have the following simple but nonlinear ODE system problem below: $$\left\{ \begin{align} &f'(t)=a\cdot (1-f(t)-g(t))^{13/10}\tag{1}\\ &g'(t)=b \cdot f(t) \tag{2}\\ &f(0)=0,\quad ...
1
vote
1answer
42 views

test function and boundary condition

For example, if we consider the Dirichlet energy $\int\frac 12 |\nabla u|^2$ and the solution space as follows: $$X=\{u\in W^{1,2}(\Omega) \text{ | } u = 0 \text{ on } \partial\Omega \}$$ , then the ...
2
votes
0answers
19 views

Does this mean a new boundary condition for ODE and how to handle it?

In order to to calculate desired light path in continuous medium with gradient refraction index, for schematic see the Figure below. $O:(0,0)$ is the disk center of light source $\odot{O}$ with ...
1
vote
0answers
63 views

Extreme of functional - Calculus of variations: Euler-Langange equation

Find all extremes of $$I[y] = y(0)^2 \int_{0}^{1} y(y')^2 dx,$$ with the given initial condition $y(1)=0$. I was thinking of writing $y(0)^2 = -\int_{0}^{1}2yy' dx$ and then what (differentiating ...
5
votes
2answers
186 views

Book on applied mathematics/analysis

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
2
votes
1answer
39 views

Find the function that minimizes $\int_{0}^{1}e^{-(y'-x)}+(1+y)y'dx$

Suppose among all the continuously differentiable functions $y(x), x\in \mathbb{R}$, with $y(0)=0$ and $y(1)=\frac{1}{2},$ the function $y_0(x)$ minimizes the functional, ...
1
vote
2answers
128 views

Calculus of Variations Problem involving mixed constraints

Motivation Let $X$ be $\mathcal{N}\Big(-\frac{\sigma^2}{2},\sigma^2\Big)$ random variable, i.e. probability density function $f(x)$ is given by \begin{equation} f(x)=\frac{1}{ \sqrt{2\pi\sigma^2} } ...
0
votes
0answers
12 views

Formulation of boundary constrained minimal surface

Using standard notation of classical surface theory how is the standard Plateau problem formulated as an iso-perimetric one minimizing area for given boundary length $$ \int \sqrt{ E \, du^2 + 2 F ...
0
votes
1answer
89 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
1
vote
1answer
44 views

Shortest path to the apex of a cone

This is something I thought about today but have no idea how to approach. We are given a right circular cone with lateral length L and angle at the base $\alpha$. A curve along the surface of the ...
1
vote
2answers
52 views

Which advanced mathematics book do you recommend for a college starter?

I'm supposed to start my preparation for the college admission math exam so I'm looking for a good explanatory/textbook/problem book which covers most of or all of precalculus topics with a hint of ...
1
vote
1answer
27 views

Any critical point $u_0\in M$ of $I|_M$ satisfies $I'(u_0)=\mu\gamma'(u_0)$

Consider $I:H\rightarrow\mathbb{R}$ defined by $$I(u)=\int_0^R\left\{\dfrac{1}{2}u_r^2-\xi u^2+\ln(1+u^2)\right\}r\,dr,$$ $\xi\in(0,1)$, where $H$ is the completion of $$X=\left\{u\in ...
0
votes
0answers
19 views

Calculus of variation: Maximize fuctional

"Among the curves of given length l, at the upper half-plane, passing through the points (-a, 0) and (a, 0) , find the one that encloses the largest surface area together with the space [-a,a] ." I ...
1
vote
0answers
19 views

Palais-Smale condition for a functional

Consider $I:H\rightarrow\mathbb{R}$ defined by $$I(u)=\int_0^R\left\{\dfrac{1}{2}ru_r^2-\gamma ru^2+r\ln(1+u^2)\right\}dr,$$ $\gamma\in(0,1)$, where $H$ is the completion of $$X=\left\{u\in ...
0
votes
0answers
7 views

Variation of a d'alambertian operator

Let $M$ be a pseudo Riemannian manyfold, $H$ be function of a scalar curvature $R$. Assume that variation of the metric tensor and it's first derivatives is zero on the border $\partial M$. Which ...
2
votes
0answers
47 views

Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf. Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \begin{equation} ...
4
votes
1answer
139 views

Obtaining an estimate for the Lagrangian $L=H^*$ from the Hamiltonian $H$

This is from C. Evans' PDE book, page 130. The convex function $H:\mathbb{R}^n\to\mathbb{R}$ is $C^2$ and satisfies $$ H\big(\frac{p_1+p_2}{2}\big) \leq \frac{1}{2}H(p_1) + \frac{1}{2}H(p_2) - ...
2
votes
2answers
48 views

How to verify this identity?

From Weinstock, "Calculus of Variations", p.24: We have the readily verifiable identity \begin{align}\frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right) = ...
1
vote
2answers
49 views

Does Constant factor rule in integration hold for functionals?

The constant factor rule in integration states the following relation is valid $$\int a f(x)dx=a \int f(x) dx$$ for all constants $a$(or $a$ that are constant functions of x, that is ...
2
votes
2answers
89 views

Why does $\mathrm{tr}(\mathrm{ln}g_{\mu\nu})$ vary as $g^{\mu\nu}\cdot\delta g_{\mu\nu}$ under $\delta g_{\mu\nu}$?

For a pseudo-Riemannian manifold, under the variation $g_{\mu\nu}\mapsto g_{\mu\nu}+\delta g_{\mu\nu}$, the determinant $g=\mathrm{det}g_{\mu\nu}$ varies as $$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}$$ ...
0
votes
1answer
41 views

Euler-Lagrange equation of energy of length function on Riemann manifold

$(M,g)$ is a Riemann manifold. $\gamma:[0,1]\rightarrow M$ is a curve.The length of $~\gamma $ is $$ L(\gamma)=\int^1_0 ||\dot\gamma (t)||_g ~dt $$ The energy is $$ E(\gamma)=\frac{1}{2}\int^1_0 ...
0
votes
1answer
46 views

Variation under constraint

I always can't compute right.$u=u(x),R=R(x)$ and $\tau$ is constant, and $M$ is compact manifold.If $u$ is the minimizer of $$ \inf\{\int_M [\tau(4|\nabla u|^2+Ru^2)-u^2\ln ...
1
vote
1answer
35 views

Variation with constraint condition

For example, let $$ I[u]=\int_M |\nabla u|^2+u dV $$ It's not hard to compute the variation of $I[u]$. If $I[u]$ reach the minimum at $u_0$, I can get that $$ i(t)=I[u_0+tv] \\ i'(0)=0 \\ ...
1
vote
0answers
50 views

Maximization of the Expectation of a function

Recently I was thinking in general on how to maximize the expectation of a function (not necessarily a utility function, but apparently this is a common case). To give an idea of the kind of problem, ...
1
vote
0answers
36 views

Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
5
votes
0answers
89 views

Does this inequality hold? Proof / Counterexample

Does the following inequality $ \int_0 ^\infty x^2 |\frac{d}{dx}f(x)|^2 dx - \int_0 ^\infty x |f(x)|^2 dx + 2\pi (\int_0 ^\infty x^2 |f(x)|^2 dx) (\int_0 ^\infty x |f(x)|^2 dx) > -\frac{1}{8\pi} ...
2
votes
1answer
21 views

Movable end points Calculus of Variation.

Given problem is $J[y] =\int_{0}^{x_1}y'^2dx$ with $y(0)=0$ and $y(x_1)=-x_1-1$. After solving Euler Lagrange equation I got $y=Ax+B$ . And using first boundry conditon I got $y=Ax$ We have ...