# Questions tagged [calculus-of-variations]

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

412
questions

**3**

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134 views

### Gelfand “Calculus of Variation” 1.7 question on definition and purpose of variational derivative

In Gelfand Calculus of Variation, chapter 1.7, the variational derivative is defined as:
$$\left.\frac{\partial J}{\partial y}\right|_{x = x_0} = \lim_{\Delta\sigma \rightarrow 0}\frac{J[y+h]-J[y]}{\...

**1**

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**0**answers

28 views

### Pontryagin's principle with Lebesgue-integrable control

Does there exist a (weak) version of Pontryagin's minimum principle in which the control is allowed to be just Lebesgue integrable? I am mostly familiar with the 1975 text of Fleming & Rishel, ...

**2**

votes

**0**answers

105 views

### Optimization of functionals with constraints

I have a minimization problem as follows:
$\min\left( \int_0^1\int_0^1\beta(t)\beta(s)G_1(t, s)dtds\right)^{1/2}+\left( \int_0^1\int_0^1\beta(t)\beta(s)G_2(t, s)dtds\right)^{1/2} $
$\texttt{s.t.}\;\;\;...

**2**

votes

**1**answer

102 views

### Fundamental Theorem of Gamma-Convergence

Let $(X, d)$ be a metric space and $F_\varepsilon, F\colon X\to [-\infty, \infty]$. Suppose $F_\varepsilon$ is an equicoercive sequence of functions on $X$, i.e. for all $t\in\mathbb{R}$ there exists ...

**5**

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124 views

### Lavrentiev phenomenon between $C^1$ and $C^2$

Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is
$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \hspace{1cm}$ or possibly $ \hspace{1cm} F(y)=...

**1**

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**1**answer

161 views

### Stability of isoperimetric inequality

Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1.
Isoperimetric inequality states that then the volume of $S$ is not greater than $V_n$,
where $V_n$ is the volume of a ball in $\mathbb{R}^n$ with ...

**2**

votes

**0**answers

69 views

### Variational problems living in two different Sobolev spaces

Is there a general reference concerning variational problems living in $W^{h,p}\times W^{k,p}$, with $h, k\in\mathbb{N}_0$ not coinciding? I'm thinking to problems of type:
$$\inf_{u,v}\int_{\Omega} ...

**1**

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**0**answers

50 views

### Higher order Leibniz rule and ordered multiindex notation

Although I think this is probably known, I am making here a short exposition on the multiindex notations I am using to make this question self-contained. I note that there is at least two different ...

**1**

vote

**1**answer

93 views

### Calculus of variations for double sum with Lagrange multiplier

This cropped up in a research question I'm tackling.
I wish to solve the following optimization problem:
$$
\text{minimize}\ \sum_{i=1}^\infty f_i \sum_{j=1}^i \sqrt{f_j}
\quad\text{subject to}\ \sum_{...

**2**

votes

**0**answers

69 views

### Minimisation in dual Sobolev space

Say we have a function $F(\lambda) = \|f(\lambda)\|_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are ...

**4**

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115 views

### Wasserstein distance and Monge-Kantorovich-Rubinstein duality

The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by
$$
W_p^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...

**1**

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88 views

### Is there a concentric map from the disk onto the ellipse with constant sum of singular values?

$\newcommand{Vol}{\text{Vol}}$
Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{...

**2**

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**0**answers

60 views

### Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$

By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy:
$$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...

**7**

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**0**answers

163 views

### On the “Collected Works” of Charles Bradfield Morrey, Jr

Why Charles Bradfield Morrey, Jr.'s "Collected works" haven't been published yet?
I've been thinking of this question for a while, at least from the first time I started to improve the ...

**5**

votes

**0**answers

47 views

### Minimizers of variational problems with less symmetry

In am interested in understanding the so-called Hartree ground states, namely, minimizers of variational problems of the form
$$\inf_{\phi\in H^1,~\|\phi\|_2=1}\left\{\int|\nabla\phi(x)|^2dx
-\int|\...

**3**

votes

**2**answers

201 views

### Gradient estimates for a boundary value problem

$\newcommand{\avint}{⨍}$
Let $B_r$ be a call of radius $r$ and centre origin and $k<1$.$w$ satisfy the following PDE:
$$
\begin{cases}
-\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\
w=0 \...

**0**

votes

**0**answers

35 views

### References for matrix variational problems on the unitary group

Let $U(d)$ denote the group of unitary $d\times d$ matrices. Let $\mathcal C_d$ denote the cone of Hermitian positive semidefinite $d\times d$ matrices. Fix an integer $r\geq 1$ and let $C(U(d),\...

**2**

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**3**answers

461 views

### How do I apply Brouwer fixed-point theorem in this claim?

Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is ...

**10**

votes

**1**answer

623 views

### How to shrink a square with minimal distortion?

$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\euc}{\mathfrak{e}}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\al}{\alpha}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\Lip}{\text{Lip}_{\text{inj}}...

**5**

votes

**2**answers

308 views

### Is an $H_0^1$ function continuous to the boundary if it is continuous in the interior?

Suppose $\Omega$ is a bounded domain in $\mathbb R^3$ with Lipchitz boundary $\partial\Omega$, and $u\in H_0^1(\Omega)\cap C(\Omega)$. Is $u$ continuous to the boundary i.e. do we have $u \in C( \...

**2**

votes

**1**answer

260 views

### Hamilton equations for Classical Field Theory

This is a second part of my previous question. I'm trying to figure it out by myself how to deduce Hamilton's equations in classical field theory as it is usually obtained in physics books.
Notation: ...

**11**

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**2**answers

581 views

### Prove/disprove $(\int_{0}^{2 \pi} \!\!\cos f(x) d x)^{2}+(\int_{0}^{2 \pi}\!\!\! \sqrt{(f'(x))^{2}+\sin ^{2} f(x)}dx)^{2}\ge 4\pi^{2}$

This problem has been posted on Math.SE but didn't receive any correct answer after a long time.
Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. ...

**1**

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**0**answers

134 views

### Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...

**1**

vote

**1**answer

167 views

### Does weak continuity of Jacobians hold for non nondegenerate maps?

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...

**1**

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**0**answers

128 views

### Variational formulation for an elliptic boundary value problem

I'm trying to determine the variational formulation of
$$
\begin{cases}
-\Delta u(\mathbf{x})=1, & \mathbf{x}\in (0,1)\times (0,1) \\
-\partial_{x}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=0, & ...

**17**

votes

**2**answers

883 views

### Example of ODE not equivalent to Euler-Lagrange equation

I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation
$$\frac{\partial L}{\...

**11**

votes

**5**answers

2k views

### Maxwell equations as Euler-Lagrange equation without electromagnetic potential

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) ...

**0**

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81 views

### What is this notion of a probability measure used in calculus of variations?

In a paper (here), the main result is the explicit form of the minimizing probability measure for a given energy, see the equation (1.7). It is written as
$$m_1 = \frac 1 \pi \delta_0 \otimes \sqrt{2-...

**3**

votes

**2**answers

135 views

### Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...

**2**

votes

**0**answers

77 views

### A problem of uniqueness

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem:
$$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$
$$ u(x,0)=f(x),\quad\forall x\...

**3**

votes

**1**answer

109 views

### Variational formulation of an elliptic pde

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, what is the variational formulation of the following problem:
$$ \text{div}(y^a\nabla_{x,y}V)=0,\quad\text{on }\mathbb{R}^n\times(0,\infty),$$
$$ V(x,...

**4**

votes

**1**answer

129 views

### Degenerate second-order Lagrangians

Let $M$ be a smooth $m$ dimensional manifold, let $\pi:E\rightarrow M$ be a smooth fibred manifold over $M$. Let us write generic fibred coordinates as $(x^i,y^\sigma)$ with $x^i$ being the base ...

**1**

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**0**answers

58 views

### Finiteness of a constrained min/max

Take a bounded subset $\Omega \subset \mathbb{R}^d$ with smooth boundary (let's say a ball), take a ball of radius $r$, $B_r \subset \Omega$, take $\alpha \in ]0,1[$. How to prove that
\begin{align*}
...

**0**

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**0**answers

95 views

### Calculus of variations

I have the following question and I wasn't sure if I can apply the calculus of variations to it. The control function is $Q$.
$$\max \int_0^1 t Q(t) dt$$
subject to:
$Q$ is weakly increasing
$Q(0) \...

**0**

votes

**1**answer

195 views

### Sufficient and necessary conditions for decomposing the sum of random variables

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and
$\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...

**2**

votes

**1**answer

85 views

### Almost geodesic on non complete manifolds

Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\...

**4**

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**0**answers

71 views

### Determination of the nature of stationary values in variational calculus

In variational calculus, when we solve the Euler-Lagrange equation $\frac{d}{dx}L_p(u',u,x)-L_z(u',u,x)$, where $L=L(p,z,x)$, to find stationary inputs of the functional
$$
I[u]=\int_0^1 L(u',u,x)dx,
$...

**1**

vote

**1**answer

115 views

### Is a locally invertible weak limit of injective maps injective almost everywhere?

This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...

**6**

votes

**1**answer

90 views

### Graph embedding that locally minimizes total edge lengths

I consider a graph $G$ (possibly infinite, but locally finite) embedded in the Euclidean plane $\mathbb{E}^2 \cup \{\infty\}$ such that each local perturbation of the embedding "increases the ...

**7**

votes

**3**answers

256 views

### When is perimeter continuous under Hausdorff convergence?

It is known that the perimeter is lower semicontinuous for the convergence of sets. Two variants are widely known:
(Golab's theorem) in $\Bbb{R}^2$ if the sets $\Omega_n$ converge to $\Omega$ in the ...

**1**

vote

**0**answers

57 views

### Relation between minimizer of regularized risk & risk in statistical learning theory

In supervised machine learning, we typically take a Risk Minimization (RM) point of view when formulating a problem. So, what we typically solve for is the following:
$$ R^L(h) = \underset{h\in\...

**4**

votes

**1**answer

182 views

### Is $L^1$ strong convergence of Jacobians valid for maps between manifolds?

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Vol}{\operatorname{Vol}}$
$\newcommand{\Det}{\operatorname{Det}}$
$\newcommand{\Volm}{\...

**9**

votes

**1**answer

650 views

### Why the least action principle is always (?) used in this particular form?

The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...

**2**

votes

**2**answers

156 views

### Classification of Lagrangians with given Euler-Lagrange equations

In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of ...

**5**

votes

**0**answers

93 views

### Does there exist an injective Lipschitz map on the disk whose gradient switches between two given matrices?

While solving a problem in calculus of variations, I came to the following question:
Let $A,B$ be two real $2 \times 2$ matrices with positive determinants, and suppose that $\operatorname{rank}(A-B)=...

**2**

votes

**0**answers

69 views

### Area of minimising surface

I am interested in calculating the least area of a surface spanning the boundary of an octant on the unit sphere; and short of precise values I am looking for upper bounds for this area.
In $\mathbf{S}...

**3**

votes

**0**answers

70 views

### Calculus variation question

Assume that we have to minimize the integral $I[y]=\int_0^1 L(x,y,y'(x))dx$ for smooth diffeomorphic mappings $y:[0,1]\to [0,1]$ with $y(0)=0$ and $y(1)=1$, where $L\in C^\infty(R\times R\times R)$, ...

**1**

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**0**answers

75 views

### Convergence of infinite linear programming

Suppose we have the following linear program (LP1),
$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...

**1**

vote

**1**answer

158 views

### Fréchet derivative of evaluation-like functional (multivariate)

I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following.
Let $H$ be ...

**2**

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**0**answers

40 views

### Variational problem where the end point cannot be held constant

I want to solve a variational problem for paths $y:[0,1]\to \mathbb{R}^n$ where $F$ is a functional on the paths and additional parameter, and the end points are fixed, $y(0)=0$ and $y(1)=b$. However, ...