Questions tagged [calculus-of-variations]

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

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5
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0answers
95 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)=...
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1answer
146 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
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0answers
65 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} ...
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0answers
38 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 ...
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1answer
91 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_{...
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41 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 ...
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86 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\...
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0answers
87 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{...
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0answers
57 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)...
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152 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
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44 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|\...
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2answers
197 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 \...
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33 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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3answers
457 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 ...
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2answers
426 views
+500

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
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2answers
297 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( \...
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1answer
255 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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2answers
576 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 $f(0)=f(2\pi)$. ...
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0answers
131 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,...
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1answer
164 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 \...
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0answers
127 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
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2answers
877 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
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5answers
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) ...
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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
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2answers
134 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 }\...
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0answers
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
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1answer
108 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
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1answer
125 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 ...
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0answers
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*} ...
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0answers
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
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1answer
192 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
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1answer
82 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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0answers
70 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
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1answer
114 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
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1answer
86 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
3answers
240 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 ...
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0answers
55 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
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1answer
178 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}{\...
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1answer
60 views

Peano Baker solution to a LTV differential equation [closed]

The solution to the following transition matrix differential equation $$ \dot{\Phi} (t,t_{0}) = A(t)\Phi (t,t_{0}), \Phi (t_{0},t_{0}) = I $$ is given by: $$ \dot{\Phi} (t,t_{0}) = I + \int_{t_{0}}^{t ...
9
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1answer
647 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
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2answers
146 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
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0answers
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
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0answers
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
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0answers
69 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 $L\in C^\infty(R\times R\times R)$, ...
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0answers
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
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1answer
152 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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0answers
38 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, ...
4
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2answers
200 views

If all points of a real function with positive values would be local minimum, can one say it is constant function?

During my studies I faced a function $f:\mathbb{R} \to \mathbb{R}^+ $ with the property: for all $x \in \mathbb{R} $ and all $y$ in open interval $(x-\frac{1}{f(x)} ,x+\frac{1}{f(x)}) $ we have $f(x) \...
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0answers
69 views

Measurability of infimum function

In Theorem 14.37 of Variational Analysis by Rockafellar and Wets, it shows that for any normal integrand $f:T\times \mathbb{R}\to\mathbb{R}$, the function $p:T\to \mathbb{R}$ given by $p(t):=\inf f(t,⋅...
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0answers
72 views

Dislocations and Random Matrix Theory

Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help. By ...

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