# Questions tagged [calculus-of-variations]

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

42
questions

**26**

votes

**3**answers

1k views

### Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?

**7**

votes

**2**answers

818 views

### Willmore minimizers for genus $\geq 2$

For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as
$$
\cal W(f) = \int _{\Sigma} \frac{1}{4} |\vec H|^2 d \mu_g,
$$
where $\vec H$ is the mean ...

**33**

votes

**3**answers

3k views

### Central limit theorem via maximal entropy

Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that:
$$\int_\mathbb{R} \rho(x)\, dx = 1$$
and
$$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$
...

**19**

votes

**2**answers

896 views

### Functional approach vs jet approach to Lagrangian field theory

Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...

**14**

votes

**1**answer

1k views

### Is $\delta(df \wedge df)=0$ an Euler-Lagrange equation?

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$\newcommand{\TM}{\operatorname{TM}}$
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$\newcommand{\M}{\mathcal{M}}$
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$\newcommand{\tr}{\...

**10**

votes

**5**answers

972 views

### Convex curves with many inscribed triangles maximizing perimeter

A classical nice result of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. Precisely, for each ...

**6**

votes

**1**answer

2k views

### Minimize Energy for Charge Distributions

I am considering [positive] charge distributions $\rho:M\rightarrow\mathbb{R}_+$ (nonnegative reals) with unit charge $\int_M\rho=1$ for convenience. Here $M$ is a nice-enough region, say a ...

**5**

votes

**0**answers

286 views

### Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result:
Theorem. All straight lines are extremals of the variational problem
$$
...

**7**

votes

**1**answer

250 views

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

Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that
$$
\inf_{y\in Lip([a,b])}F(y)<\inf_{...

**2**

votes

**0**answers

171 views

### Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?

Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...

**2**

votes

**0**answers

162 views

### Variational formulation for elliptic interface problem

Where can I find a paper that deals with the following interface problem with variational methods? In particular, what is the correct variational formulation of the problem (that is, a functional ...

**6**

votes

**5**answers

2k views

### Variation of curvature with respect to immersion?

Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by
$$ f(t) = f_0 + tuN_0, $$
where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and ...

**16**

votes

**1**answer

1k views

### What braking strategy is most fuel-efficient?

You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...

**11**

votes

**1**answer

1k views

### Invariance of the l.h.s. of Euler-Lagrange equation

Let $M^n$ be a smooth manifold equipped with a nondegenerate Lagrangian $L:TM\to\mathbb R$, $L=L(x,y)$, $x\in M$, $y\in T_xM$. The stationary points of the corresponding integral functional on curves ...

**13**

votes

**1**answer

595 views

### Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?

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$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\...

**9**

votes

**4**answers

1k views

### Boundedness of nonlinear continuous functionals

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
...

**8**

votes

**2**answers

414 views

### In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?

In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Let $\pi:E\rightarrow M$ be a fiber bundle ...

**8**

votes

**1**answer

1k views

### Coordinate-free derivation of the Einstein's field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here):
$$S = \int_M R \mu_g,$$
is given by ...

**8**

votes

**1**answer

352 views

### Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?

**2**

votes

**0**answers

125 views

### Formulation of contour variational problem

I am having difficulty formulating a problem, which involves optimizing a contour shape, into a well-posed variational form that would give a reasonable answer.
Within a bounded region on the $xy$ ...

**0**

votes

**3**answers

875 views

### Names of certain surfaces

Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated.
Surface I. Implicit ...

**11**

votes

**2**answers

788 views

### For what metrics are circles solutions of the isoperimetric problem?

A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...

**8**

votes

**1**answer

743 views

### Maximal tetrahedra inscribed in ellipsoid

Pietro Majer quoted the theorem of Michel Chasles in his MO question,
"Convex curves with many inscribed triangles maximizing perimeter,"
which states that the triangles of maximum perimeter inscribed ...

**6**

votes

**0**answers

161 views

### The distributional gradient of the closest isometry to the differential of a smooth map

The setting-a "linear algebra" fact:
Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...

**2**

votes

**1**answer

372 views

### Unusual problem of calculus-of-variations. Attempt 2

I already tried to ask the same question here Unusual problem of calculus-of-variations, but I did a huge mistake and have no time to understand it. In the past post, I tried to simplify the problem, ...

**7**

votes

**3**answers

1k views

### Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla f|^{2}...

**6**

votes

**0**answers

222 views

### a variational problem related to weighted logarithmic capacity

Consider the following multiple contour integral:
$$ \Phi_\lambda := \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j^{-1} - z_k^{-1}) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n ...

**5**

votes

**1**answer

298 views

### Questions about the regularity of the “norm” associated to a convex set

Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...

**3**

votes

**3**answers

526 views

### Functional derivatives on Banach spaces

Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. ...

**2**

votes

**0**answers

37 views

### First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space

The curve $\Gamma$ in $\mathbb{R}^2$ is defined by a continuous and monotonically increasing function $f(x)\in\text{C}[0,1]$, where $f(0)=0$, $f(1)=1$.
Let $(X,Y)$ is jointly and uniformly ...

**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 ...

**1**

vote

**0**answers

192 views

### Unusual problem of calculus-of-variations

I have a domain $D:\quad$ $-1<x<1$ and $-f(x)<y<f(x)$.\
There is function $u(x,y)$ which is obey the equation $\Delta u(x,y)=0$, $\forall (x,y)\in D$ with the Dirichlet boundary condition ...

**7**

votes

**1**answer

1k views

### Functional minimization problem

Is there a smooth solution to minimize this:
$$
\int_0^1{x \over {1+k^2f'(x)^2}}dx, f(0)=1, f(1)=0, f'(x)\leq 0, k^2>0.
$$
I could "solve" it using a numeric approximation (my algorithm converged ...

**5**

votes

**1**answer

158 views

### An extension of Hadamard maximum determinant problem

Consider the Vandermonde product $\prod_{1\le j < k \le n} |z_j - z_k|$. It is well-known that under the constraint $|z_j| \le 1$ for all $j$, the product is maximized at a picket fence ...

**4**

votes

**3**answers

103 views

### Find distribution that minimises a function of its moments

Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive).
How can I find $f(x)$ that ...

**4**

votes

**0**answers

432 views

### Optimal transport between two distributions in a Markov chain

In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the ...

**3**

votes

**2**answers

277 views

### Representing a nonlinear elliptic PDE as an energy minimization problem

I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...

**1**

vote

**1**answer

167 views

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

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

**1**

vote

**1**answer

603 views

### Variational proof for minimum curvature of cubic splines

Background: Give an increasing set of points $(x_i)_{i=0}^n \subset \mathbb [a,b]$, a cubic spline $S(x)$ is a piecewise cubic polynomial with continuous second derivative. One can also prove, roughly,...

**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\...

**1**

vote

**1**answer

99 views

### question about $TGV^2$ space

Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and
$$
TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in C_c^\infty(I),\,\|\phi\|_{L^{\infty}(...

**0**

votes

**0**answers

135 views

### Is this has anything to do with Riesz representation?

The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...