The calculus-of-variations tag has no usage guidance, but it has a tag wiki.

**0**

votes

**0**answers

19 views

### Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy,
$$
\int_{\Omega}|\nabla u |^2\, -\, \int_{\...

**1**

vote

**1**answer

138 views

### Shannon's proof of the entropy power inequality

In Shannon's paper on information theory, found here, he asserts the entropy power inequality in appendix 6, found on page 52. I was reading his proof and it seems like there is a gap. Through his ...

**1**

vote

**1**answer

189 views

### Question on Harmonic maps between Riemannian manifolds

In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows:
$$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$
In many Books such ...

**4**

votes

**1**answer

105 views

### Calculus of variations when functional involves inverse of the function

Typically the Euler-Lagrange equations are defined for the functional
$$ J[u] = \int_a^b L(x,u,u') dx. $$
However, I was wondering if anyone knows if they can be solved when the expression involves ...

**2**

votes

**0**answers

58 views

### Currents with mean curvature

so let $T=\tau(M,\theta,\xi)$ a rectifiable current in $U\subset\mathbb{R}^{n+k}$, then $T$ has mean curvature vector $H$ if for every $X\in C^1_c(U\setminus\partial T,\mathbb{R}^{n+k})$ the variation ...

**1**

vote

**0**answers

46 views

### First variation on double integral [closed]

Currently I am trying to fully understand the paper of munk1921.
In the derivation of the minimum induced drag theorem it is at one point stated (p.378) that in order to minimize drag the following ...

**0**

votes

**1**answer

36 views

### Existence of stationary tangent cones

My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42
So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density $\theta^n(\mu_V,x)&...

**2**

votes

**0**answers

44 views

### decomposition of codim 1 currents

Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\...

**2**

votes

**0**answers

87 views

### Willmore functional

Let $(M^2,g)$ and $(\bar{M},\bar{g})$ be two Riemannian manifolds. Suppose that $\mathcal{W}$ is the Willmore functional on the set of immersion functions from $M^2$ to $\bar{M}$.
We know that $(\...

**1**

vote

**1**answer

54 views

### Extremal of an L^1 continuous functional on a compact bounded set

Please, I need a small help with a reference.
Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and ...

**5**

votes

**3**answers

340 views

### Euler-Lagrange equations and Bellman's principle of optimality

One method to optimize the integral
$$\int_{\mathcal T} L(t,x,\dot{x}) \; dt $$
of a functional over a curve is the calculus of variations, which leads to ordinary differential equations: the Euler-...

**3**

votes

**0**answers

64 views

### Invariant Lagrangians of a connection and its derivatives: how do they look like?

Let
$$
L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma)
$$
be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...

**4**

votes

**0**answers

49 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 : \mathbb{R}...

**5**

votes

**1**answer

141 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 ...

**0**

votes

**1**answer

104 views

### Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation:
$i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields:
$i(u_t,v) + \beta (u_x,v_x) = 0$ ...

**1**

vote

**1**answer

97 views

### The space of loops as a Banach space [closed]

Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a ...

**2**

votes

**1**answer

109 views

### Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...

**2**

votes

**1**answer

110 views

### Reference Request: Variational Problem

I want to solve approximately the following variational problem:
Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and $\...

**2**

votes

**2**answers

138 views

### Total absolute variation of brownian motion, with different sampling rates

Let $(B_t)$ be a brownian motion on [0,1]. For the following, let $\omega$ be fixed.
Let's compute the total absolute variation when sampling period = $\delta$ is fixed:
$$V(\delta) = \sum_{i=0}^{N-...

**1**

vote

**0**answers

100 views

### Interchange of integral and infimum

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum:
$
\inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt,
$
where $ U\...

**23**

votes

**2**answers

923 views

### How can you compute the maximum volume of an envelope(used to enclose a letter)?

It's obvious that the volume of a envelope is 0 when flat and non-0 when you open it up. However, if you were to fill it with liquid, there must be some shape where it has a maximum volume. Is there a ...

**3**

votes

**0**answers

111 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 - z_k) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n z_j^{-\...

**1**

vote

**2**answers

196 views

### The term for problems “like” Brachistocrone?

Is there a commonly-accepted umbrella term for infinite-dimensional calculus problems where the goal is to compute an optimal geometric path between a pair of points? Three examples of this would be ...

**1**

vote

**1**answer

180 views

### Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...

**1**

vote

**0**answers

43 views

### Numerical techniques for nonlinear, coupled integro-differential equations

The gist of the problem I have is I want to be able to find a numerical solution to these three coupled, rather unpleasant looking integro-differential equations
(1):
$$ \frac{d^2 x(t)}{dt^2} = \frac{...

**0**

votes

**0**answers

60 views

### Well-posedness of gradient flows

For a convex lower-semicontinuous functional on a Hilbert space $I\colon H\rightarrow\mathbb{R}$, it is shown in Evans' PDE that the Hilbert-space-valued ODE
$$\begin{cases}\mathbf{u}'(t)\in-\partial ...

**0**

votes

**1**answer

98 views

### gradient descent in space of functions

Differential equations of the form
$$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$
can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, ...

**4**

votes

**1**answer

390 views

### A continuous path between two Sobolev functions

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in H^1(\...

**0**

votes

**1**answer

111 views

### Double Integral Equations

In my research I've come across a handful of double integral equations, and I'm nearly at a total loss for how to derive anything useful from such things.
I've been lead to believe that even single ...

**4**

votes

**1**answer

202 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 ...

**5**

votes

**0**answers

76 views

### What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...

**5**

votes

**0**answers

101 views

### Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory?
I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...

**7**

votes

**2**answers

309 views

### Closed curve whose neighborhood is as large as possible

Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this:
(ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB)
...

**4**

votes

**0**answers

67 views

### Minimisers and critical points of variational integrals

In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral:
\begin{equation*}
I[u]=\int_{\...

**6**

votes

**1**answer

308 views

### electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...

**1**

vote

**0**answers

80 views

### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & \mathrm{in}\hspace{...

**4**

votes

**1**answer

186 views

### Fair surfaces - general mathematical theory

Fairness measures for surfaces are, in general, functionals containing more complicated terms thatn the usual bending energy, and may depend not only on the mean curvature but also on principal ...

**3**

votes

**0**answers

68 views

### Can Mumford-Shah functional be adapted to lower $L^1$ space?

The well know Mumford-Shah functional functional
$$
F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1
$$
where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...

**2**

votes

**0**answers

22 views

### Is this function $u\in SBV(\Omega)$ also belongs to $L^\infty(\Omega)$?

Some early discussion can be found here.
My question:
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Also $\mathcal H^{N-1}(\partial\Omega))<\infty$.
Let $u\in SBV(\...

**0**

votes

**0**answers

48 views

### Nonlinear optimization problem with inequality constraints

Consider a real valued function $g(x_i)=\frac{1}{a_1+ \frac{a_2}{x_i}}, \forall i=\{1,2,3,...,n\}$.
The objective function $H$ is
$H=\sum_{i=1}^{n}\frac {1}{g(x_i)-a_3x_i}$
The optimization ...

**1**

vote

**1**answer

81 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

120 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 ...

**0**

votes

**1**answer

96 views

### Constrained optimal control problem

I am trying to solve an optimization problem which is probably reminiscent of optimal control theory but all of this is not exactly my field of expertize and I am a little bit lost in translation. If ...

**0**

votes

**0**answers

53 views

### How to modify a SBV convergence sequence to obtain uniform integrability?

Given $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Assume $(u_n)\subset SBV(\Omega)$ a sequence of functions such that $u_n\to u_0$ weakly in $SBV$ for some function $u_0\in SBV(\...

**1**

vote

**1**answer

144 views

### Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ then exponent of v or -v is not summable?

I have the following Question:
1) Is it true that
if $\Omega\subset\mathbb R^3$, $\Omega$ - bounded, $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ implies $\int\limits_{\Omega}{e^vdx}=+\infty$ ...

**0**

votes

**1**answer

130 views

### How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
$...

**8**

votes

**1**answer

200 views

### Which domain maximizes the energy of the Lebesgue measure?

This could be asked in more generality, but let me stick to a concrete case.
Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$...

**3**

votes

**0**answers

58 views

### Morse index in variational calculus

Let $f(t,x,v)$ be a smooth function on $\mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n$, $\mathcal{C}_{x,y}$ be the set of smooth paths $c:[0,1]\to \mathbb{R}^n$ with $c(0)=x,c(1)=y$, $E(c)$ be a ...

**0**

votes

**1**answer

115 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

**6**

votes

**1**answer

380 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 ...