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

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### $\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ for $|f|\leq1$ on a unit disk

Let $f:\mathbb{R^2}\mapsto\mathbb{R}$ be continuous and have partial derivatives in $D=\{(x,y):x^2+y^2\leq1\}$, and let $\mathscr{H}$ the set of such functions for which $\sup_D |f|\leq1$.
Could ...

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47 views

### Alternative characterization of epi-convergence

I am struggling with the proof of a property of epi-convergence.
We need the following definitions:
For a sequence of sets $(C^\nu)_\nu$ in $\mathbb R^n$, the outer limit is the set $\limsup_\nu C^\...

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143 views

### The Poincaré inequality for the Sobolev space on a domain with a non Lipschitz boundary

Let $\Omega$ be a bounded open Lipschitz domain in $\mathbb{R}^{d}, d \geq 3$. Assume that $L$ is a straight segment such that $L \subset int(\Omega)$.
Let $v \in V:= \overline{ \{ \phi \in C^{\infty}...

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58 views

### Solve simple stochastic variational inequality

Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\...

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225 views

### Bounded deformation vs bounded variation

Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...

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46 views

### Volume of critical points decreases under symmetric decreasing rearrangement

In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious ...

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66 views

### Hölder isoperimetric problem

Denote by $S_r$ the usual circle of radius $r$, with the path metric ($d(x,y) = r\theta$, where $\theta$ is the angle between the vectors $x$ and $y$), and let $\alpha \in (1/2,1)$. Consider the ...

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35 views

### Tools/techniques for a problem in variational calculus (coming from discrete geometry)

I'm working on a problem in discrete geometry, more specifically on visibility of polygons. The easiest instance of this problem reduces to the following.
Among all density functions $f:[0,1]\to R$, ...

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**1**answer

114 views

### Intuition for coercive functions

I have been working with $\Gamma$-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions?

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93 views

### Isoperimetric inequality inside a regular polygon

Let $\mathcal{P}_n$ be a fixed $n$-sided regular polygon with area $A:=\vert \mathcal{P}_n\vert>0$. For any $c\in (0,A)$, I would like to find the shape of the domain $D\subset \mathcal{P}_n$ such ...

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113 views

### Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?

Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$u(t) \leq \psi(t) ...

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110 views

### Calculus of variation with discontinuous solutions?

I'm thinking of the following question:
Consider a function $f: [0,L]\rightarrow\mathbb{R}$ and an energy functional $$F=\int_{0}^{L}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\mathrm{d}x.$$ The ...

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94 views

### Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]

Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$
$$
f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),
$$
...

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20 views

### Convergence of auxiliary problem w Lagrange-Multipliers

Let $V = H^1_0(\Omega)$ and $V_k$ the FE-space of all piecewise affine and continuous functions. Let $(u_k, \lambda_k) \in V_k \times L^\infty(\Omega)$ be the solution to
\begin{align*}
\int_\Omega ...

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91 views

### Lower semi-continuity of integration

I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....

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86 views

### Non-linear translation invariant functionals on $L^1$

I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...

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191 views

### Minimizing the expectation of a functional of probability distribution subject to an entropy constraint

Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
F(\pi) = \mathbb{E}_\pi |x-y| $$
It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the ...

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147 views

### Countable (?) dependent choice

In some circumstances I've been using a form of choice over the first uncountable ordinal knowing a priori that only a countable number of choices were going to be made (without any a priori upper ...

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85 views

### A one-dimensional integral minimization problem

Let $\mathscr F$ be the collection of smooth functions $f \colon
\mathbb R \to \mathbb R$ such that
$f \in C^\infty_c(\mathbb R)$, with $\text{supp } f \subset [-1,1]$;
$\int_0^1 x f(x) dx ...

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842 views

### A variational problem - some guidance

This is a problem I'm thinking about, to learn some more advanced calculus of variations on my own. I would appreciate some help, or a solution, just to have a sample to compare in the future.
Let
$\...

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83 views

### Infimum of an integral functional involving a symmetric matrix

I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int_{B_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du,
...

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180 views

### dual to Hodge theory

Let $(M,g)$ be a closed Riemannian manifold.
In my understanding Hodge theory shows that any de Rham cohomology class can be represented uniquely by a harmonic form. Moreover the harmonic form ...

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77 views

### A system similar to the Keller-Segel one in $\mathbb{R}^N$

Can you point out references on existence and uniqueness of solutions to the following system?
$$u_t = \nabla(h(u)\nabla v) $$
$$v = \Delta u$$
$$u(0,\cdot) = u_0(\cdot)$$
in $(0,\infty) \times \...

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41 views

### Gamma convergence for control problems with changing domains

$\Gamma$-convergence is a notion of convergence for functionals which has the nice property that if $x_\varepsilon$ are minimisers for a family of functionals $\{F_\varepsilon, \varepsilon > 0\}$ ...

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77 views

### Family of large deviation principles

The following question may be a bit imprecise in its formulation, I guess however the problem I have in mind is clear. Although to me it looks like a fairly standard question, I couldn't find any ...

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359 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, ...

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162 views

### Does there exist energy-minimizing immersions?

This is a cross-post.
Let $M,N$ be $d$-dimensional oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E_d:C^{\infty}(M,N) \to \mathbb{R}$ be the $d$-energy, i.e.
$$ E_d(f)=\...

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186 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 ...

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79 views

### Variation in Einstein-Hilbert action [closed]

In this page there are calculations of variation of Einstein-Hilbert action.
I see variations of terms like this:
$\delta {R^{\rho }}_{{\sigma \mu \nu }}$
where the term is not a functional, and ...

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232 views

### Inf of Jensen's inequality

I'm reading a monograph that considers the following problem:
$$\inf_{z(t) \in C^1} \int_0^1 c\bigg(\frac{dz(t)}{dt}\bigg) dt\\ z(0) = x, z(1) = y$$
Here $c$ is a convex function, $z(t)$ are paths ...

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128 views

### How to make sense of the Euler Lagrange equations for an infinite action?

The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional
$S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t$
Say we have an ...

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### Conformal $L^p$ rigidity of Riemannian manifolds

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\CO}[1]{\text{CO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\g}{\mathfrak{g}}...

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51 views

### Lagrangian sub manifolds versus solutions of variational problems

Is the only relation between Lagrangian submanifolds and solutions of variational problems that they are both related to Lagrange?

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82 views

### Has this logarithmic volume functional been studied?

$\newcommand{\M}{\mathcal{M}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\VolM}{\text{Vol}_{\M}}
\newcommand{\VolN}{\text{Vol}_{\N}}$
This question is mainly a reference request. (It is a cross-post ...

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33 views

### Harnack type Estimates for a p-Poisson equation with constant source term

Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE
$$
-\Delta_p u=1\,\,\mbox{in $B$}
$$
Let another function $f$ be such that
$$
\begin{cases}
-\Delta_p f =1 \;\;\mbox{in $B$}\\
f=0 \...

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243 views

### Sophus Lie's contribution to solution of problems of variational type as in Euler and Lagrange

The original impetus for Sophus Lie's work was apparently to streamline the solution of certain problems of variational type such as those treated in the work of Euler and Lagrange. This presumably ...

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### Is there Calculus for (Almost) Continuous functions?

So I asked a similar question to this on Math Stack Exchange a couple of weeks ago, did a bounty, and I didn't receive any answers.
I am struggling a bit with a part of my research (on CS).
Suppose ...

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85 views

### Optimal conditional density choice

Suppose $X$ is a random variable whose density is denoted by $p(x)$. We can observe a realization of $X$ and we want to take an action $Y$ to minimize
$$E[(Y-X)^{2}]$$
What is the optimal choice ...

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188 views

### Can one obtain this ODE as an Euler-Lagrange equation?

Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...

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### Is this space compactly contained in $L^p((0,\infty),rdr)$ for all $p\geq 2$?

Some Background: A typical problem in mathematical physics is the existence of positive radially symmetric solutions to a nonlinear Schrodinger type equation over $\mathbb{R}^{2+1}$. Such a problem ...

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168 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_{...

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127 views

### Lavrentiev Phenomenon

Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in Lip([a,b])}F(y)>\inf_{y\in W^{1,1}([a,b])...

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401 views

### How is the Euler-Lagrange equation derived without local coordinates?

The Euler-Lagrange equation states that the time flow is given by a vector field such that the vector field contracted with the symplectic form gives dL, where L is the Lagrangian function on the ...

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63 views

### Optimal curves on planar annular regions

I am looking for references on optimal curves in planar annular regions
More precisely, on simple closed curves of class $C^2$ which simultaneously minimizes a finite number of functionals from the ...

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110 views

### A discrete-to-continuous approach to the Dirichlet principle?

Dirichlet principle: Let $\Omega \subset R^n$ be a compact set with $C^1$ boundary. Then, there exists a unique solution $f$ satisfying $\Delta f = 0$ in $\Omega$ and $f=g$ on $\partial \Omega$.
We ...

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57 views

### Maximizing the $\alpha$-moment of a distributution

Given $\alpha$ and constant $\mu$,
$$\begin{array}{ll} \text{maximize} & \displaystyle\int_0^\infty p(x)x^\alpha \,\mathrm d x\\ \text{subject to} & \displaystyle\int_0^\infty p(x)\,\mathrm d ...

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147 views

### One side Harnack inequality for Subharmonic function

It is well known that for any non negative Harmonic function w ($\Delta w=0$, $w\geq 0$) in a ball, $B_1(0)$, $\exists$, C>0 such that $\forall y\in B_{1/2}(0)$
$$
Cw(0)\leq w (y)
$$
It is a clear ...

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46 views

### Optimal contour shape for variational problem over captured area

Let's assume we have a continuous and finite scalar function $f(x,y)$ over the $xy$ plane ($\mathbb{R}^{2}$) and this function is to be integrated over a bounded area (surface) $A\subset\mathbb{R}^{2}...

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### A simple proof that all the symmetries of the Dirichlet energy are conformal

This is a cross-post.
It seems to be folklore knowledge that all the (source) symmetries of the $d$-Dirichlet energy are conformal maps.
Specifically, I have found this nice proof for the following ...

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111 views

### Has this functional been studied?

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
This is a cross-post from MSE.
Let $\M,\N$ be Riemannian ...