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Questions tagged [calculus-of-variations]

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

2
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3answers
109 views

Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional $$ f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\}, $$ which ...
2
votes
2answers
114 views

Example of convex functions fulfilling a (strange) lower bound

I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
1
vote
1answer
60 views

Uniqueness of minimizers in the Calculus of Variations

Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by $$ f(x,y):= (x^+)^2 + (y^+)^2 $$ where $a^+ = \max\{a,0\}$ for any real number $a$. Given a Lipschitz regular domain $\Omega \...
6
votes
1answer
130 views

Anisotropic perimeter and regularity of anisotropic minimal surfaces

1. Introduction. By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set. Question. What are the known regularity results for ...
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0answers
41 views

Elliptic Dirichlet problems with measure boundary data

Can you point out any references on the Dirichlet problem for divergence-type elliptic operators with a Radon measure as boundary data?
2
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1answer
73 views

convergence and a mean curvature condition imply convexity

I have a question regarding the proof of theorem 4.6 in https://arxiv.org/abs/1007.3899 (Hall's conjecture). Let $S^2$ be the class of Borel subsets in $\mathbb{R}^2$ with finite and positive ...
0
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0answers
27 views

limit derivative convex function exist

Let $U$ be an open convex subset of $\mathbb{R}^{n}$ and $f(y,t)$ are real continuous convex function on $U$. We assume that $x_{n}\rightarrow x$ and $f(x_{n},t_{n})$ is diffrentaible with respect to $...
4
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0answers
164 views

Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\...
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0answers
50 views

Optimizing a complex functional with respect to the Lexicographic Ordering?

I'm wondering if the following argument is correct: Consider optimizing a complex functional $S[x(t)]$. Since $S$ is complex, it only has an optimum with respect to the lexicographic order of the ...
0
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1answer
91 views

$\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ 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 ...
1
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2answers
58 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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0answers
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 $\...
5
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2answers
237 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 ...
2
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0answers
47 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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0answers
72 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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0answers
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 $f:[0,1]\to R$, ...
1
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1answer
134 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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0answers
96 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 ...
2
votes
1answer
150 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) ...
1
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1answer
193 views

Calculus of variation with discontinuous solutions?

I'm thinking of the following question: Consider a function $f: U\rightarrow\mathbb{R}$ where $U=[0,L_1)\cup(L_1,L]$, and an energy functional $$F=\int_{U}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\...
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1answer
95 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 $x \in \mathbb{R}^+ \cup\{0\}$ $$ f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right), $$ ...
3
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0answers
94 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 $\mathbb{R}^d$....
4
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1answer
92 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 $...
2
votes
1answer
199 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 ...
6
votes
1answer
159 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 ...
5
votes
1answer
86 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 ...
7
votes
1answer
848 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 $\...
3
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1answer
87 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, ...
12
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0answers
188 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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0answers
43 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 $\{F_\varepsilon, \varepsilon > 0\}$ ...
2
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1answer
79 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 ...
2
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1answer
360 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, ...
3
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1answer
165 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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0answers
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 ...
0
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1answer
80 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 ...
4
votes
1answer
233 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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0answers
137 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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0answers
62 views

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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0answers
52 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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0answers
87 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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0answers
34 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 \...
8
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1answer
245 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 ...
2
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0answers
312 views

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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0answers
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 ...
3
votes
2answers
199 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 ...
2
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1answer
137 views

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 ...
4
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1answer
174 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_{...
3
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0answers
143 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])...
4
votes
2answers
420 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 ...
2
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0answers
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 ...