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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How can I prove that the following function is increasing according to x1?

Suppose that $0 \le {X_1} < {X_2} < {X_3}$ . How is it possible to prove the following function is increasing based on ${X_1}$ in the range of $0 \le {X_1} < {X_2}$ ? $f({X_1},{X_2},{X_3})...
Hamed's user avatar
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A particular semi-linear equation on Riemannian manifolds

Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation $$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
Ali's user avatar
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4 votes
1 answer
297 views

On convex functions which are non constant on every segment

I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
Y.B.'s user avatar
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Pohozaev identity and related non-existence result for a nonlinear problem

Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional $$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \...
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3 votes
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Solution singular PDE

I've been studying the following singular PDE $$ \mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$ in $\Omega \subset \mathbb{R}^{2}$. Do you know any reference, ...
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2 votes
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111 views

Laplacian variational problem with asymptotically quadratic term

Consider the functional $$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$ where $\Omega$ is a bounded smooth domain. The problem has been solved for example if $F$ is (1) subquadratic, or (2) ...
Riku's user avatar
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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 ...
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Solving the Euler-Lagrange equation for the brachistochrone problem with friction

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how ...
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5 votes
3 answers
546 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 ...
Y.B.'s user avatar
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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 ...
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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 \...
Y.B.'s user avatar
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7 votes
1 answer
454 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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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?
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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 ...
TheAppliedTheoreticalOne's user avatar
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275 views

Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as the radius $$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)=\...
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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 ...
Chennes's user avatar
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2 answers
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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^\...
Philipp Wacker's user avatar
1 vote
0 answers
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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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6 votes
2 answers
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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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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 ...
big_huong's user avatar
8 votes
0 answers
109 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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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 $f:[0,1]\to R$, ...
Gelasio Salazar's user avatar
2 votes
1 answer
1k 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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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) ...
M.L's user avatar
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1 answer
263 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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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 $x \in \mathbb{R}^+ \cup\{0\}$ $$ f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right), $$ ...
James's user avatar
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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 $\mathbb{R}^d$....
ABIM's user avatar
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4 votes
1 answer
171 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 $...
Y.B.'s user avatar
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3 votes
1 answer
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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 ...
Igor Markov's user avatar
11 votes
1 answer
587 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 ...
Nicola Gigli's user avatar
6 votes
1 answer
206 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 ...
Y.B.'s user avatar
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7 votes
1 answer
1k 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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3 votes
1 answer
168 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, ...
Y.B.'s user avatar
  • 391
13 votes
0 answers
257 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 ...
Thomas Rot's user avatar
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2 votes
1 answer
240 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 $\{F_\varepsilon, \varepsilon > 0\}$ ...
snar's user avatar
  • 129
3 votes
1 answer
101 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 ...
Tobsn's user avatar
  • 289
2 votes
1 answer
393 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, ...
Peter's user avatar
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3 votes
1 answer
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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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1 vote
0 answers
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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 ...
Peter's user avatar
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0 votes
1 answer
186 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 ...
asv's user avatar
  • 111
4 votes
1 answer
338 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 ...
yoshi's user avatar
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2 votes
0 answers
447 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 ...
R Mary's user avatar
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4 votes
0 answers
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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}}...
Asaf Shachar's user avatar
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1 vote
0 answers
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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 ...
Asaf Shachar's user avatar
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1 vote
0 answers
46 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 \...
Harish's user avatar
  • 251
7 votes
1 answer
281 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 ...
Mikhail Katz's user avatar
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2 votes
0 answers
330 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 ...
AspiringMat's user avatar
1 vote
0 answers
89 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 of ...
Galor's user avatar
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3 votes
2 answers
540 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 ...
alexa's user avatar
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3 votes
1 answer
160 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 ...
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