Questions tagged [calculus-of-variations]
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
546
questions
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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})...
1
vote
0
answers
61
views
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
$$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
4
votes
1
answer
297
views
On convex functions which are non constant on every segment
I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
2
votes
0
answers
152
views
Pohozaev identity and related non-existence result for a nonlinear problem
Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional
$$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \...
3
votes
1
answer
351
views
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, ...
2
votes
0
answers
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) ...
2
votes
0
answers
206
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 ...
0
votes
0
answers
481
views
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 ...
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 ...
3
votes
2
answers
201
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 ...
3
votes
1
answer
211
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 \...
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 ...
0
votes
0
answers
65
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?
1
vote
1
answer
177
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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 ...
4
votes
0
answers
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)=\...
0
votes
1
answer
118
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
vote
2
answers
188
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^\...
1
vote
0
answers
69
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 $\...
6
votes
2
answers
341
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
votes
0
answers
78
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 ...
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 ...
1
vote
0
answers
50
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$, ...
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?
3
votes
0
answers
181
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
vote
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 $$F=\int_{U}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\...
-6
votes
1
answer
137
views
Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]
Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$
$$
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),
$$
...
4
votes
0
answers
218
views
Lower semi-continuity of integration
I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
4
votes
1
answer
171
views
Non-linear translation invariant functionals on $L^1$
I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...
3
votes
1
answer
1k
views
Minimizing the expectation of a functional of probability distribution subject to an entropy constraint
Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
F(\pi) = \mathbb{E}_\pi |x-y| $$
It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the ...
11
votes
1
answer
587
views
Countable (?) dependent choice
In some circumstances I've been using a form of choice over the first uncountable ordinal knowing a priori that only a countable number of choices were going to be made (without any a priori upper ...
6
votes
1
answer
206
views
A one-dimensional integral minimization problem
Let $\mathscr F$ be the collection of smooth functions $f \colon
\mathbb R \to \mathbb R$ such that
$f \in C^\infty_c(\mathbb R)$, with $\text{supp } f \subset [-1,1]$;
$\int_0^1 x f(x) dx ...
7
votes
1
answer
1k
views
A variational problem - some guidance
This is a problem I'm thinking about, to learn some more advanced calculus of variations on my own. I would appreciate some help, or a solution, just to have a sample to compare in the future.
Let
$\...
3
votes
1
answer
168
views
Infimum of an integral functional involving a symmetric matrix
I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int_{B_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du,
...
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 ...
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 $\{F_\varepsilon, \varepsilon > 0\}$ ...
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 ...
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, ...
3
votes
1
answer
180
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.
$$ E_d(f)=\...
1
vote
0
answers
203
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Unusual problem of calculus-of-variations
I have a domain $D:\quad$ $-1<x<1$ and $-f(x)<y<f(x)$.\
There is function $u(x,y)$ which is obey the equation $\Delta u(x,y)=0$, $\forall (x,y)\in D$ with the Dirichlet boundary condition ...
0
votes
1
answer
186
views
Variation in Einstein-Hilbert action [closed]
In this page there are calculations of variation of Einstein-Hilbert action.
I see variations of terms like this:
$\delta {R^{\rho }}_{{\sigma \mu \nu }}$
where the term is not a functional, and ...
4
votes
1
answer
338
views
Inf of Jensen's inequality
I'm reading a monograph that considers the following problem:
$$\inf_{z(t) \in C^1} \int_0^1 c\bigg(\frac{dz(t)}{dt}\bigg) dt\\ z(0) = x, z(1) = y$$
Here $c$ is a convex function, $z(t)$ are paths ...
2
votes
0
answers
447
views
How to make sense of the Euler Lagrange equations for an infinite action?
The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional
$S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t$
Say we have an ...
4
votes
0
answers
83
views
Conformal $L^p$ rigidity of Riemannian manifolds
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\CO}[1]{\text{CO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\g}{\mathfrak{g}}...
1
vote
0
answers
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views
Has this logarithmic volume functional been studied?
$\newcommand{\M}{\mathcal{M}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\VolM}{\text{Vol}_{\M}}
\newcommand{\VolN}{\text{Vol}_{\N}}$
This question is mainly a reference request. (It is a cross-post ...
1
vote
0
answers
46
views
Harnack type Estimates for a p-Poisson equation with constant source term
Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE
$$
-\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 \...
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 ...
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 ...
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 ...
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 ...
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 ...