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Results tagged with calculus-of-variations
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user 36721
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
1
vote
Accepted
Does a minimiser exist for this Gaussian-like functional?
$\newcommand\R{\mathbb R}$For any $a\in\R$, there is no minimizer of
\begin{equation*}
I(f):=\int_\R xe^{f(x)-x^2}\,dx \tag{-1}\label{-1}
\end{equation*}
over all $f\in F_a$, where $F_a$ is the se …
- 111k
0
votes
Accepted
Derivatives of infimum in variational problem
Let $t:=\lambda$, so that
$$R(t):=\inf_{x\in Y}F(t,x).$$
Suppose that $\int_{\partial_e Y}f\,d\mu_x$ is lower-semicontinuous in $x$ (with respect to the appropriate topology, which you appear to assum …
- 111k
3
votes
Accepted
Infimum of an integral functional involving a symmetric matrix
$\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\ …
- 111k
8
votes
Accepted
A one-dimensional integral minimization problem
$\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\ …
- 111k
2
votes
Variational problem: how to minimise the second moment?
Let $X$ be a positive random variable (r.v.) with probability density function $f$. By the Cauchi--Schwarz inequality, $x_1^2=(EX)^2$ is a lower bound on $x_2=EX^2$, and this lower bound is attained i …
- 111k
1
vote
Accepted
Determine $\alpha \in (0,1)$ such that $J_{\alpha}(\phi):=\int \psi/\phi^{\alpha}$ exists?
$\newcommand\al\alpha\newcommand\R{\mathbb R}$Let us prove the following generalization of your desired statement:
Let $\psi\colon\R\to\R$ be a continuous function with compact support $S$. Let $\phi …
- 111k
2
votes
Optimization on non-convex set
Yes (mainly). That $u_*:=1_{f<0}$ is a minimizer (of $\int uf$ over all $u\in X(\Omega)$) follows because, if $0\le u\le1$, then
$$uf-u_*f=(u-u_*)f\ge0, \tag{1}\label{1}$$
whence
$$\int u_*f\le\int uf …
- 111k
3
votes
Accepted
Why this function is monotonic?
The claim is incorrect. See e.g. this image of a Mathematica notebook:
The function $f$ is not increasing for $a=6/5>1$, $\alpha=-2<0$, and $\beta=3/4>0$.
- 111k
1
vote
Accepted
Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r...
If $r=0$, then $f(x)=0$ for all $x\ge0$. If $r>0$, then
$$f'(x)=-\frac{8 r (x+1) (8 r x+18 r+24 x+27)}{(2 x+3) (4 x+3) (2 r+2 x+3) (2 r+4 x+3)}<0$$
for $x\ge0$, $f(0)=2 \log \left(\frac{2 r}{3}+1\rig …
- 111k
3
votes
Accepted
Is $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$ differentiable?
From the context, here $AC[0, 1]$ should be the set of all absolutely continuous functions from $[0,1]$ to $\mathbb R^n$ with the norm $W^{1,1}$.
Note that for all $x=(x_1,\dots,x_n)\in AC[0, 1]$ we h …
- 111k
4
votes
Accepted
Example of convex functions fulfilling a (strange) lower bound
Let $G$ be the set of functions $g\colon\mathbb R\to\mathbb R$ such that for some strictly positive real $a$ and $b$ and all real $x$ we have $g(x)=-ax$ if $x\le0$ and $g(x)=bx$ if $x\ge0$. Let $l_1,\ …
- 111k
1
vote
Non convex optimization problem in $W_0^{1,2}$
$\newcommand{\al}{\alpha}$In leo monsaingeon's answer, for the the value $J(\al)$ of the infimum it was shown that
\begin{equation*}
J(\al)\le9
\end{equation*}
and conjectured that
\begin{equation …
- 111k
3
votes
Accepted
Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
$\newcommand{\vpi}{\varphi}\newcommand\R{\mathbb R}$
Claim 1: The map $F$ is not Lipschitz if $p>1$.
Claim 2: The map $F$ is $1$-Lipschitz if $p=1$: For all $f,g$ in $D_p$,
\begin{equation*}
W_ …
- 111k
2
votes
Accepted
Continuity of generalised Legendre transform
$\newcommand\vpi\varphi$The answer is no. E.g., suppose that $X=Y=\mathbb R$, $c=0$, and $\vpi_a(x)=\min(a,\max(0,x-a))$ for $a\ge0$ and all real $x$. Then, as $a\to\infty$, we have $\vpi_a\to0$ unifo …
- 111k
4
votes
Accepted
Maximizing the $\alpha$-moment of a distributution
Let us consider the closely related problem: maximize $EX^\alpha$ over all nonnegative random variables (r.v.'s) $X$ with $EX=\mu$. To avoid trivialities, assume that $\mu\in(0,\infty)$. Consider the …
- 111k
2
votes
Find distribution that minimises a function of its moments
Let $X$ be a positive random variable (r.v.) with probability density function $f$. The exact lower bound on
$$r(X):=\frac{x_3+x_1^3-2x_1x_2}{(x_2-x_1^2)^2}$$
is $0$, and it is not attained at any $f …
- 111k
7
votes
Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?
$\newcommand{\R}{\mathbb R}$Your conjecture is true.
Indeed,
\begin{equation*}
\begin{aligned}
f(x)&=F(a,u):=a\frac{\sin u}u+4\cos u, \\
g(x)&=G(a,u):=a\frac{\cot(u/2)}{u^2}\,h(u)-4\frac{\sin …
- 111k
3
votes
How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$
$\newcommand{\al}{\alpha}\newcommand{\be}{\beta}\newcommand{\vpi}{\varphi}\newcommand{\thh}{\theta}\newcommand{\I}{\mathscr I}\newcommand{\J}{\mathscr J}$The conjunction of your conditions,
\begin{equ …
- 111k
3
votes
Accepted
Which set of functions admits the existence of the minimizer?
The answer to (3) is yes. Indeed, then all the conditions of what you call "special version of Tonelli’s theorem" (proved in this answer) are satisfied.
The answer to (2) is no. Indeed, for natural $n …
- 111k
1
vote
Find distribution that minimises a function of its moments
This is to complete Mateusz Kwaśnicki's answer by proving that
$$EY^2(1+Y)\ge(EY^2)^2\tag{1}$$
if $Y\ge-1$ and $EY=0$.
Since $Y\ge-1$, for any real $v$ we have
\begin{align}
Y^3=(Y+1)(Y-v)^2&+(2v-1 …
- 111k
1
vote
Accepted
On some convergence theorems by Felix E. Browder (1967)
Concerning Lemma 6: Clearly, this lemma is false if $F=\emptyset$ and, say, $u_n=nu$ for some nonzero $u\in H$. Assume therefore that $F\ne\emptyset$.
The inequality $\|u_n\|\le\|u_1\|+\|f\|$ will not …
- 111k
2
votes
Accepted
How do I integrate this inequality that appears in a paper of Rabinowitz?
For any unit vector $u$ and real $t>0$, let
\begin{equation}
h(t):=H(tu).
\end{equation}
Then
\begin{equation}
h'(t)=H'(tu)\cdot u=\frac{H'(tu)\cdot(tu)}t\ge\frac{\mu H(tu)}t=\frac{h(t)}t.
\e …
- 111k
6
votes
Accepted
Lipschitz property of the symmetric rearrangement
$\newcommand{\R}{\mathbb R}$By the continuity of measure, the nonincreasing function $\mu$ is right-continuous. The function $u^*$ is radial, that is,
\begin{equation*}
u^*(x)=U(|x|)
\end{equation …
- 111k
8
votes
Accepted
Prove or disprove the linearity of expectiles
The $\tau$-expectile, say $E_\tau X$, of a random variable (r.v.) $X$ is the root $t$ of the equation
$$r_X(t)=\rho(\tau),$$
where
$$r_X(t):=\frac{E(X-t)_+}{E(t-X)_+}, \quad \rho(\tau):=\frac{1-\tau …
- 111k
1
vote
Accepted
The regularity theorem, a non-regular minimizer problem
$\newcommand\ol\overline$The steps were:
Show that $f$ is infinitely differentiable, $\xi \mapsto f(x,\xi)$ is convex and $f_{\xi \xi} (x,\xi) > 0$ holds for all $x$ except for $x=0$.
Show that the …
- 111k
1
vote
Accepted
What is the maximum possible coefficient of variation for data taking values within a specif...
Indeed, for any random variable (r.v.) $X$ with values in $[a,b]$ and mean $\mu\in[a,b]$,
$$Var\,X\le Var\,X_{a,b;\mu}=(b-\mu)(\mu-a), \tag{1}$$
where $X_{a,b;\mu}$ is any r.v. with the unique distrib …
- 111k
1
vote
Accepted
Optimal Transport: how is this transport map Borel measurable?
$\newcommand\p\partial\newcommand{\R}{\mathbb R}\newcommand\ep\varepsilon$Let $h\colon\R^d\to\R$ be a strictly convex function.
Consider the set $S:=2^{\R^d}$ of all subsets of $\R^d$ endowed with the …
- 111k
4
votes
Accepted
Special version of Tonelli’s theorem
$\renewcommand\bar\overline$Indeed, it is not obvious why "$u'_{n_k} \to \overline{u}'$ in the sense of $L^r[a,b]$".
Look at this example: $[a,b]=[0,2\pi]$, $u_n(x)=\dfrac{\sin nx}n$, $\bar u=0$. Then …
- 111k