# Questions tagged [calculus-of-variations]

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

362
questions

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votes

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45 views

### Inf sup condition for discrete Stokes problem

I am considering the discrete inf-sup condition for the discrete Stokes problem
$ \inf_{q \in Q_h} \sup_{v \in V_h} \frac{(q, \nabla \cdot v)}{\| q \| \| \nabla v\|} \geq \beta > 0$
This means ...

**4**

votes

**1**answer

84 views

### minimizing weighted length of closed curves

Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...

**3**

votes

**1**answer

152 views

### Reference request: Variational techniques for complex “iterated” Lagrangians

I am interested in solving variational problems of the form
$$
\min_u \int \Big\{L(x,y,u(x,y)) + \phi\Big(\int J(z,y,u(z,y))\,dz\Big)\Big\} p(x,y)\,dx\,dy.
$$
for some known, smooth functions $L,J,\...

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votes

**1**answer

55 views

### Minimum mean over all random variables subject to logarithm constraint

Does the following problem have a solution?
$$
\min_X \mathbb{E} X
\quad\text{subject to}\quad
\mathbb{E} \log X = C.
$$
Here, the minimization is with respect to all integrable random variables $X$ ...

**1**

vote

**0**answers

91 views

### Correct translation of the title from French

It is not a research question. But it is a question to working researchers, who is aware of old and new terminology of calculus of variations. That is why I ask it here.
I want to figure out correct ...

**1**

vote

**0**answers

52 views

### Is there a flaw in this proof of the validity of the Palais-Smale condition?

In Chapter 3 of his monograph (available on Researchgate), Kavian applies the Mountain Pass Theorem to a semilinear elliptic equation. To this aim, he needs to check that a functional satisfies the ...

**3**

votes

**1**answer

134 views

### Ricci flow proof of isoperimetric inequality

It is well-known in geometric analysis that one can use curve-shortening flow to prove the isoperimetric inequality (where the general result requires curve-shortening flow for non-convex curves).
I ...

**2**

votes

**0**answers

52 views

### Analogous $H^1$-space for pseudo inner products

Perhaps this is a naive question but I could not find anything related to this.
Imagine we are on a bounded and regular open subset $\Omega$ of $\mathbb{R}^3_1$, i.e, $\mathbb{R}^4$ is considered ...

**-2**

votes

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51 views

### the subdifferential at points of differentiability in infinite dimensional space

Let $ f:X \longrightarrow (-\infty,\infty] $ that $X$ is infinite dimensional space and $f$ be a proper convex function and $ x\in int(dom(f))$.
Is it the case that: if $f$ is differentiable at $x$, ...

**2**

votes

**0**answers

32 views

### First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space

The curve $\Gamma$ in $\mathbb{R}^2$ is defined by a continuous and monotonically increasing function $f(x)\in\text{C}[0,1]$, where $f(0)=0$, $f(1)=1$.
Let $(X,Y)$ is jointly and uniformly ...

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vote

**2**answers

48 views

### Non-parametric regression and curvature

Given a finite set of points $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ in the plane, Linear Regression tells us how to find the straight line "$y=a+bx$" best approximating the given points, in the ...

**2**

votes

**1**answer

119 views

### Poincaré inequality under weighted average condition

Let $\Omega=[0,1]^2$ be the unit square and $a>0$.
1) I would like to know one estimate of the constant $C(a)$ such that
$$
\forall u\in W^{1,2}(\Omega)\quad \int_\Omega u^2\le C(a)\int_\Omega |\...

**3**

votes

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101 views

### Optimal Poincaré constants under combined boundary and average conditions

Let $\Omega=[0,1]^2$ be the unit square, $\Gamma_1=[0,1]\times\{0;1\}$ its horizontal boundary and $\Gamma_2= \{0;1\}\times[0,1]$ its vertical boundary.
I would like to know the optimal Poincaré ...

**3**

votes

**1**answer

133 views

### Prove or disprove the linearity of expectiles

For expectation (mean), there are many useful properties such as Linearity of Expectation:
$\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$
$\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$
(The 2 equations ...

**1**

vote

**1**answer

45 views

### Variational problem: how to minimise the second moment?

This is a neater version of a question I posted here, on which I'm also stuck.
The problem: Say I have a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-...

**4**

votes

**3**answers

97 views

### Find distribution that minimises a function of its moments

Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive).
How can I find $f(x)$ that ...

**3**

votes

**3**answers

377 views

### Functional derivatives on Banach spaces

Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. ...

**4**

votes

**2**answers

192 views

### $| f_n |^p - | f |^p - | f_n-f |^p$ converges in distribution sense if $f_k$ converges almost everywhere and weakly to $f$?

Let $1<p<\infty$ and $f_n$ be a sequence in $L^p(S^1)$ that converges weakly to some $f$. Here $S^1$ is the circle so we are dealing with periodic functions.
Let us see if $| f_n |^p - | f |^p -...

**2**

votes

**0**answers

46 views

### Variational forms of non-convex functions

I am trying to understand what kind of variational forms exist for non-convex functions. Alternatively, are there conjugate forms which attain strong duality? For a non-convex function $f$, I am ...

**1**

vote

**0**answers

26 views

### Commonly used metrics to compare two Young measures

Let $\Omega\subset \mathbb{R}^n$ be a bounded open set, $K\subset \mathbb{R}^d$ be a compact set, and $M_1(K)$ be the set of probability measures on $K$. Then a Young measure is defined as a $\textrm{...

**-2**

votes

**1**answer

270 views

### Why this function is monotonic?

Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \...

**2**

votes

**2**answers

148 views

### Representing a nonlinear elliptic PDE as an energy minimization problem

I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...

**5**

votes

**1**answer

191 views

### Do there exist any variational principles on the space of braids (or knots)?

This is very speculative question and I do not know where to start looking up the literature, or if what I am looking for is even mathematically possible/meaningful.
Q: I am interested in finding out ...

**9**

votes

**1**answer

501 views

### Do these surfaces intersect?

For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$
with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$,
does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...

**2**

votes

**1**answer

121 views

### Finite energy solution for Allen -Cahn equation

I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional
$$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...

**1**

vote

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40 views

### Infimum of odd path functional

Let $c>0$, $A_c\triangleq \left\{ f \in C_0([0,1]:\mathbb{R}^d): \|f_t\| >c \mbox{and} \dot f \mbox{ exists-a.e.}\, \mbox{ for some } t \in [0,1]
\right\}$, and set $g(x)=\left(\max\{x_i,0\} \...

**16**

votes

**2**answers

651 views

### Functional approach vs jet approach to Lagrangian field theory

Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...

**1**

vote

**1**answer

69 views

### Integral of inverse of a function [closed]

Is it possible to obtain an expression to the integral of the inverse of a function, maybe dependent of its primitive.
$\int_{0}^t \frac{1}{f(\tau)} d \tau = g(t) - g(0)$
Above the corresponding ...

**2**

votes

**0**answers

52 views

### Prove the equicontinuity of a maximizing sequence

Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...

**0**

votes

**0**answers

75 views

### Area of Disc that Intersects Another under Smooth Flow

The following question can be asked in any $\mathbb{R}^n$ for $n > 1$, but the case of interest is (thankfully) the case $n = 2$. The formulation of the problem with discs isn't actually critical ...

**3**

votes

**1**answer

118 views

### Energy Decay of the functional $\int_{B_1} |Du|^2 +Au^2$

Suppose $u \in C^1(B_1)$ with $B_1 \subset \mathbb{R}^n$ such that $\Delta u =0$ weakly. We would have the energy decay estimate
$$\int_{B_r} |Du|^2 \leq r^n \int_{B_1} |Du|^2.$$
Now suppose $u \in C^...

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votes

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36 views

### A classic uniqueness problem in a constraint minimization problem

Consider the following constraint minimization problem
$$
\inf_{\| u \|_p = 1} \int_{\mathbb{R}^N} |\nabla u|^2 + V(x)u^2 \,dx
$$
where $\| \cdot \|_p$ is the $L^p$ norm, $2 < p < \frac{2N}{N-2}...

**0**

votes

**0**answers

28 views

### Controlling oscillation of a p-harmonic function in a small ball

Given $\Omega\subset \mathbb R^N$ open. And $u:\Omega\rightarrow \mathbb R$ be a $p$-harmonic function. That is it minimizes the functional:
$$
\min _{v\in W_{\varphi}^{1,p}(\Omega)}\int_{\Omega}|\...

**0**

votes

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77 views

### Help showing F is weakly lower semicontinuous

Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...

**0**

votes

**0**answers

27 views

### Given a sequence of self-adjoint operator $L_w$, find the parameter $w$ maximizing the spectral gap of $L_w$

Let $(E,\mathcal E,\mu)$ be a probability space, $$L^2_0(\mu):=\left\{g\in L^2(\mu):\int g\:{\rm d}\mu=0\right\}$$ and $k\in\mathbb N$. Suppose for each $w\in L^2(\mu;\mathbb R^k)$, there is a ...

**0**

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44 views

### interchange of minimization and integration on a product space

Let
$(\Omega,\mathcal A,\mu)$ be a measure space
$E$ be a $\mathbb R$-Banach space
$K$ be a closed subspace of $E$
$\Gamma:=\left\{x:\Omega\to E\mid x\text{ is }\mathcal A\text{-measurable and }x(\...

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votes

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40 views

### Optimization of an integral functional when the multiplier rule yields no useful information

Let
$(E,\mathcal E,\lambda)$ be a measure space
$\mu\ll\lambda$ be a probability measure on $(E,\mathcal E)$
$p\in[1,\infty)$
$k\in\mathbb N$
$f:E\times\mathbb R^k\times\mathbb R^k\to[0,\infty)$ such ...

**2**

votes

**0**answers

74 views

### Link between Yamabe invariant and Yamabe equation

I am trying to understand the solution to the Yamabe problem as presented by Lee and Parker. It seems to me that the constant $\lambda$ which appears in the Yamabe equation $$\square\varphi = \lambda \...

**1**

vote

**0**answers

71 views

### Can we reduce the maximization of this integral to the maximization of the integrand?

I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...

**7**

votes

**2**answers

324 views

### Do Minkowski sums have anything like calculus?

Is there anything resembling differential calculus over the space of (nicely behaved) regions in $\mathbb{R}^d$, where addition is interpreted in terms of Minkowski sums? For example, it is known ...

**1**

vote

**1**answer

162 views

### Converting an integral equation into a differential equation

Let $a, b \in \mathbb R^n$ and $f, g \in L^1 [0,1]$. Assume for all $h \in AC[0,1 ]$ (space of absolutely continuous functions) following integral equality holds
$$ \int_{0}^{1} \langle f(t) , h(t) \...

**1**

vote

**0**answers

77 views

### Energy functional continuous with respect of time $t$

I am studying a paper of Liu Yacheng which named "On potential wells and applications to semilinear hyperbolic equations and parabolic equations" it considers a nonlinear parabolic equation
\begin{...

**2**

votes

**1**answer

151 views

### Properties of the topology of sequential convergence $\tau_{seq}$

Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_{seq}$ has the following ...

**3**

votes

**0**answers

196 views

### Maximize an $L^p$-functional subject to a set of constraints

Let
$(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces
$f\in L^2(\lambda)$
$I$ be a finite nonempty set
$\varphi_i:E'\to E$ be bijective $(\mathcal E',\mathcal E)$-measurable ...

**0**

votes

**0**answers

51 views

### When is $ \int_{0}^{1} F (t , x(t) , x' (t)) \; dt$ finite for all trajectories $x(t)$?

Let $f : AC[0, 1] \to [- \infty , + \infty]$ be defined by $ f(x(.)) := \int_{0}^{1} F (t , x(t) , x' (t)) \; dt$. Where, $AC[0, 1]$ is the set of absolutely continuous functions with the norm $W^{1,1}...

**2**

votes

**1**answer

173 views

### Is $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$ differentiable?

Let $f : AC[0, 1] \to R$ be defined by $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$. Where, $AC[0, 1]$ is the set of absolutely continuous functions with the norm $W^{1,1}$, and $F: R^n \to R$ is ...

**2**

votes

**0**answers

158 views

### Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?

Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...

**2**

votes

**0**answers

43 views

### Variational inference: Does the natural gradient follow (Fisher-Rao) geodesics locally?

Amari's natural gradient descent is a well-known optimisation algorithm for functionals defined on statistical manifolds. It consists of preconditioning the vanilla gradient descent update rule with ...

**2**

votes

**0**answers

39 views

### Extremal field implies minimality — low regularity assumptions?

In A stability theorem for minimal foliations on a torus, Moser is studying variational integrals of the following form:
\begin{equation*}
\mathcal{F}(u) = \int F(x,u(x),Du(x)) \, dx
\end{equation*}
...

**1**

vote

**0**answers

132 views

### Gradient of squared riemannian distance on complete manifold

Let $\theta: M \times M \to \mathbb{R}$ the squared distance function $\theta(x,y)=d(x,y)^{2}$ on complete Riemannian manifold $M$. I would like to calcule the gradient of $d^{2}$, where $d^{2}_{y}(x)=...