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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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
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1answer
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
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1answer
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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1answer
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 $X$ ...
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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 ...
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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
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1answer
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 ...
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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 ...
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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
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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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2answers
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 ...
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1answer
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 |\...
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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
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1answer
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 ...
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1answer
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-...
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3answers
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 ...
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3answers
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
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2answers
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 -...
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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 ...
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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{...
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1answer
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 \...
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2answers
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
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1answer
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 ...
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1answer
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
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1answer
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 $$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...
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0answers
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\} \...
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2answers
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 ...
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1answer
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 ...
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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 ...
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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
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1answer
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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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}...
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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}|\...
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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 $\...
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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 ...
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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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0answers
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 ...
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0answers
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 \...
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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\...
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2answers
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 ...
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1answer
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) \...
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0answers
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{...
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1answer
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 ...
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0answers
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 ...
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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}...
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1answer
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
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0answers
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
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0answers
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
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0answers
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*} ...
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0answers
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)=...

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