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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Minimal norm problem with linear combination of translation operator to be estimated

Follow up question from this one Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form $$ H = H(\alpha_1,...
user8469759's user avatar
2 votes
0 answers
37 views

A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional

I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann,...
Elio Li's user avatar
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3 votes
0 answers
54 views

About the naturality of Krasnoselskii genus on Variational Methods

I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
Jacaré's user avatar
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Optimal orbital tranfers: reference for statements & proofs

I know that Pontryagin's contribution to optimal control in the 1950'ies was allegedly inspired by the then-nascent rocket industry and the question of how to get a rocket into orbit with minimum fuel....
5th decile's user avatar
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13 votes
3 answers
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How to get to the earliest time zone?

You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take? Formally, fix spherical coordinates $(\theta, \...
Nate River's user avatar
  • 4,802
0 votes
1 answer
69 views

Equi-coercivity of functionals on a metric space

Definition: A family of functionals $\{F_n: X\to\bar{\mathbb R}\}$ on a metric space $X$ is said to be equi-coercive if, for every $\alpha \in \mathbb{R}$, there is a compact set $K_\alpha$ of $X$ ...
Guy Fsone's user avatar
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1 vote
1 answer
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How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let $$ \begin{matrix} F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\ (x,z,p) \mapsto F(...
maxematician's user avatar
2 votes
0 answers
64 views

Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem

Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
Elio Li's user avatar
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2 votes
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37 views

Functional of fully nonlinear equations

Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
Davidi Cone's user avatar
3 votes
2 answers
445 views

A problem about how dominated convergence is used in the analysis of variation

I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...
Elio Li's user avatar
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1 vote
0 answers
70 views

On Talenti's proof of optimal constant in Sobolev inequality

I'm reading the paper by Giorgio Talenti on the best constant for the Sobolev inequality. The main theorem states that for $u:\mathbb{R}^m\rightarrow \mathbb{R}$ sufficiently smooth (eg. Lipschitz) ...
user519428's user avatar
5 votes
1 answer
702 views

Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces

It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
leo monsaingeon's user avatar
1 vote
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79 views

Periodic orbits in planar smooth billiard table with large periods

Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period. Formulation of my question: We are considering ...
XYC's user avatar
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17 votes
1 answer
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What variational problem does the parabolic suspension bridge solve?

(Posted to MSE here, no answers) The catenary curve $y(x)$ minimizes the gravitational potential energy $$\int \rho g y\,ds=\int \rho g y \sqrt{1+y'^2}dx,$$ subject to a fixed length, $L=\int \sqrt{1+...
thedude's user avatar
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3 votes
1 answer
127 views

Is every area-minimizing cone a level set of a least-gradient function?

Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons ...
Leo Moos's user avatar
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8 votes
1 answer
381 views

Is there an infinite dimensional Stein's lemma?

Classical Stein's lemma says that if $\mathbf{X}$ is a centered Gaussian random vector and $g$ is a function which is nice enough, we have $$ \mathbb{E} \, X_i \, g ( \mathbf{X} ) = \sum_k \...
tsnao's user avatar
  • 462
6 votes
1 answer
759 views

A Poincaré-like inequality

Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have $$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx \le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
Iosif Pinelis's user avatar
0 votes
1 answer
105 views

Integral inner product with exponential function

Suppose on some unknown interval $[0, I]$ we have non-negative functions $f, g : [0, I] \rightarrow \mathbb{R}^{\geq 0}$. If we know that \begin{aligned} \int_0^I f & = c \\ \int_0^I e^f & = e^...
Lewwwer's user avatar
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1 vote
0 answers
86 views

Definition of stable solution of elliptic PDE and the classification of the solution (as the critical points of energy functional)

My questions arise from Here, it seems that I didn't give a clear question, so I rephrase my questions here. For example, for $$ -\Delta u=f(u) \quad \text { in } \Omega, $$ we call a solution is ...
Elio Li's user avatar
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2 votes
0 answers
153 views

A naive question about the stable solution and Morse index of elliptic PDE

For example, for $$ -\Delta u=f(u) \quad \text { in } \Omega, $$ we call a solution is stable if $$ Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \...
Elio Li's user avatar
  • 719
2 votes
0 answers
76 views

relative entropy, Fisher information, and metric slope for non-convex domains

$\newcommand{\R}{\mathbb R}$ If $\Omega\subset \R^d$ is a convex domain it is well-known that the relative entropy $$ \mathcal H(\rho)= \int_{\Omega}\rho\log\rho \ \mathrm{d}x \qquad \mbox{for }\rho=...
leo monsaingeon's user avatar
3 votes
0 answers
111 views

Is Stokes equation a saddle point problem or a minimum problem?

Consider $\Omega \subset \mathbb{R}^2 $ (or $\mathbb{R}^3$). The well known stationary Stokes equations in the incompressible case are \begin{equation} \begin{cases} - \Delta u + \nabla p = f \text{ ...
tommy1996q's user avatar
2 votes
1 answer
113 views

Generalize the conception of 'stable' solution and 'stable outside a compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold

I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold. I'm reading $\Delta u +e^u=0$...
Elio Li's user avatar
  • 719
2 votes
1 answer
209 views

Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$

Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
Elio Li's user avatar
  • 719
3 votes
1 answer
410 views

Integrability of Schroedinger's equation

Consider the periodic nonlinear Schrödinger equation $$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$ where $\mathbb{T}:= \mathbb{R}/\...
kvicente's user avatar
  • 171
4 votes
1 answer
153 views

Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians

Consider a macroscopic free energy functional of the form $$\mathcal{F}_\beta(\mu):= \frac{1}{\beta}\int_{\mathbb{R}^d}\log(\mu)\mu dx + \int_{\mathbb{R}^d}V(x)\mu(x)dx + \iint_{(\mathbb{R}^d)^2}g(x-y)...
Matt Rosenzweig's user avatar
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0 answers
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A dilemma in a lemma regarding Hexagonal Fuzzy approximation of parabolic fuzzy number

I was reading Nayagam & Murugan - Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM paper regarding hexagonal fuzzy approximation. In that, the statement of Lemma 3.1 and ...
vidyarthi's user avatar
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0 votes
1 answer
63 views

Continuity of generalised Legendre transform

Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$]. For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, we consider ...
fsp-b's user avatar
  • 421
1 vote
1 answer
95 views

Optimization on non-convex set

Let $\Omega$ be an open bounded subset of $\mathbb{R}^2$ and $f\in L^2(\Omega)$ be a given function. Consider the optimization problem $$\mathrm{min} \int_\Omega u(x) f(x) \,dx\,,$$ where a minimum is ...
mlogm's user avatar
  • 11
1 vote
1 answer
96 views

Linear response for SDE

Consider a family of stochastic processes $dX^h_t=(g(X^h_t)+h(s))\,dt+dW_t$ and a functional $I_f:h(s) \rightarrow E[f(X_t^h)] $. I would like to compute the kernel of the derivative of this ...
Vash's user avatar
  • 13
2 votes
0 answers
67 views

Extremizing the integral part of an integro-differential equation

Consider the problem of finding a continuously twice-differentiable function $x(t)$ which extremizes the convergent improper integral \begin{equation} I=\int_{-\infty}^{t} f(x,s)\mathop{ds} \end{...
UNOwen's user avatar
  • 79
1 vote
1 answer
146 views

How to find critical points of functionals when there is a boundary?

Banach spaces have a relatively uninteresting topology, because they are contractible. This prevents the direct application of Morse-like min-max arguments to establish the existence of critical ...
Leo Moos's user avatar
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3 votes
1 answer
262 views

Thin-Plate-Spline understanding and solution

This is a migrated question from: Thin-Plate-Spline understanding and solution. If I need to delete one of the questions let me know. I was suggested to post it here as well. As I understand it a Thin-...
user8469759's user avatar
-1 votes
2 answers
322 views

Conditional expectation: commuting integration and supremum

Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
Vokram's user avatar
  • 109
2 votes
0 answers
40 views

$1$-parameter family of metrics preserving the normal direction

Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
Eduardo Longa's user avatar
2 votes
1 answer
283 views

How to solve an optimization problem whose optimization variable is a function?

I would like to find an optimal probability density function (PDF) $f$. Given $b$, $$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\...
Erik's user avatar
  • 21
3 votes
0 answers
152 views

A variant of the Laplace principle

$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
leo monsaingeon's user avatar
1 vote
0 answers
86 views

Nonlocal elliptic problem - what is its associated energy?

It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem: $$...
Bogdan's user avatar
  • 1,330
0 votes
0 answers
63 views

Euler-Lagrange equation with extra variable under integral. (Reference request)

Assume that we have a functional to optimize of the form $$\iiint L[x, f_y(y, z), f_z(y,z)]dxdydz.$$ As one may notice, the optimized function $f$ depends only on $y$ and $z$, which means that writing ...
Dmitri Scheglov's user avatar
5 votes
0 answers
101 views

Varifold convergence of images of Sobolev maps

Suppose I have a sequence of maps $\{f_k:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\}$ such that: $f_k\rightharpoonup f_*$ weakly in $W^{1,p}(\Omega,\mathbb{R}^{n+1})$, The images $\Sigma_k:...
sobol's user avatar
  • 151
5 votes
0 answers
99 views

What normal variational vector fields are allowed for the area to be preserved?

Let $(M^3,g)$ be a Riemannian manifold with boundary and let $\varphi : \Sigma \to M$ be a two-sided proper embedding of a compact surface with boundary, with unit normal $N$. If $\varphi$ is not ...
Eduardo Longa's user avatar
1 vote
2 answers
206 views

How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$

Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that \begin{equation}\tag{1}\label{1} \int_a^b \...
Hyperbolic PDE friend's user avatar
1 vote
0 answers
84 views

Reference request: theory for local minimizers in the calculus of variations

Let $F: \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be the Lagrangian. We say that $f \in X$ is a local minimizer of the variational integral if for all compact sets $C \subset \...
Franlezana's user avatar
2 votes
0 answers
59 views

Defining minimality 'through deformations'

Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if ...
Leo Moos's user avatar
  • 4,968
2 votes
0 answers
52 views

A variational problem

I was studying the paper "Non-holonomic Euler-Poincaré equations and stability in Chaplygin's sphere" by D. Schneider (2002). A non-holonomic constraint - see equation (24) - is expressed as ...
Simon's user avatar
  • 91
3 votes
1 answer
280 views

Are all Helmholtz decompositions related?

Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...
MrPie 's user avatar
  • 185
3 votes
1 answer
137 views

Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?

Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
Analyst's user avatar
  • 637
1 vote
1 answer
228 views

Integral identity for critical points of the Ginzburg-Landau functional

I am reading a paper of Comte and Mironescu [CM96], where they discuss critical points $v = v_{\epsilon}: G \to \mathbf{C}$ of the (non-magnetic) Ginzburg–Landau functional $E_\epsilon(v) = \frac{1}{2}...
Leo Moos's user avatar
  • 4,968
12 votes
0 answers
387 views

A model of pillows

(The same system with slightly different questions has been asked in MSE.) Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...
Daniel Castro's user avatar
0 votes
0 answers
105 views

Wasserstein compactness of sublevel sets of relative entropy

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\...
pseudocydonia's user avatar

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