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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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Elliptic regularity theory in $\mathbb{R}^2$

I recently encountered two papers discussing elliptic PDEs and variational methods. The first paper claims that according to regularity theory, the solution to $-\Delta u = ug(u^2)$ in $\mathbb{R}^2$ ...
sorrymaker's user avatar
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25 views

Show that functional $J$ satisfies Palais-Smale condition iff every $P-S$ sequence is bounded

Let $H$ be a Hilbert space and let $K: H \rightarrow \mathbb{R}$ be $C^1$ and such that $\nabla K: H \rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H \rightarrow \...
YuerWu's user avatar
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1 vote
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43 views

Finding thin plate spline subjected to boundary conditions

I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing. This question is related to : Thin-Plate-Spline ...
user8469759's user avatar
0 votes
0 answers
23 views

Prove that the inf of the quotient functional are the same for all $\lambda $

Let $q: \mathbb{R}^N \rightarrow \mathbb{R}$ be continuous, strictly positive and such that $\lim _{|x| \rightarrow \infty} q(x)=0$. For $\lambda \geq 0$ and $p \in\left(2,2^*\right)$, consider the ...
YuerWu's user avatar
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27 views

A question to the proof of Lemma 9 in "Multiple solutions for the Brezis-Nirenberg problem"

I'm currently puzzled by the final portion of the proof for Lemma 9 in the paper "Multiple solutions for the Brezis-Nirenberg problem"(DOI:10.57262/ade/1355867873). In particular, I'm unsure ...
sorrymaker's user avatar
6 votes
0 answers
139 views

Can an ellipse roll down a tilted sine curve without jumping?

Background Assume that we have a solid ellipse with uniform density, and that it rolls along a curve. In the following MO question, I asked along what curve an ellipse rolls down fastest. It was ...
Max Muller's user avatar
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1 vote
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20 views

Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)

Intro Suppose we have the following static linear equations (e.g. of an elastostatic problem): $$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$ We want a multipoint constraint of the type $$\boldsymbol{\...
Breno's user avatar
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0 answers
32 views

Express the inf of a functional via Rayleigh quotient

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^N$, with $N \geq 3$. Take $p \in\left(2,2^*=\frac{2 N}{N-2}\right]$ and define $$ S_p=\inf _{\substack{u \in H_0^1(\Omega) \\ u \neq 0}} \frac{...
YuerWu's user avatar
  • 411
1 vote
1 answer
248 views

Decay estimate of energy minimizer and the linear ODE

Suppose $a$ is constant, $b(x)=C_b(1-x)$ and $c(x)=C_c(1+x^2)$ for some positive constants, we consider the minimizer of the energy $$E(u)=\int_{\mathbb{R}}\left[a^2(u'(x))^2/2+c(x)\left(\dfrac{1}{c(x)...
mnmn1993's user avatar
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A problem about how to understand the existence of derivative of level set in Mountain-pass theorem

I'm confused about the Mountain pass theorem in Lemma4 of here. Background : $$ \begin{gathered} I_\lambda(u)=\frac{1}{2} \int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g-\frac{\lambda}{2 m} \log ...
Elio Li's user avatar
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How can we calculate the Euler-lagrange equations?

In this paper https://arxiv.org/pdf/1907.09605.pdf \ let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
Mohamed's user avatar
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3 votes
0 answers
67 views

Surface terms in the calculus of variations on jet bundles

Let $\pi:N\rightarrow M$ be a fibered manifold with $m=\dim M$ and $m+n=\dim N$. The variational bicomplex on the infinite jet space $J^\infty(\pi)$ is denoted $(\Omega^{k,l}(\pi),\delta,\mathbf d)$ ...
Bence Racskó's user avatar
-3 votes
1 answer
66 views

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
57 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
61 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
  • 131
1 vote
0 answers
47 views

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
2k views

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
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0 votes
1 answer
85 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
  • 1,033
1 vote
1 answer
90 views

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
68 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
  • 729
2 votes
0 answers
39 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
489 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
  • 729
1 vote
0 answers
79 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
6 votes
2 answers
1k 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
0 answers
84 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
1k views

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
  • 1,499
3 votes
1 answer
146 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
394 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
  • 600
6 votes
1 answer
768 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
110 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
  • 129
1 vote
0 answers
88 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
  • 729
2 votes
0 answers
154 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
  • 729
2 votes
0 answers
82 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
127 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
118 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
  • 729
2 votes
1 answer
215 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
  • 729
3 votes
1 answer
414 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
  • 181
4 votes
1 answer
156 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
0 votes
0 answers
44 views

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
  • 2,079
0 votes
1 answer
68 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
  • 461
1 vote
1 answer
98 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
98 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
68 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
157 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
  • 4,968
3 votes
1 answer
349 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
344 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
310 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
159 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
88 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
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