# Questions tagged [calculus-of-variations]

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

555
questions

2
votes

0
answers

83
views

### 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 $\mathbb{R}^2$ ...

0
votes

0
answers

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 \...

1
vote

0
answers

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 ...

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 ...

0
votes

0
answers

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 ...

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 ...

1
vote

0
answers

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{\...

0
votes

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{...

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)...

0
votes

0
answers

50
views

### 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 ...

1
vote

0
answers

47
views

### 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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...

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 $(\Omega^{k,l}(\pi),\delta,\mathbf d)$ ...

-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,...

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,...

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 ...

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....

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, \...

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 $X$ ...

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(...

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 ...

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 ...

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 ...

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) ...

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\...

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 ...

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+...

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 ...

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 \...

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
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$

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^...

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 ...

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, \...

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=...

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{ ...

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 $\Delta u +e^u=0$...

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 ...

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}/\...

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)...

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 ...

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 ...

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 ...

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 ...

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{...

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 ...

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-...

-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 ...

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 $S^2_0(M)$ ...

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\...

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 ...

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:
$$...