# Questions tagged [critical-point-theory]

The critical-point-theory tag has no usage guidance.

35
questions

**6**

votes

**0**answers

175 views

### Optimal configurations on the flat torus

I'm studying (with my colleagues R.Piergallini and S.Isola) configurations of points on the flat torus which minimize an attractive or repulsive potential depending on the distance. Two model cases ...

**3**

votes

**1**answer

138 views

### Number of critical points of sum of two functions

I ran into the following "simple" question and I am wondering whether there are any references, which might help me. I am coming from statistics, so I am not so aware which branch of math ...

**1**

vote

**0**answers

58 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 ...

**4**

votes

**1**answer

207 views

### Minimizing sequence $\implies$ Palais–Smale sequence

Set $F:\mathbb{R}^n\rightarrow \mathbb{R}$ a $C^2$-function that is bounded from below. Set $x_n$ a minimizing sequence, i.e., $F(x_n)\to \alpha = \inf F$. I want to prove that under the assumption of ...

**2**

votes

**0**answers

109 views

### Laplacian variational problem with asymptotically quadratic term

Consider the functional
$$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$
where $\Omega$ is a bounded smooth domain.
The problem has been solved for example if $F$ is (1) subquadratic, or (2) ...

**2**

votes

**0**answers

58 views

### Differences among various index theories in critical point theory

Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones?
the ...

**0**

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**0**answers

88 views

### Generalized Sard's lemma

Let $f: X \to \mathbb{R}$ be a $C^{1,1}$ (that is $C^1$ with Lipschitz differential) function on a manifold $X$. Suppose that $f$ is smooth at all points of a subset $C \subset \text{Crit}f$ of ...

**5**

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73 views

### Is there an upper bound on the number of critical points of a spherical harmonic on a local scale?

Take a spherical harmonic $y_d$ of degree $d$ on the sphere $\mathbb{S}^2$ and a spherical disk of radius $\frac{1}{d^2}$ centered at any point (let's say the north pole).
Is there an upper bound, ...

**5**

votes

**0**answers

245 views

### Pullback of Morse form satisfies Palais Smale

Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...

**3**

votes

**1**answer

171 views

### Smoothing a periodic function of two variables

Let $F \colon \mathbb{R}^2 \to \mathbb{R}^n$ be a $C^{1}$-function 1-periodic in each variable, so it can be considered as a function on the flat torus $\mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$. We ...

**5**

votes

**2**answers

397 views

### Critical values of analytic functions of several variables

Let $f:\mathbb{R}^d\to \mathbb{R}$ be real analytic. Define $S=\{x\in\mathbb{R}^d, \nabla f (x)=0\} $. Is it true that for any compact set $K\subset \mathbb{R}^d$, $f(S\cap K)$ is a finite set ?

**19**

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2k views

### Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$

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$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\...

**3**

votes

**2**answers

213 views

### Finding the critical points of a degree $5$ Blaschke product

The derivative of a degree $5$ polynomial $p\in\mathbb{C}[z]$ is a degree four polynomial $p'\in\mathbb{C}[z]$, and as such, the zeros of $p'$ may be found explicitly using the quartic formulae.
One ...

**3**

votes

**1**answer

174 views

### C^1 fractals in statistical mechanics

It is well-known--even famous--that the Schramm-Loewner curves appear as domain boundaries between phases at second-order critical points like the critical Ising model or percolation in two dimensions....

**2**

votes

**1**answer

253 views

### Bijection of critical points on two manifolds

Suppose that $f$ and $g$ are two smooth functions defined on $R^n$. Assume that $(a-\epsilon, b+\epsilon)$ contains no critical point of $g$. Then $g^{-1}[a,b]$ it homomorphic to $g^{-1}(a)\times [a,b]...

**0**

votes

**1**answer

135 views

### Applying min-max to find a critical point in a ball

Let $\mathbb B^n$ be an open unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\frac{1}{2}\mathbb B^n\subset \mathbb B^n$ be an open ball or radius $\frac{1}{2}$. Let $\mathbb B^...

**2**

votes

**1**answer

228 views

### Polynomial with subset of critical points and values prescribed

Motivated by this question I wish to pose the following question:
Given $k$ points $(x_1, y_1), \ldots (x_k, y_k)$ with (WLOG) $x_i < x_{i+1}$, can we find a polynomial $p(x)\in\Bbb R[x]$ ...

**0**

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**0**answers

112 views

### When is a critical value of a map contained in the interior of the image?

Let $M^n$ be a compact manifold, and $F\colon M \to \mathbb{R}^n$ a smooth map. The inverse function theorem implies that every regular value of $F$ lies in the interior of $F(M)$, hence every point ...

**2**

votes

**0**answers

276 views

### Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...

**2**

votes

**2**answers

553 views

### Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities

This question is an expansion of another question that I asked over at Math Stack Exchange.
In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ [...

**36**

votes

**3**answers

2k views

### Height function on 2-torus with only 3 critical points

It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number ...

**3**

votes

**0**answers

124 views

### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & \mathrm{in}\hspace{...

**1**

vote

**0**answers

62 views

### Estimation of the number of local extrema

I have a question about a simple proposition, I suppose that this is something
well-known or a special case of something well-known:
Let $D\subset\mathbb{R}^{2}$ be the closed unit disk in the plane ...

**2**

votes

**1**answer

544 views

### Lusternik-Schnirelmann Theorem

In various paper i found this:
But i don't find this theorem of Lusternik-Schnirelmann, have you an idea where i can find this theorem, the condition?
Thank you.

**0**

votes

**1**answer

1k views

### Set of critical values is compact [closed]

Let $M\subseteq \mathbb{R}^n$ be a compact manifold with $\partial M=\emptyset$ and let $f: M\rightarrow S^p$ be a smooth map. I want to unerstand the proof of the following lemma which occurs in ...

**10**

votes

**2**answers

554 views

### Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial

The Gauss--Lucas Theorem states that all zeros of derivative of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros ...

**3**

votes

**1**answer

237 views

### Does this squared distance functional have a unique critical point on geodesically convex manifolds?

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow \...

**2**

votes

**0**answers

204 views

### Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here:
http://www.mtm.ufsc.br/...

**0**

votes

**0**answers

146 views

### Categorizing saddle points of real multivariate polynomials

I have a multivariate polynomial function of N variables
$f(x_1,x_2,…,x_N) = x_1 x_2 x_3 .. x_N \left( 1 + \sum_i^N (a_i x_i^2 - x_i) \right)$,
where $a_i > 0$ are real positive numbers.
By ...

**4**

votes

**1**answer

567 views

### Failure of Palais-Smale Condition C and the Mini-Max Principle

To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is Condition ...

**0**

votes

**1**answer

131 views

### Continuity of critical points with respect to a parameterisation.

Hello.
I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in ...

**1**

vote

**1**answer

848 views

### How to explain the condition (C) in critical point theory?

Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which $\|\nabla f\|$is not bounded away from zero, contains a critical point of f.
How to see the meaning of " $\|\...

**3**

votes

**0**answers

320 views

### What is the analog of “monotonic” for scalar functions on surfaces?

"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...

**1**

vote

**1**answer

213 views

### Critical points in Hilbert space [closed]

Let $f$ be a $C^1$ functional on a Hilbert space $X$, and $Y$ a closed subspace of $X$.
Suppose the restriction of $f$ on $Y$ has a critical point $x_0 \in Y$.
Q: Is $x_0$ a critical point of $f$?

**4**

votes

**1**answer

751 views

### Do the focal points of a submanifold $M$ in $\mathbb{R}^k$ form a closed subset?

Let $M$ be a submanifold in an euclidean space $\mathbb{R}^k$, and $\nu(M)$ the normal bundle to $M$, let us denote $\phi$ the restriction to $\nu(M)$ of the exponential map for $\mathbb{R}^k$.
A ...