Questions tagged [critical-point-theory]

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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
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1answer
174 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 ...
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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) ...
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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 ...
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82 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 ...
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68 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, ...
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225 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
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1answer
141 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
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2answers
323 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 ?
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1k views

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

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\...
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2answers
196 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
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1answer
160 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
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1answer
251 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]...
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1answer
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^...
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1answer
212 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 $p(x)\in\Bbb R[x]$ ...
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110 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 ...
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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 ...
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2answers
496 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 $ [...
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3answers
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 ...
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122 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{...
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59 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
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1answer
520 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.
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1answer
921 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 ...
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2answers
519 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
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1answer
214 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 \...
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200 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/...
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139 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
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1answer
550 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
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1answer
126 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
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1answer
829 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 " $\|\...
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315 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
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1answer
207 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 $f$?
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1answer
732 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 ...