Questions tagged [critical-point-theory]
The critical-point-theory tag has no usage guidance.
22 questions with no upvoted or accepted answers
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Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
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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 ...
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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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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 ...
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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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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 ...
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Characterization of critical point of an integral operator
I have an integral operator and I wonder how I can characterize the critical point.
I'll give a simplified example so maybe people can comment on and I can maybe generalize in another question.
...
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Is a function looking like a cubic cusp globally equivalent to the cubic cusp?
Let's consider a family of smooth odd functions $\phi_v(u)\colon \mathbb{R}^2\to\mathbb{R}$, which ,looks like' a family of functions $f_y(x)=x^3-yx$ in the vicinity of $(0,0)$: $\phi_v(u)$ has no ...
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Critical points of the area functional restricted to CMC embeddings
For fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,\alpha}$ embeddings $f, f' : M \to N$ are said to be equivalent if there exists $\varphi \in \operatorname{Diff}(M)$ such that $f' = f \...
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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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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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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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How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
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Locality and restriction properties for self-avoiding and loop-erasing random walks
This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa
I ...
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what are definitions of born or die (birth-death point) and crossing point?
in this paper we have :
A presentation for the mapping class group of a closed orientable surface.by Hatcher.W.Thurston
...(a) $f_{t_{0}}$ has exactly one degenerate critical point, of the form $f_{t}(...
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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 ...
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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 ...
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Sufficient conditions for a polynomial function to have the same critical points as its symmetrized version
Are there any sufficient conditions known for a polynomial function (of many variables) to have the same critical points as its symmetrized version (with respect to all variables)?
This question has ...
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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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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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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 ...
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