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Results tagged with perturbation-theory
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user 46290
5
votes
1
answer
211
views
Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?
While analyzing a variational problem, I came to the following question:
Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ …
- 6,741
9
votes
2
answers
875
views
Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank?
Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and tha …
- 6,741
0
votes
Accepted
Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?
It seems that the answer is negative for dimension $n=2$ . I am not sure if higher dimensions can be reduced to the $2D$ case.
Here is the argument for $n=2$:
Suppose that there exist $f_k \in C^{\i …
- 6,741
4
votes
1
answer
491
views
Is the kernel of a Fredholm operator stable under perturbation?
This is a follow-up of this question.
In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator?
Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space …
- 6,741