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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 $f: \mathbb D^n \to \mathbb{R}^n$ …
Asaf Shachar's user avatar
  • 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 …
Asaf Shachar's user avatar
  • 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 …
Asaf Shachar's user avatar
  • 6,741