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Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and let $f \in C^1(\overline{\Omega})$. Following Topological degree theory and applications, we define the degree of $f$ with respect to some regular point $p \notin f(\partial \Omega)$ by $$\deg(f,\Omega,p) = \sum\limits_{x \in f^{-1}(p)} \text{sign}(J_f(x))$$ where $J_f(x)$ denotes the jacobian determinant of $f$ at $x$. In fact, if $f \in C^2(\overline{\Omega})$, $\deg(f,\Omega,\cdot )$ is locally constant so the degree of $f$ can be defined at a singular point $p$ by taking a regular point $q$ closed to $p$: $$\deg(f,\Omega,p)=\deg(f,\Omega,q), \ \|p-q\| < d(p,f(\partial \Omega))$$ A fundamental property is that if $H \in C^2([0,1] \times \overline{\Omega})$ and $p \notin H_t(\partial \Omega)$ for all $t \in [0,1]$, then $\deg(H_t,\Omega,p)$ does not depend on $t$ (homotopy invariance).

My question is: how to prove it?

Seirios
  • 33,157

1 Answers1

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Proposition 1.2.2 in Topological degree theory and applications can be generalized without changing the proof:

Proposition: Let $\Omega \subset \mathbb{R}^n$ be a bounded open set, $f \in C^1(\overline{\Omega})$ and $p \notin f(\partial \Omega)$ a regular point of $f$. Then there exists $\epsilon_0>0$ such that for any function $\phi : \overline{\Omega} \to \mathbb{R}$ satisfying $\int_{\Omega} \phi=1$ and $\phi(x)=0$ if $\| x \|>\epsilon_0$, $$\deg(f,\Omega,p):=\sum\limits_{x \in f^{-1}(p)} \text{sign}(J_f(x))= \int_{\Omega} \phi(f(x)-p)J_f(x)dx$$

Then, lemma 2 and theorem 3 in An elementary analytic theory of the degree of mapping in n-dimension space can be proved in almost the same way; just use Sard's lemma to justify that $p$ can be supposed regular.

Another important detail: in Heinz's article, it is prove that we can take $\epsilon_0= d(p,f(\partial \Omega))$ whereas we don't know how evolve $\epsilon_0$ with $f$ in the previous proposition. However, such an information is needed to prove lemma 2. So we have to add some details about this point in the previous proof.

Seirios
  • 33,157