Topological degree theory

Question

Let $D$ be a region in $R^n$. If $f:D\to R^n$ is continuous, nonzero on $\partial D$ and of Brower degree 0, does there exists a continuous function $g=f$ on $\partial D$ and $g\neq 0$ on $D$?

Answer 1

If $D$ is a smooth compact manifold with boundary, then any $f: \partial D \to \mathbb{R}^n \setminus 0$ of degree $0$ can be extended to a map $D \to \mathbb{R}^n \setminus 0$. Proof: A theorem of Hopf asserts that homotopy classes of maps $\partial D \to S^{n-1}$ are determined by their degree. Thus your map $f|_{\partial D}$ is nullhomotopic. Use the nullhomotopy to extend $f$ over a small interior collar of $\partial D$ inside $D$. On the interior boundary of the collar, it is constant and can be extended by the constant map to all of $D$.

Answer 2

I assume that the setting is as described by Pietro in the comments above, so $D$ is a connected open set and $f(\partial D)$ does not contain $0$. You can perturb $f$ so that $f|_D$ is smooth and $0$ is a regular value. Then $f^{-1}(0)$ consists of finitely many points $x_1,\ldots, x_{2n}$ contained in the open domain $D$ and $f$ is a local diffeomorphism at each $x_i$. Since the degree is zero, half of these local diffeomorphisms are orientation-preserving, say on $x_1,\ldots, x_n$. The others are orientation-reversing. Choose $n$ disjoint smooth arcs in $D$ that connect $x_i$ to $x_{i+n}$. Take a small regular neighborhood of each arc: it is an open ball with smooth boundary. On each such ball $B$ the map $f|_B$ has degree zero and you can apply Ebert's argument to modify $f|_B$ such that it avoids $0$.