Question: Let $f \colon \Omega \to \mathbb{R}^n$ be such that $f$ is continuously differentiable where $\Omega$ is a bounded connected set in $\mathbb{R}^n$. For each $t \in \mathbb{R}$ define $f_t(x) := f(x) + t x$. Let $D$ be any open connected set in $\Omega$ such that $\overline{D} \subset \Omega$. Show that $f_t$ is injective on $D$ for sufficiently large $t$ and $f_t(\partial D) = \partial(f_t(D))$. Also, suppose $g\colon \Omega \to \mathbb{R}^n$ that is continuously differentiable with $f(x) = g(x)$ for $x \in \partial D$. Show that
$$ \int_D |J_g|\, dx = \int_D |J_f|\, dx$$ where $J_g, J_f$ are Jacobians of $f$ and $g$ respectively.
This questions seems to require the use of inverse function theorem due to the injectivity involved, however I not too sure how to tackle $f_t(\partial D) = \partial(f_t(D))$ and well as the second part of the question involving Jacobians. Also, is it necessary for $\Omega$ to be connected?
Best Answer
I will just give you some hints and leave the details to you.
The inverse function theorem will only give you local invertibility and not global invertibility. I assume that $\Omega$ is open. Since the closure of $D$ is compact, it has positive distance from the boundary of $\Omega$. If $2d$ is the distance, we can enlarge $D$, and the open set $$A=\{x\in \Omega: dist(x,\partial D)<d\}$$ has closure contained in $\Omega$. Since $f$ is continuously differentiable, it’s gradient is bounded in the closure of $A$. Use this to prove that $f$ is Lipschitz continuous in the smaller set $D$ with Lipschitz constant $L$.
Now assume by contradiction that $f_t$ is not injective. Then there exist $x,y$ in $D$ such that $f_t(x)=f_t(y)$. It follows that $$t(x-y)=-f(x)+f(y).$$ Taking the norm on both sides, and using the fact that $f$ is Lipschitz, you get a contradiction provided $t>L$.
The fact that $\partial f_t(D)=f_t(\partial D)$ comes from the fact that, by the inverse function theorem, the function $f_t$ is open, so it maps open sets into open sets.
The last part should follow by using $f_t$ as a change of variables.