Convergence of the inverse in Sobolev spaces

Question

Assume we have a sequence $f_k$ which converges to $f$ in the Sobolev space $H^p(D)$, where $D\subset\mathbb{R}^N$ ($N\geq 2$) is relatively compact and $p\geq 1$ is an integer. We also assume that $$ \| f^{-1}_k \|_{H^q(D)}\leq c\qquad \mbox{ for all } k\geq 0.$$
where $q\geq 1$ is an integer and $c>0$ is a constant (independent of $k$). What are the lowest $p$ and $q$ such that $$ \| f_k^{-1} - f^{-1}\|_{H^{q-1}(D)} \to 0 \quad ?$$ Or is it a difficult question?

Answer 1

Since $H^q(D)$ is a Hilbert space, you get a subsequence of $\{f_k\}$ (without relabeling), such that $f_k^{-1} \rightharpoonup v$ in $H^q(D)$ for some $v \in H^q(D)$. Using the compact embedding from Rellich, we have $f_k^{-1} \to v$ in $H^{q-1}(D)$.

Next, check $v = f^{-1}$. Again there are subsequences (without relabeling) such that $f_k \to f$ pointwise a.e. and $f_k^{-1} \to v$ pointwise a.e. Thus, we get $v = f^{-1}$.

Finally, one can use a standard subsequence-subsequence argument to get the convergence of the entire sequence $\{f_k^{-1}\}$ towards $f^{-1}$ in $H^{q-1}(D)$.