Limits of sequences of volume preserving homeomorphisms in $W^{1,p}$, for $0<p<1$ (2505.02482v1)

Abstract: Douady referred in a personal communication to Peetre that, if $0<p\<1$, the canonical mapping from $W^{1,p}(\mathbb{R})$ to $L^p(\mathbb{R})$ is not injective. Peetre \cite{peetre} went further, proving that the continuous and linear function $\pi=(\pi_1,\pi_2):W^{1,p}(\mathbb{R})\rightarrow L^p(\mathbb{R})\times L^p(\mathbb{R})$ such that, for $f\in C^\infty(\mathbb{R})$, $\Phi(f)=(f,f')$, is an isomorphism, showing the ``complete disconnection'' between $\pi_1$ and $\pi_2$. This means that given $g,h\in L^p(\mathbb{R})$ there exists a sequence $(f_n)_n$ of $C^1$ functions such that $\left(f_n,f_n'\right)_n$ converge to $(g,h)$ in $L^p(\mathbb{R})\times L^p(\mathbb{R})$. In this paper we obtain results of this type, but for homeomorphisms of an open bounded $\Omega \subseteq \mathbb{R}^d$, that are volume preserving, in the case $d\geq 2$. The convergence of the sequences will be more restricted as we consider it in $L^\infty(\Omega)\times L^p(\Omega)$. As a particular case, we will show that if $SO(d)$ is the special orthogonal group and $H$ is a Riemann integrable function from $\Omega$ to $SO(d)$, then there exists a sequence $(f_n)_n$ of orientation and volume preserving $C^\infty$ homeomorphisms of $\Omega$ converging in $L^\infty(\Omega)$ to the identity and such that $\left(Df_n\right)_n$ converges to $H$ in $L^p(\Omega)^{d^2}$. The same is true if we substitute the identity function by any $f$, $C^1$ homeomorphism of $\Omega$ such that $Df(x)\in SO(d)$ for all $x\in\Omega$. In the case $d=1$, where the volume preserving condition has no interest, we will prove that a pair $(f,F)\in C^1(I)\times L^p(I)$, such that $f', (f^{-1})'\in L^r(I)$ for some $r\>1$, admits a sequence $(f_n)_n$ of $C^1$ homeomorphisms converging in $L^\infty(I)$ to $f$ and such that $(f_n')_n$ converges in $L^p(I)$ to $F$, if and only if $0\leq F\leq g'$.