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

Abstract

If $\Omega$ is an open subset of $\mathbb{R}$ and $p>0$ then the elements of $W^{1,p}(\Omega )$ can be seen as the pairs $(f,F)\in L^p(\Omega )\times (L^p(\Omega ){)}^d$ such that there exists a sequence $(f_n{)}_n$ of $C^1$ functions converging to $f$ in $L^p(\Omega )$ such that $(\nabla f_n{)}_n$ converges to $F$ in $(L^p(\Omega ){)}^d$. If $p\ge 1$ the pair $(f,F)$ is defined by $f$ as $F$ must be the distributional gradient of $f$. If $0<p<1$, there is, in general, a disconnection between $f$ and $F$. For instance, Peetre (see \cite{peetre}) proved that, if $d=1$, this disconnection is complete, as any pair $(f,F)\in L^p(\Omega )\times L^p(\Omega )$ is an element of $W^{1,p}(\Omega )$. So $F$ is not defined by $f$ in any sense, as it can be any element of $L^p(\Omega )$. In this paper we obtain results of this type, concerning $C^1$ homeomorphisms of $\Omega$ that are volume preserving if $d\ge 2$. We will show, in particular, that if $H:\Omega \to SO(d)$ is a Riemann integrable function, then there exists a sequence $(f_n{)}_n$ of orientation and volume preserving $C^{\infty }$ homeomorphisms of $\Omega$ uniformly converging to the identity of $\Omega$ and such that ${\left(D{f}_n\right)}_n$ converges to $H$ in $L^p(\Omega {)}^{d^2}$. If $d=1$ and $I$ is a bounded interval, we will prove that a pair $(f,F)\in C^1(I)\times L^p(I)$, such that $f^{\prime }\ne 0$, $f^{\prime },(f^{-1}{)}^{\prime }\in L^r(I)$ for some $r>1$, admits a sequence $(f_n{)}_n$ of $C^1$ homeomorphisms uniformly converging to $f$ and such that $(f_n^{\prime }{)}_n$ converges in $L^p(I)$ to $F$, if and only if $0\le \frac{F}{f^{\prime }}\le 1$.

19 pages, 2 figures