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 $p\geq 1$ 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 $d\geq 2$. We will show, in particular, that if $H:\Omega\rightarrow SO(d)$ 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 $\left(Df_n\right)_n$ 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 $f'\neq 0$, $f', (f^{-1})'\in L^r(I)$ for some $r>1$, admits a sequence $(f_n)_n$ of $C^1$ homeomorphisms uniformly converging to $f$ and such that $(f_n')_n$ converges in $L^p(I)$ to $F$, if and only if $0\leq \frac{F}{f'}\leq 1$.