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

TL;DR

The paper investigates the limits of volume-preserving $C^1$-homeomorphisms in the Sobolev space $W^{1,p}$ for $0<p<1$, where the natural pairing of a function with its gradient can be highly non-unique. It introduces the framework $\mathcal{M}^{1,p}(\Omega)$ and $\mathcal{M}^{1,p}_\lambda(\Omega)$ to study pairs realized as uniform limits of volume- preserving maps with $Df_n\to F$ in $L^p$, and provides constructive results: in 1D, a complete characterization $0\le F/f'\le1$ for admissible pairs, and in higher dimensions ($d\ge2$), that every $H:\Omega\to SO(d)$ that is Riemann integrable can be approached by $Df_n$ of identity-perturbing, volume-preserving diffeomorphisms with $Df_n\to H$ in $L^p$. The work develops technical tools (local rotation-diffeomorphisms, localization maps, transition jumps, and Vitali coverings) to patch local data into global maps, yielding a robust method to realize prescribed gradients under volume preservation and illuminating the structure of $W^{1,p}$ when $p<1$.

Abstract

If $Ω$ is an open subset of $\mathbb{R}$ and $p>0$ then the elements of $W^{1,p}(Ω)$ can be seen as the pairs $(f,F)\in L^p(Ω)\times (L^p(Ω))^d$ such that there exists a sequence $(f_n)_n$ of $C^1$ functions converging to $f$ in $L^p(Ω)$ such that $(\nabla f_n)_n$ converges to $F$ in $(L^p(Ω))^d$. 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 $(f,F)\in L^p(Ω)\times L^p(Ω)$ is an element of $W^{1,p}(Ω)$. So $F$ is not defined by $f$ in any sense, as it can be any element of $L^p(Ω)$. In this paper we obtain results of this type, concerning $C^1$ homeomorphisms of $Ω$ that are volume preserving if $d\geq 2$. We will show, in particular, that if $H:Ω\rightarrow SO(d)$ is a Riemann integrable function, then there exists a sequence $(f_n)_n$ of orientation and volume preserving $C^\infty$ homeomorphisms of $Ω$ uniformly converging to the identity of $Ω$ and such that $\left(Df_n\right)_n$ converges to $H$ in $L^p(Ω)^{d^2}$. 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 $0\leq \frac{F}{f'}\leq 1$.

Figures (2)

• Figure 1: In the left, the graphic of $f_3$ (with $a_3=\frac{1}{9}$ and $g_3(x)=54x^2(1-6x)$). In this case $f_3$ is a $C^1$ function. In the right, a zoom of the graphic of $f_3$ around $x=\frac{1}{3}$.
• Figure 2: In blue the graphic of $\phi_5'$ as defined in the proof of Proposition \ref{['l1']}, relatively to the step function in red $\frac{1}{4}\chi_{[0,\frac{2}{5}]}+\frac{1}{2}\chi_{[\frac{2}{5},\frac{3}{5}]}+\frac{1}{6}\chi_{[\frac{3}{5},1]}$.

Theorems & Definitions (27)

• Remark 1.1
• Theorem A
• Theorem B
• Definition 2.1
• Example 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof