\documentclass[reqno]{amsart} \usepackage{mathrsfs} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Two nonlinear days in Urbino 2017,\newline \emph{Electronic Journal of Differential Equations}, Conference 25 (2018), pp. 77--85.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \setcounter{page}{77} \title[\hfilneg EJDE-2018/conf/25\hfil Differentiability vs. approximate differentiability] {Differentiability versus approximate differentiability} \author[Luigi D'Onofrio \hfil EJDE-2018/conf/25\hfilneg] {Luigi D'Onofrio} \address{Luigi D'Onofrio \newline University of Napoli \lq\lq Parthenope \rq\rq, Italy} \email{donofrio@uniparthenope.it} \dedicatory{In memory of Anna Aloe} \subjclass[2010]{46E35} \keywords{Sobolev homeomorphism; Lusin condition; \hfill\break\indent approximate differentiability} \begin{abstract} One of the main tools in geometric function theory is the fact that the \emph{area formula} is true for Lipschitz mapping; if $f$ is differentiable a.e.\ (in the classic sense) then $f$ can be exhausted up to a set of zero measure; the restriction of $f$, set by set, is Lipschitz \cite[Theorem 3.18]{F}. %[6] The aim of this survey is to clarify the regularity assumptions for a map to be differentiable a.e., and to give some auxiliary results when it is not, using the notion of approximate differentiability. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, and let $f:\Omega \to \Omega'=f(\Omega)$ be a homeomorphism ($f\in \mathrm{Hom}(\Omega, \Omega')$ in short). We recall two classical notions: \begin{itemize} \item $f$ satisfies the Lusin condition (also called condition (N), and denoted as $f\in (N)$) if \begin{equation}\label{Lusin} E\subset \Omega \text{ with $|E|=0$ implies }|f(E)|=0 \end{equation} where $|\cdot|$ denotes the Lebesgue measure. \item $f$ is a \emph{Sobolev homeomorphism} if $f$ belongs to $W^{1,1}_{\rm loc}$. For $n\geq 2$, a mapping $f$ is a \emph{bi-Sobolev map} if $f$ and $f^{-1}$ are Sobolev homeomorphisms. \end{itemize} For a homeomorphism $f:\Omega\stackrel{\rm onto}{\longrightarrow} \Omega'$, condition \eqref{Lusin} holds if and only if $f$ maps measurable sets to measurable sets. Moreover, if $f$ is differentiable at every point $x$ of the Borel set $B\subset \Omega$ and $J_f(x)$ is the Jacobian determinant of $f$ at $x$, then the \emph{weak area formula} holds on $B$; that is, \begin{equation}\label{weakarea} \int_{B}\eta(f(z))|J_f(z)|\mathrm d z\leq \int_{f(B)}\eta(w)\mathrm d w \end{equation} for any nonnegative Borel-measurable function $\eta$ on $\mathbb R^n$. Note that for $n\geq 3$, there is a homeomorphism of class $W^{1,n-1}_{\rm loc}((-1,1)^n,\mathbb{R}^n)$ such that both $f$ and $f^{-1}$ are nowhere differentiable \cite[Example 5.2]{CHM}. For this reason the aim of this survey (based mainly on results contained in \cite{DSS,DSS1}) is to understand when dealing with mappings of $W^{1,p}$ with $p0$ and $E'$ is a non measurable set contained in $P'_0$ (recall that every set of Lebesgue positive measure contains a non measurable subset) then $f^{-1}(E')$ is contained in the null set $N_0$ hence it is measurable. The question of the differentiability in the classical sense of a homeomorphisms has a rather simple positive answer in the case $n=2$ thanks to a classical Theorem by Gehring-Lehto (see \cite{GL,LV,Me}). \begin{theorem} \label{thm2.1} Let $\Omega$ and $\Omega'$ be bounded domains in the plane and suppose that $f\in Hom(\Omega,\Omega')$ has finite partial derivatives a.e.\ in $\Omega$, then $f$ is differentiable a.e.\ in $\Omega$. \end{theorem} As a consequence, if $f=(u,v): \Omega \subset \mathbb R^2 \stackrel{\rm onto}{\longrightarrow} \Omega'\subset \mathbb R^2$ is a Sobolev-homeomor\-phism, then $f$ and $f^{-1}$ are differentiable a.e. (\cite{HKO}). The fairly well-known Theorem of Gehring-Lehto is one of the few facts from real analysis that carry geometric information up from the infinitesimal level. Its proof uses properties of the plane, in fact the Theorem at this stage of generality ($f$ a $BV$-homeomorphism or $f$ a $W^{1,1}$-homeomorphism) is false in higher dimension. In the general case $n\geq 2$, the minimal integrability conditions on the partial derivatives of a Sobolev homeomorphism $f\in W^{1,1}_{\rm loc}(\Omega, \mathbb R^n)$ needed to guarantee a.e.\ differentiability have been found by J. Onninen \cite{O}, generalizing a classical result of Stein \cite{S}. It turns out that if $f\in W^{1,1}(\Omega, \mathbb R^n)$ and $|Df|\in L^{n-1,1}(\Omega)$ (where the Lorentz space $L^{p,1}(\Omega)$, $1\leq p <\infty$ is defined as the class of all measurable functions $u: \Omega \to \mathbb R$ such that $$ \| u \|_{L^{p,1}(\Omega)}=p \int_0^\infty |\{ z\in \Omega: |u(z)|>t\}|^{1/p} \mathrm{d}t $$ is finite) then the homeomorphisms $f$ and $f^{-1}$ are differentiable a.e. This is sharp after an example of a $W^{1, n-1}$-homeomorphism $f$ ($n \geq 3$) which is bi-Sobolev (that is, $f^{-1} \in W^{1,1}$) and both $f, f^{-1}$ are nowhere differentiable \cite{CHM}. Let us prove the following useful result which generalizes \cite[Lemma 3.4]{LV}. \begin{proposition}[\cite{DSS}] \label{positivejacobian} If $f$ is a Sobolev homeomorphism such that $J_f\geq 0$, then $f^{-1}$ satisfies condition (N), if and only if, $J_f(z)>0$ for a.e.\ $z\in \Omega$. \end{proposition} \begin{proof} Suppose first that $f^{-1}\in (N)$ and denote by $\tilde{\Omega}$ a subset of $\Omega$ of full measure such that the area formula \eqref{federer2} with $B=\tilde{\Omega}$ holds true. Hence, $$ |f(\{z\in \tilde{\Omega}:J_f(z)=0\})|=0 $$ and by condition (N), for $f^{-1}$ we have $$ |\{z\in \Omega:J_f(z)=0\}|=|\{z\in \tilde{\Omega}: J_f(z)=0\}\cup(\Omega\setminus \tilde{\Omega})|=0. $$ Conversely, suppose $J_f(z)>0$ a.e.\ and let us prove that $f^{-1}\in (N)$. Assuming by contradiction that there exists $|N'_0|=0$, $N_0'\subset \Omega'$ with $|f^{-1}(N_0')|>0$, then we have $$ \int_{f^{-1}(N_0')}J_f\leq |f(f^{-1}(N_0'))|=|N_0'|=0. $$ Hence $J_f=0$ on the positive set $f^{-1}(N_0')\subset\Omega$ and this is a contradiction. \end{proof} Let us prove a simple characterization of condition $(N)$ for a function $f$. \begin{proposition}[\cite{DSS}] \label{fineagosto} If $f:\Omega \stackrel{\rm onto}{\longrightarrow} \Omega'$ is a Sobolev homemorphism, $J_f\geq 0$ and $$ \int_BJ_f=|f(B)| $$ for any Borel set $B\subset \Omega$, then $f\in (N)$ on every Borel set $B\subset\Omega$. \end{proposition} \begin{proof} By contradiction, assume that there exists a subset $E\subset B:|E|=0$ and $|f(E)|>0$. Then $$ \int_BJ_f=\int_{B\setminus E}J_f\leq |f(B\setminus E)|=|f(B)|-|f(E)|<|f(B)| $$ which is a contradiction. \end{proof} An interesting application of condition (N) is the following result on the inverse of an a.e.\ differentiable homeomorphism, which in the plane and has an interesting counterpart. \begin{proposition}[\cite{DSS}] \label{prop1} Let $f\in Hom(\Omega,\Omega')$ be differentiable a.e.. If $f$ satisfies condition {\rm (N)}, then the inverse $f^{-1}$ is differentiable a.e.. \end{proposition} \begin{proof} We notice that the area formula \eqref{area} holds on each set on which $f$ satisfies condition $(N)$; in particular it holds on $\mathcal R_f\cup \mathcal Z_f$, that is the set where $f$ is differentiable: \begin{equation} \int_{\mathcal R_f \cup \mathcal Z_f}\eta(f(z)) |J_f(z)|\mathrm d z =\int_{f(\mathcal R_f \cup \mathcal Z_f)}\eta(w)\mathrm d w. \end{equation} In particular, we have the following version of Sard Lemma, \begin{equation}\label{Sard} |f(\mathcal Z_f)|=0. \end{equation} Since $f$ is differentiable a.e., $\mathcal E_f$ has measure zero and by condition (N), $f(\mathcal E_f)$ has measure zero. We note that $f^{-1}$ is differentiable in $f(\mathcal R_f)$ which is a subset of full measure of $f(\Omega)$; indeed, $$ f(\Omega)\setminus f(\mathcal R_f)= f(\mathcal Z_f)\cup f(\mathcal E_f) $$ has measure zero by \eqref{Sard} and condition (N). \end{proof} By $A \triangle B$, we denote the set $(A \cup B) \setminus (A\cap B)$. By $A=B$ a.e.\ we mean $|A\triangle B|=0$. \begin{proposition} \label{prop2.5} Let $f\in \mathrm{Hom}(\Omega,\Omega')$ and assume that $f$ and $f^{-1}$ are differentiable a.e. and both satisfy condition {\rm (N)}, then $f$ essentially maps $\mathcal E_f$ to $\mathcal Z_{f^{-1}}$ and $f^{-1}$ maps $\mathcal E_{f^{-1}}$ to $\mathcal Z_{f}$ in the sense that $$ |f(\mathcal E_f) \triangle \mathcal Z_{f^{-1}}|= |f(\mathcal E_{f^{-1}}) \triangle \mathcal Z_{f}|=0. $$ \end{proposition} \section{Differentiability versus approximate differentiability} As we have already observed the notion of differentiability can not guaranteed for homeomorphism that are in $W^{1,n-1}$. To avoid this problem the notion of approximate differentiability comes to the play, so we would like to know if some results presented in the previous section are still valid. For example, let us consider the following issue: let $\Omega$ and $\Omega'$ be domains in $\mathbb{R}^n$, if $f: \Omega \stackrel{\rm onto}{\longrightarrow} \Omega' $ is a homeomorphism approximately differentiable at $x_0\in\Omega$ with nonzero Jacobian, $J_f (x_0) \neq 0$ , is it true that $f^{-1}$ is approximately differentiable at $y_0= f(x_0)$ and that \begin{equation}\label{classical} J_{f^{-1}}(y_0)= \frac{1}{J_f(x_0)}? \end{equation} This is true for $n=1$, because a homeomorphism $h: [a,b]\subset \mathbb{R} \stackrel{\rm onto}{\longrightarrow} [a', b']\subset \mathbb{R}$ is strictly monotone, hence approximate differentiability is equivalent to ordinary differentiability \cite{HKlect}. Recall that $f$ is approximately differentiable at $x_0\in \Omega$ with approximate gradient $Df(x_0)$, if there is a set $A \subset \Omega$ of \emph{density} one at $x_0$, i.e.\ \begin{equation}\label{densityeq} \lim_{r \to 0} \frac{|A \cap \mathbb{B}_r(x_0)|}{|\mathbb{B}_r(x_0)|}=1\,, \end{equation} where $\mathbb{B}_r(x)$ denotes the closed ball of center $x$ and radius $r$, such that \begin{equation}\label{apprdifferentiability} \lim_{y \to x_0, \, y \in A} \frac{|f(y)- f(x_0)- D f(x_0) (y-x_0)|}{|y-x_0|}=0. \end{equation} Lebesgue's density Theorem guarantees that almost every point of a measurable set $A \subset \Omega$ is a point of density one for $A$ \cite[p. 129]{Sa}. In view of the notion of approximate differentiability, we decompose the set $\Omega$ in a different way than in Section 2: $$ \Omega= \mathscr{R}_f \cup \mathscr{Z}_f \cup \mathscr{E}_f\,, $$ where \begin{gather*} \mathscr{R}_f=\{ x\in \Omega : f \text{ is approximately differentiable at $x$ and } J_f(x) \neq 0 \},\\ \mathscr{Z}_f=\{ x\in \Omega : f \text{ is approximately differentiable at $x$ and } J_f(x) = 0 \}, \\ \mathscr{E}_f=\{ x\in \Omega : f \text{ is not approximately differentiable at $x$}\} \end{gather*} The following version of chain rule was established in \cite{FMS,H0,HKlect}. \begin{theorem}\label{gifa} Let $f: \Omega \subset \mathbb{R}^n \stackrel{\rm onto}{\longrightarrow} \Omega'\subset \mathbb{R}^n$ be a bi-Sobolev map. Then $f$ and $f^{-1}$ are approximately differentiable a.e.\ and there exists a Borel set $B \subset \mathscr R_f$ with $|\mathscr R_f\setminus B|=0$ such that $f(B)\subset \mathscr R_{f^{-1}}$ where \begin{equation}\label{Rf-1} \mathscr R_{f^{-1}}=\{ y\in\Omega': f^{-1} \text{ is approximately differentiable at $y$ and } J_{f^{-1}}(y)\neq 0\} \end{equation} with $|\mathscr R_{f^{-1}}\setminus f(B)|=0$. Also we have $$ Df^{-1}(y)=\left(Df(f^{-1}(y))\right)^{-1}\quad \forall y \in f(B). $$ \end{theorem} The proof in \cite{HKlect} relies on the fact that if $g: \Omega \to \mathbb{R}^n$ is a Lipschitz map and $J_g(x_0)\neq 0$ for a point $x_0\in \Omega$, then \begin{equation} A \text{ being a set of density 1 at $x_0$ implies $g(A)$ is a set of density 1 at } g(x_0). \end{equation} So we are naturally induced to consider the following problem: how density points of measurable sets are transformed under Sobolev or bi-Sobolev maps? Actually a result of Buczolich \cite{Bu} guarantees the preservation of density points of $\Omega$ under any bi-Lipschitz map $f: \Omega \subset \mathbb{R}^n \stackrel{\rm onto}{\longrightarrow} \Omega' \subset \mathbb{R}^n$. Note that if $f: \Omega \stackrel{\rm onto}{\longrightarrow} \Omega' $ preserves density points, then $f$ satisfies the Lusin $(\mathcal N)$-property. Let us emphasize that for almost every $x\in \Omega$ the density of $E$ at $x$ is one if and only if the density of $f(E)$ at $f(x)$ is one. Here we want to prove that, under suitable assumptions on $f$, for \emph{ all } $x\in \Omega$ we have preservations of density points. \subsection{Density points for $n=1$} We shall consider here $Q$-quasiminimizers, $Q\geq 1$, of the one-dimensional Dirichlet integral \begin{equation}\label{Dirichlet} u\to\int|u'|^pdx \quad p>1 \end{equation} whose definition goes back to Giaquinta-Giusti \cite{GG1,GG2,MS}). Let $(a,b)$ be an open interval in $\mathbb{R}$ and $h\in W^{1,p}_{\rm loc}((a,b))$; then $h$ is a $Q$-quasiminimizer of \eqref{Dirichlet} if for all $[c,d]\subset (a,b)$ \begin{equation}\label{MartioSbordone} \int_c^d|h'|^p dx \leq Q\int_c^d|k'|^p dx \end{equation} whenever $k\in h+W^{1,p}_0 ((c,d))$. We say that $h$ is a quasiminimizer of a Dirichlet integral if there exists $p>1$ such that $h$ is a quasiminimizer of \eqref{Dirichlet}. Note that if $ h: (a,b)\to (a', b')$ satisfies the bi-Lipschitz condition, i.e.\ there exists $L>1$ such that \begin{equation}\label{lip} \frac{1}{L} \leq |h'(x)| \leq L \end{equation} for a.e.\ $x\in (a, b)$, then for any $p>1$ it satisfies \eqref{MartioSbordone} with $Q= L^2$. Indeed, by \eqref{lip} we have $$ \frac{1}{d-c} \int_c^d (h')^2 \leq \frac{L}{d-c} \int_c^d h' \leq \Big( L \hbox{--}\hskip-9pt\int_c^d h'\Big) L \hbox{--}\hskip-9pt\int_c^d h'. $$ Hence $$ \hbox{--}\hskip-9pt\int_c^d (h')^2 \leq L^2 \Big( \hbox{--}\hskip-9pt\int_c^d h' \Big)^2\,. $$ In \cite{DSS} the authors proved that quasiminimizers (which are far away from being bi-Lipschitz mappings) are $W^{1,r}$-biSobolev maps (that is $f,f^{-1}\in W^{1,r}$ for a certain $r>1$) and preserve density points. We emphasize that this can be interpreted also as a regularity result for one dimensional $Q$-quasiminima. \begin{theorem}[\cite{DSS}]\label{Preserve} Let $h\in W^{1,p}_{\rm loc}((a,b))$, $p>1$, be a non-constant $Q$-quasimini\-mizer of \eqref{Dirichlet}. Then there exists $r>1$ such that $h$ is a $W^{1,r}$-biSobolev map which preserves density points together with its inverse. \end{theorem} \subsection{Density points for $n=2$} Now, let us consider a special class of bi-Sobolev maps: the quasiconformal mappings. We will say that $f: \Omega \stackrel{\rm onto}{\longrightarrow} \Omega'$ is a $K$-quasiconformal map, $K\geq 1$, if $f$ is a bi-Sobolev map such that \begin{equation}\label{conformal} |Df(x)|^2\leq K J_f(x) \end{equation} for a.e. $x \in \Omega$. Since $f$ and $f^{-1}$ satisfy the $(\mathcal{N})$-condition of Lusin, i.e. they map sets of measure zero onto sets of measure zero (see \cite{AIM}, \cite{HKlect} ), then for any $E\subset \Omega$ measurable, $f(E)$ is measurable; hence for a.e.\ $x\in \Omega$, $x$ is a density point for $E$ if and only if $f(x)$ is a density point for $f(E)$. We remark that this holds for \emph{all} $x\in \Omega$, according to a result by Gehring-Kelly \cite{GK}. \begin{theorem}[\cite{GK}] \label{GehringKelly} If $f: \Omega \stackrel{\rm onto}{\longrightarrow} \Omega'$ is a $K$-quasiconformal map, then $f$ and $f^{-1}$ preserve density points. \end{theorem} This is a consequence of the following invariant form of the area distortion Theorem: there is a constant $C(K)$ such that for any $K$-quasiconformal map $f$, we have \begin{equation}\label{six} \frac{1}{C(K)} \Big(\frac{|E|}{|Q|} \Big)^{K} \leq \frac{|f(E)|}{|f(Q)|} \leq C(K) \Big(\frac{|E|}{|Q|} \Big)^{1/K} \end{equation} for any cube $Q\subset \Omega$ and for any subset $E\subset Q$ \cite[Theorem 13.1.5)]{AIM}. \subsection{Density points for $n \geq 2$} We say that a homeomorphism $f: \Omega\subset \mathbb{R}^n \stackrel{\rm onto}{\longrightarrow} \Omega'\subset \mathbb{R}^n$ satisfies the \emph{uniform area inequality} if there exist constants $C, \alpha$ with $0< \alpha \leq 1 \leq C$ such that \begin{equation}\label{distortionarea} \frac{|f(E)|}{|f(B)|} \leq C \Big( \frac{|E|}{|B|} \Big)^{\alpha} \end{equation} for any ball $B\subset \Omega$ and for any measurable set $E\subset B$. \begin{theorem}\label{arean} Let $f: \Omega \stackrel{\rm onto}{\longrightarrow} \Omega'$ be a continuous mapping satisfying the uniform area inequality \eqref{distortionarea}. If $f$ is differentiable at $x_0$ with $J_f(x_0)\neq 0$ and $A$ is a set of density one at $x_0$, then the density of $f(A)$ at $f(x_0)$ is one. \end{theorem} Our aim is to give the chain rule formula \eqref{classical} assuming only that $f$ is an a.e.\ approximately differentiable homeomorphism not necessarily of Sobolev class and with no assumptions on $f^{-1}$. In this sense the following result is a generalized version of the inverse function Theorem. \begin{theorem}[\cite{DSS1}] \label{chainapp} Let $f: \Omega \subset \mathbb{R}^n \stackrel{\rm onto}{\longrightarrow} \Omega'\subset \mathbb{R}^n$ be a homeomorphism. If $f$ is approximately differentiable a.e.\ then there exists a Borel set $B \subset \mathscr R_f$ with $|B|= |\mathscr R_f|$ such that $f(B)\subset \mathscr R_{f^{-1}}$ with $|f(B)|= |\mathscr R_{f^{-1}}|$ and $$ Df^{-1} (f(x)) Df(x)= Id \quad J_{f^{-1}}(f(x)) J_f (x)= 1 \quad \text{for all } x \in B. $$ Moreover, if $|B|>0$ then $|f(B)|>0$. \end{theorem} As a consequence, we have the following corollary. \begin{corollary}[\cite{DSS1}] \label{eqmes} Let $f: \Omega \subset \mathbb{R}^n \stackrel{\rm onto}{\longrightarrow} \Omega'\subset \mathbb{R}^n$ be a homeomorphism. If $f$ is approximately differentiable a.e.\ then \begin{equation}\label{measu} |\mathscr R_{f^{-1}}|= | f(\mathscr R_{f})|. \end{equation} \end{corollary} \begin{corollary}[\cite{DSS1}] \label{coro=3.7} Let $f: \Omega \subset \mathbb{R}^n \stackrel{\rm onto}{\longrightarrow} \Omega'\subset \mathbb{R}^n$ be a homeomorphism. If $f$ is approximately differentiable almost everywhere and $f$ satisfies the Lusin $(\mathcal{N})$ condition, then $f^{-1}$ is approximately differentiable almost everywhere. \end{corollary} In general, it is not true that $ f(\mathscr R_{f}) \subset \mathscr R_{f^{-1}}$, as it happens when $f$ is differentiable in the classical sense. D'Onofrio-Sbordone-Schiattarella \cite{DSS1} provided the following example. \begin{example}\label{esempio} \rm Let $n\geq 2$. There is a bi-Sobolev homeomorphism $f: \mathbb{B}_1(0)\stackrel{\rm onto}{\longrightarrow} \mathbb{B}_1(0)$ with $f\in W^{1,n}\left( \mathbb{B}_1(0), \mathbb{R}^n \right)$ such that $f$ is not differentiable at $0$, $f$ is approximately differentiable at $0$, $J_f(0) \neq 0$ and $f^{-1}$ is not approximately differentiable at $f(0)=0$. 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