\input amstex \documentstyle{amsppt} \loadmsbm \magnification=\magstephalf \hcorrection{1cm} \vcorrection{6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1995/06\hfil Non-uniformly elliptic systems. \hfil\folio} \def\leftheadline{\folio\hfil F. Leonetti \& C. Musciano \hfil EJDE--1995/06} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations\hfil\break Vol. {\eightbf 1995}(1995), No. 06, pp. 1-14. Published June 7, 1995.\hfil\break ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfil\break ftp (login: ftp) 147.26.103.110 or 129.120.3.113 \bigskip}} \topmatter \title REGULARITY FOR NON-UNIFORMLY ELLIPTIC SYSTEMS AND APPLICATION TO SOME VARIATIONAL INTEGRALS \endtitle \thanks \noindent {\it 1991 Mathematics Subject Classifications:} 35J60, 49N60.\hfil\break {\it Key words and phrases:} Regularity, weak solutions, minimizers, ellipticity, variational integrals. \hfil\break \copyright 1995 Southwest Texas State University and University of North Texas.\hfil\break Submitted: September 20, 1994.\hfil\break This work has been supported by MURST, GNAFA-CNR and INdAM. \endthanks \author Francesco Leonetti\\ and \\ Chiara Musciano \endauthor \address Francesco Leonetti \newline\indent Dipartimento di Matematica \newline\indent Universit\`a di L'Aquila \newline\indent Piazzale Aldo Moro 5, 00185 Roma, Italy \newline\indent E-mail: leonetti\@vxscaq.aquila.infn.it \newline\indent \quad \newline\indent Chiara Musciano \newline\indent Istituto Nazionale di Alta Matematica Francesco Severi \newline\indent Citt\`a Universitaria \newline\indent Piazzale Aldo Moro 5, 00185 Roma, Italy \endaddress \abstract This paper deals with higher integrability for minimizers of some variational integrals whose Euler equation is elliptic but not uniformly elliptic. This setting is also referred to as elliptic equations with $p,q$-growth conditions, following Marcellini. Higher integrability of minimizers implies the existence of second derivatives. This improves on a result by Acerbi and Fusco concerning the estimate of the (possibly) singular set of minimizers. \endabstract \endtopmatter \document \def\loc{\hbox{\sevenrm loc}} \define\nonexist{{/} \negthickspace \negthickspace \exists} \heading 0. Introduction \endheading Let $\Omega$ be a bounded open set of $\Bbb R^n$, $n \geq 2$, $u$ be a (possibly) vector-valued function, $u : \Omega \to \Bbb R^N, N \geq 1$, $F$ be a continuous function, $F : \Bbb R^{nN} \to \Bbb R$; we consider the integral $$ I(u) = \int_\Omega F(Du(x)) \, dx\,, \tag0.1 $$ where $$ |F(\xi)| \leq c \, ( 1+|\xi|^p)\,, \tag0.2$$ $u\in W^{1,p}(\Omega), 2\leq p$. Regularity of minimizers has been widely studied when $$ \hat m \, ( 1+|\xi|^{p-2}) |\lambda|^2 \leq DDF(\xi)\lambda\lambda\,, \qquad\qquad\qquad\qquad 0<\hat m\,, \tag0.3 $$ $$ |DDF(\xi)| \leq c \, ( 1+|\xi|^{p-2}), \tag0.4 $$ see \cite{24}, \cite{14}, \cite{16}, \cite{17} (and \cite{10}, \cite{12}, \cite{18}, \cite{11}, \cite{20}, where (0.3) is weakened in order to consider quasi-convex integrals but (0.4) is still present). We refer to (0.3), (0.4) as {\it uniform ellipticity condition}. When dealing with $$ \hat J(u) = \int_\Omega \{a|Du|^2 + a|Du|^p + \sqrt{1+(\text{det}\, Du)^2}\} \, dx\,. \tag0.5$$ where $2\leq n \leq p < 2n$, $u:\Bbb R^n \to \Bbb R^n$, $a>0$, it turns out that (0.4) does not hold true any longer; conversely, the following growth condition applies: $$ |DDF(\xi)| \leq c \, ( 1+|\xi|^{2n-2})\,. \tag0.6 $$ Moreover, if $a$ is large enough \cite{13}, namely $a\geq a(n)=2n^4[(n-2)!]$, then (0.3) is still true: we are lead to consider integrals (0.1) verifying (0.2), (0.3) and $$ |DDF(\xi)| \leq c \, ( 1+|\xi|^{q-2})\,, \tag0.7 $$ for some $q>p$. We refer to (0.3), (0.7) as {\it nonuniform ellipticity condition} \cite{13}, {\it nonstandard growth condition} \cite{22}, or {\it $p,q$-growth condition} \cite{23}. In this paper we prove higher integrability and differentiability for minimizers of integrals verifying the nonuniform ellipticity (0.3), (0.7). Our results apply to the model integral (0.5) in this way: assume that $u:\Bbb R^n \to \Bbb R^n$, $u\in W^{1,p}(\Omega)$, $ \Omega \subset \Bbb R^n$ is bounded and open, $2\leq n \leq 2n-2 < p < 2n$; if $u$ minimizes $\hat J$ and $a\geq a(n)$, then $$ DDu \text{ and } D(|Du|^{(p-2)/2}Du) \in L^2_{\loc}(\Omega). \tag0.8 $$ We can also apply a partial regularity theorem contained in \cite{1}, see also \cite{15}, in order to get $$Du \in C^{0,\alpha}_{\loc}(\Omega_0), \qquad \forall \alpha \in (0,1)\,, \tag 0.9$$ for some open $\Omega_0 \subset \Omega$, with $$ |\Omega \setminus \Omega_0 | = 0\,, \tag0.10 $$ where $|E|$ is the $n$-dimensional Lebesgue measure of $E \subset \Bbb R^n$. Now we are able to improve on (0.10), because of our result (0.8): $$ \Cal H^{n-2+\epsilon} (\Omega \setminus \Omega_0) = 0 \qquad \forall \epsilon >0\,, \tag0.11$$ where $\Cal H^{n-2+\epsilon}$ is the $(n-2+\epsilon)$-dimensional Hausdorff measure. \heading 1. Notation and main results \endheading Let $\Omega$ be a bounded open set of $\Bbb R^n$, $n \geq 2$, $u$ be a (possibly) vector-valued function, $u : \Omega \to \Bbb R^N, N \geq 1$, $F$ be a function $F : \Bbb R^{nN} \to \Bbb R$. We consider the integral $$ I(u) = \int_\Omega F(Du(x)) \, dx\,, \tag1.1 $$ where $$F\in C^1(\Bbb R^{nN})\tag1.2$$ and, for some positive constants $c,p,m$, $$ |F(\xi)| \leq c\, ( 1+|\xi|^p)\,, \tag1.3 $$ $$ |DF(\xi)| \leq c \, ( 1+|\xi|^{p-1})\,, \tag1.4 $$ $$ m \, ( |\xi| + |\hat\xi|)^{(p-2)}|\xi -\hat\xi|^2 \leq (DF(\xi) - DF(\hat\xi)) (\xi -\hat\xi)\,, \tag1.5 $$ for every $\xi$, $\hat\xi \in\Bbb R^{nN}$. About $p$, we assume that $$ 2 \leq p\,. \tag1.6 $$ We say that $u$ minimizes the integral (1.1) if $u:\Omega\to\Bbb R^N$, $u \in W^{1,p}(\Omega)$ and $$ I(u) \leq I(u + \phi )\,. \tag1.7 $$ for every $\phi :\Omega\rightarrow \Bbb R^N$ with $\phi\in W_0^{1,p} (\Omega)$. We will prove the following higher integrability result for $Du$: \proclaim{Theorem 1} Let $u\in W^{1,p}(\Omega)$ minimize the integral (1.1) and $F$ satisfy (1.2--1.6); then $$Du \in L^\sigma_{\loc}(\Omega), \qquad \forall \sigma < p\frac{n}{n-1}\,. \tag1.8$$ \endproclaim The higher integrability result (1.8) for $Du$ allows us to get existence of second weak derivatives under additional conditions on $F$. Now we assume that $$F\in C^2(\Bbb R^{nN})\tag1.9$$ and, for some constants $c,p,q,\hat m , \mu$, $$ |F(\xi)| \leq c \, ( 1+|\xi|^p)\,, \tag1.10 $$ $$ |DF(\xi)| \leq c \, ( 1+|\xi|^{p-1})\,, \tag1.11 $$ $$ \hat m \, ( \mu +|\xi|^{p-2}) |\lambda|^2 \leq DDF(\xi)\lambda\lambda\,, \quad \qquad \qquad \qquad 0<\hat m\,, \quad 0\leq \mu\,, \tag1.12 $$ $$ |DDF(\xi)| \leq c \, ( 1+|\xi|^{q-2}), \tag1.13 $$ $$ 2 \leq p < q < p\frac{n}{n-1}, \tag 1.14 $$ for every $\xi, \lambda \in \Bbb R^{nN}$. Let us remark that (1.12) implies (1.5): compare with Corollary 2.8 in the next section 2. \proclaim{Theorem 2} Let $u\in W^{1,p}(\Omega)$ minimize the integral (1.1) and $F$ satisfy (1.9--1.14); then $$D(|Du|^{(p-2)/2}Du) \in L^2_{\loc}(\Omega). \tag1.15 $$ Moreover, if (1.12) holds true with $0 < \mu$, then $$DDu \in L^2_{\loc}(\Omega). \tag1.16$$ \endproclaim In this setting we can apply the partial regularity result contained in \cite{1}, see also \cite{15}, in order to get $$Du \in C^{0,\alpha}_{\loc}(\Omega_0), \qquad \forall \alpha \in (0,1), \tag 1.17$$ for some open $\Omega_0 \subset \Omega$, with $$ |\Omega \setminus \Omega_0 | = 0. \tag1.18 $$ Now Theorem 2 allows us to improve on the estimate (1.18) of the (possibly) singular set. This is achieved in the following: \proclaim{Theorem 3} Let $u\in W^{1,p}(\Omega)$ minimize the integral (1.1) and $F$ satisfy (1.9--1.14); moreover, we assume that (1.12) holds true with $0 < \mu$: then, there exists an open set $\Omega_0$, $\Omega_0 \subset \Omega$, such that $$Du \in C^{0,\alpha}_{\loc}(\Omega_0), \qquad \forall \alpha \in (0,1) \tag 1.19$$ and $$ \Cal H^{n-2+\epsilon} (\Omega \setminus \Omega_0) = 0, \qquad \forall \epsilon >0, \tag1.20$$ where $\Cal H^{n-2+\epsilon}$ is the $(n-2+\epsilon)$-dimensional Hausdorff measure. \endproclaim \phantom{...} A model functional for the previous theorems is $$ J(u) = \int_\Omega \{a|Du|^2 + a|Du|^p + h(\text{det}\, Du)\} \, dx, \tag1.21 $$ where $u:\Bbb R^n \to \Bbb R^n$, $h:\Bbb R \to \Bbb R$, $h\in C^2(\Bbb R)$, and for some positive constants $c_1, d$, $$1\leq d< 2, \tag 1.22$$ $$|h(t)|\leq c_1 \, (1+|t|)^d, \tag1.23$$ $$|h^\prime (t)|\leq c_1 \, (1+|t|)^{d-1}, \tag1.24$$ $$0\leq h^{\prime\prime} (t) \leq c_1, \tag 1.25$$ for every $t\in\Bbb R$. Under these assumptions if $a$ is large enough, see \cite{13}, $$ a\geq a(n,d,c_1)=c_1 n^4 [(n-2)!] \{1+[n!]^{d-1}\} \tag1.26$$ and $$ 2\leq n \leq 2n-2 < p < 2n, \qquad nd \leq p, \tag1.27$$ then (1.9), \dots , (1.14) hold true with $q=2n$ and $\mu = 1$ in (1.12). For example, we can take $h(t) = \sqrt{1+t^2}$, $d=1$, $c_1=1$; the resulting functional is $$ \hat J(u) = \int_\Omega \{a|Du|^2 + a|Du|^p + \sqrt{1+(\text{det}\,Du)^2}\} \, dx. \tag1.28$$ \phantom{...} In order to deal with weak solutions of non-uniformly elliptic systems which are not Euler equations of variational integrals, we find out that Theorem 1 remains true; with regard to Theorem 2, we need a more restrictive range of $q$. More precisely, we consider $A:\Bbb R^{nN} \to \Bbb R^{nN}$ and the system of partial differential equations $$ \text{div}\, \big( A(Du(x)) \big) = 0, \tag1.29 $$ where $$ A \in C^0(\Bbb R^{nN}), \tag1.30 $$ and for some positive constants $c, p, m$, $$ |A(\xi)| \leq c \, (1+|\xi|^{p-1}), \tag1.31 $$ $$ m \, ( |\xi| + |\hat\xi|)^{(p-2)}|\xi -\hat\xi|^2 \leq (A(\xi) - A(\hat\xi)) (\xi -\hat\xi), \tag1.32 $$ for every $\xi$, $\hat\xi \in\Bbb R^{nN}$. About $p$, we keep on assuming $$ 2 \leq p. \tag1.33 $$ We say that $u$ is a weak solution of (1.29) if $u : \Omega \to \Bbb R^N$, $u \in W^{1,p}(\Omega)$ and $$ \int\limits_\Omega A(Du(x)) \, D\phi(x) \, dx = 0, \tag1.34 $$ for every $\phi : \Omega \to \Bbb R^N$ with $\phi \in W^{1,p}_0(\Omega)$. We have the following higher integrability result for $Du$: \proclaim{Theorem 4} Let $u\in W^{1,p}(\Omega)$ be a weak solution of (1.29) and $A$ satisfy (1.30--1.33); then $$ Du \in L^\sigma_{\loc}(\Omega), \qquad \forall \sigma < p\frac{n}{n-1}\,. \tag1.35 $$ \endproclaim \phantom{...} As in the case of minimizers, the higher integrability of $Du$ allows us to get higher differentiability; let us remark that, when dealing with elliptic systems that are not the Euler equation of some variational integral, we do not know any longer that the bilinear form $(\lambda,\xi) \to DA \, \lambda \, \xi$ is symmetric: this lack of information is responsible for the more restrictive range of $q$ in the following (1.40). Now we assume that $$ A \in C^1(\Bbb R^{nN}), \tag1.36 $$ and, for some constants $c, p, q, \hat m, \mu$, $$ |A(\xi)| \leq c\, (1+|\xi|^{p-1}), \tag1.37 $$ $$ \hat m \, ( \mu +|\xi|^{p-2}) |\lambda|^2 \leq DA(\xi)\lambda\lambda, \quad \qquad \qquad \qquad 0<\hat m, \quad 0\leq \mu , \tag1.38 $$ $$ |DA(\xi)| \leq c \, ( \mu +|\xi|^{q-2}), \tag1.39 $$ $$ 2 \leq p < q < p\frac{2n-1}{2n-2}\,, \tag 1.40 $$ for every $\xi, \lambda \in \Bbb R^{nN}$. Let us remark that (1.38) implies (1.32). \proclaim{Theorem 5} Let $u\in W^{1,p}(\Omega)$ be a weak solution of (1.29) and $A$ satisfy (1.36--1.40); then $$ D(|Du|^{(p-2)/2}Du) \in L^2_{\loc}(\Omega). \tag1.41 $$ Moreover, if (1.38), (1.39) hold true with $0 < \mu$, then $$ DDu \in L^2_{\loc}(\Omega). \tag1.42 $$ \endproclaim \subheading{Remark} The most important result of this paper is Theorem 1: in our framework, the main step towards regularity is the improvement from $Du \in L^p$ to $Du \in L^\sigma$, $\sigma 0$ $$ \sum^n_{s=1} \int\limits_{B_{\rho}} |\tau_{s,h} f(x)|^2 dx \leq C |h|^{2d}, $$ for every $h$ with $|h|< \rho$, then $f\in L^r (B_{\rho/4})$ for every $ r < 2n/(n-2d)$. \endproclaim \demo{Proof} The previous inequality tells us that $f\in W^{b,2} (B_{\rho/2})$ for every $b-1$, for every $k\in \Bbb N$ there exist positive constants $c_2, c_3$ such that $$ c_2 (|v|^2 + |w|^2)^{\gamma /2} \leq \int\limits_0^1 |v+tw|^{\gamma} dt \leq c_3 (|v|^2 + |w|^2)^{\gamma /2} \tag2.1 $$ for every $v, w \in \Bbb R^k$. (See {\rm [}{\bf 2} {\rm ]}.) \endproclaim Lemma 2.6 allows us to easily get the following Corollaries. \proclaim{Corollary 2.7} For every $p \geq 2$, for every $k\in \Bbb N$ there exists a positive constant $c_4$ such that $$ c_4 \int\limits_0^1 |\lambda+t( \xi - \lambda )|^{p-2} dt \leq (|\lambda| + |\xi|)^{p-2} \tag2.2 $$ for every $\lambda, \xi \in \Bbb R^k$. \endproclaim \proclaim{Corollary 2.8} Let $F$ be a function $F : \Bbb R^{nN} \to \Bbb R$ of class $C^2(\Bbb R^{nN})$ and $p\geq 2$; if there exists $\hat m >0$ such that $$ \hat m \, |\xi|^{p-2} |\lambda|^2 \leq DDF(\xi)\lambda\lambda, $$ for every $\xi, \lambda \in \Bbb R^{nN}$, then there exists $m>0$ such that $$ m \, ( |\xi| + |\hat\xi|)^{(p-2)}|\xi -\hat\xi|^2 \leq (DF(\xi) - DF(\hat\xi)) (\xi -\hat\xi), $$ for every $\xi$, $\hat\xi \in\Bbb R^{nN}$. \endproclaim \proclaim{Corollary 2.9} For every $p \geq 2$, for every $k\in \Bbb N$ there exists a positive constant $\hat c$ such that $$ |\lambda - \xi |^p \leq \hat c \left| |\lambda|^{\frac{p-2}{2}} \lambda - |\xi|^{\frac{p-2}{2}} \xi \right|^2 \tag2.3 $$ for every $\lambda, \xi \in \Bbb R^k$. \endproclaim \heading 3. Proof of Theorem 1 \endheading Since $u$ minimizes the integral (1.1) with growth conditions as in (1.3), (1.4), $u$ solves the Euler equation, $$ \int\limits_\Omega DF(Du (x)) D\phi (x)\, dx =0,\tag3.1 $$ for all functions $\phi : \Omega\rightarrow \Bbb R^N$, with $\phi\in W_0^{1,p} (\Omega)$. Let $R>0$ be such that $\overline{B_{4R}} \subset \Omega$ and let $B_\rho$ and $B_R$ be concentric balls, $0 < \rho < R $. Let $\eta: \Bbb R^n \rightarrow \Bbb R$ be a ``cut off'' function in $C_0^{\infty} (B_R)$ with $ \eta \equiv 1 \text{ on } B_\rho$, $0\leq\eta \leq 1$. Fix $s \in \{1,\dots ,n\}$, take $0<|h|< R$. Using $\phi = \tau_{s, -h} (\eta^2\tau_{s,h} u)$ in (3.1) we get, as usual $$ (I) = \int\limits_{B_R} \eta^2 \tau_{s,h} \left( DF(Du) \right) \tau_{s,h} D u \, dx = - \int\limits_{B_R} \tau_{s,h} \left( DF(Du) \right) 2 \eta D\eta \, \tau_{s,h} u \, dx = (II) \tag3.2 $$ We apply (1.5) so that $$ m \int\limits_{B_R} (|Du(x+he_s)|+|Du(x)|)^{p-2} |\tau_{s,h}D u(x)|^2 \eta^2 (x) \, dx \leq (I). \tag3.3 $$ Now we use Lemma 2.5 and Corollary 2.7 in order to get, for some positive constant $c_5$, independent of $h$, $$\multline c_5 \int\limits_{B_R} \left| \tau_{s,h} \left( \left|Du(x)\right|^{(p-2)/2}Du(x) \right)\right|^2 \eta^2 (x) \, dx \\ \leq m \int\limits_{B_R} (|Du(x+he_s)|+|Du(x)|)^{p-2} |\tau_{s,h}D u(x)|^2 \eta^2 (x) \, dx. \endmultline \tag3.4$$ In order to estimate $(II)$, we first use the growth condition (1.4): $$ \aligned |\tau_{s,h} \left( DF(Du(x)) \right)| =& |DF(Du(x+he_s)) - DF(Du(x))| \\ \leq & |DF(Du(x+he_s)) | + | DF(Du(x))| \\ \leq & c \,( 1+ |Du(x+he_s)|^{p-1}) + c \, (1 + |Du(x)|^{p-1})\,. \endaligned \tag3.5 $$ We apply inequality (3.5) and the properties of the ``cut off'' function $\eta$, then H\"older inequality, finally Lemma 2.1 and 2.4: $$ \multline (II) \leq c_6 \int\limits_{B_R} (1+ |Du(x+he_s)|^{p-1}+|Du(x)|^{p-1}) |\tau_{s,h} u(x)| \, dx \\ \leq c_7 \left( \int\limits_{B_R} (1+ |Du(x+he_s)|^p+|Du(x)|^p) \, dx \right)^{(p-1)/p} \left( \int\limits_{B_R} |\tau_{s,h} u(x)|^p dx\right)^{1/p} \\ \leq c_8 \left( \int\limits_{B_{2R}} (1+ |Du(x)|^p) \, dx \right)^{(p-1)/p} \left( \int\limits_{B_{2R}} |D_s u(x)|^p dx\right)^{1/p} |h| \leq c_9 |h|, \endmultline \tag3.6$$ for some positive constants $c_6, c_7, c_8, c_9$ independent of $h$. Collecting the estimates for $(I)$ and $(II)$ yields, for some positive constant $c_{10}$, independent of $h$, $$ \int\limits_{B_R} \left| \tau_{s,h} \left( \left|Du(x)\right|^{(p-2)/2}Du(x) \right)\right|^2 \eta^2 (x) \, dx \leq c_{10} |h|, \tag3.7 $$ for every $s=1,\dots ,n$, for every $h$ with $|h|0$), thus giving (1.15) (and (1.16), provided $\mu >0$). This ends the proof. \qed \subheading{5. Proof of Theorem 3} We can use the partial regularity result contained in \cite{1}, see \cite{15} too, in order to get $$Du \in C^{0,\alpha}_{\loc}(\Omega_0), \qquad \forall \alpha \in (0,1), $$ for the open set $\Omega_0$ defined as follows $$ \Omega_0 = \left\{ x \in \Omega :\, \lim_{r \to 0} (Du)_{B(x,r)} \in \Bbb R^{nN}, \quad \lim_{r \to 0} \, r^{-n}\int\limits_{B(x,r)} |Du(y) - (Du)_{B(x,r)}|^p dy = 0 \right\}. $$ where $$ (g)_{B(x,r)} = |B(x,r)|^{-1} \int\limits_{B(x,r)} g(y) \, dy. $$ So, for the singular set, we have $$ \Omega \setminus \Omega_0 \subset S_1 \cup S_2, $$ where $$ S_1 = \left\{ x \in \Omega : \qquad \nonexist \lim_{r \to 0} (Du)_{B(x,r)} \qquad \text{ or } \qquad \lim_{r \to 0} |(Du)_{B(x,r)}|=\infty \right\}, $$ $$ S_2 = \left\{ x \in \Omega : \limsup_{r \to 0} \, r^{-n}\int\limits_{B(x,r)} |Du(y) - (Du)_{B(x,r)}|^p dy > 0 \right\}. $$ Let us take $\xi \in \Bbb R^{nN}$ such that $| \xi |^{\frac{p-2}{2}} \xi = \left( |Du|^{\frac{p-2}{2}}Du \right)_{B(x,r)}$; then $$ r^{-n}\int\limits_{B(x,r)} |Du(y) - (Du)_{B(x,r)}|^p dy \leq 2^p r^{-n}\int\limits_{B(x,r)} |Du(y) - \xi|^p dy = (V); $$ we can use Corollary 2.9 with $\lambda = Du(y)$ and, if we keep in mind the particular choice of $\xi$ and Poincar\`e inequality, we get $$ \align (V) \leq & \hat c \, 2^p \, r^{-n}\int\limits_{B(x,r)} \left| | Du(y) |^{\frac{p-2}{2}} Du(y) - | \xi |^{\frac{p-2}{2}} \xi\right|^2 dy \\ = & \hat c \, 2^p \, r^{-n}\int\limits_{B(x,r)} \left| | Du(y) |^{\frac{p-2}{2}} Du(y) - (| Du |^{\frac{p-2}{2}} Du)_{B(x,r)}\right|^2 dy \\ \leq & \tilde c \, \hat c \, 2^p \, r^{2-n} \int\limits_{B(x,r)} \left|D \left( | Du(y) |^{\frac{p-2}{2}} Du(y) \right) \right|^2 dy\,. \endalign $$ Thus $$ S_2 \subset \left\{ x \in \Omega : \quad \limsup_{r \to 0} \, r^{2-n}\int\limits_{B(x,r)} \left| D \left( | Du(y) |^{\frac{p-2}{2}} Du(y) \right) \right|^2 dy>0 \right\}. $$ Since we have proven that $$DDu \text{ and } D(|Du|^{(p-2)/2}Du) \in L^2_{\loc}(\Omega), $$ we can use standard technique \cite{19}, \cite{14}, in order to get (1.20). This ends the proof. \qed \heading 6. Proof of Theorem 4 and 5\endheading Theorem 4 is proven just in the same way as Theorem 1, so we skip it and we go to Theorem 5. Arguing as in Theorem 2 we get $$ \multline \int\limits_{B_R} \int\limits_0^1 DA(Du+ t \tau_{s,h}Du) \, \eta \, \tau_{s,h}Du \, \eta \, \tau_{s,h}Du \, dt \, dx = (I) \\ =(II) = \int\limits_{B_R} \int\limits_0^1 - 2 DA(Du+ t \tau_{s,h}Du) \, \eta \, \tau_{s,h}Du \, D\eta \, \tau_{s,h} u \, dt \, dx. \endmultline \tag6.1 $$ Since the bilinear form $(\lambda , \xi ) \to DA \, \lambda\, \xi$ is no longer symmetric, we cannot use Cauchy-Schwartz inequality as we did in (4.3). Let us remark that $q< p(2n-1)/(2n-2) < pn/(n-1)$, so we can use the higher integrability result proven in Theorem 4: $$ Du \in L^\sigma_{\loc}(\Omega), \qquad \forall \sigma < p\frac{n}{n-1}. \tag1.35 $$ We apply the nonuniform ellipticity conditions (1.38) and (1.39), then we use (1.35) with $\sigma =q$: $$ 0\leq \hat m \, \int\limits_{B_R} \int\limits^1_0 ( \mu + |Du + t \tau_{s,h} Du |^{p-2} ) |\tau_{s,h} Du|^2 \, \eta^2 \, dt \, dx = \hat m \, (IV) \leq (I) <\infty. \tag6.2 $$ Let us estimate $(II)$. First of all we use the growth condition (1.39): $$ \multline | 2 DA(Du+ t \tau_{s,h}Du) \, \eta \, \tau_{s,h}Du \, D\eta \, \tau_{s,h} u| \\ \aligned \leq & c_{17} \, ( \mu + |Du + t \tau_{s,h} Du |^{q-2} ) |\eta \, \tau_{s,h} Du| \, |\tau_{s,h} u| \\ \leq & \epsilon \, ( \mu + |Du + t \tau_{s,h} Du |^{p-2} ) |\eta \, \tau_{s,h} Du|^2 \\ &+ \frac{c_{17}^2}{\epsilon} ( \mu + |Du + t \tau_{s,h} Du |^{2q-p-2} ) | \tau_{s,h} u|^2, \qquad \forall \epsilon >0, \endaligned \endmultline \tag6.3 $$ for some positive constant $c_{17}$ independent of $h$ and $\epsilon$, so that $$ |(II)| \leq \epsilon (IV) + \frac{c_{17}^2}{\epsilon} \int\limits_{B_R} \int\limits_0^1( \mu + |Du + t \tau_{s,h} Du |^{2q-p-2} ) |\tau_{s,h} u|^2 \, dt \, dx. \tag6.4 $$ Because of (1.40), $p