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\begin{document}

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%% titulni strana

\thispagestyle{empty}
\ 
\vfill

\centerline{\Large\sc Regularita PDR}
\bigskip
\centerline{z�pisky z  p�edn�ky doc. J. Star�, ZS 2003/2004}


\vfill
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\newpage
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%% obsah
\thispagestyle{empty}
{\bf Obsah.}

\vskip 1cm

Prostory funkc� a rovnice \dotfill  \pageref{uvod}

Technika diferenc� \dotfill \pageref{diference}

U�it� Morreyov�ch a Campanatov�ch prostor� \dotfill \pageref{camp-morr}

��ste�n� regularita \dotfill \pageref{partial}

Syst�my bl�zk�  $-\Delta$. \dotfill \pageref{blizke}


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\thispagestyle{empty}

{\bf Zna�en�.}

\vskip 1 cm

$L^p(\om)$ ... Lebesgue�v prostor $p$-integrovateln�ch funkc�

$\norm{\cdot}{p}{}$ ... norma v prostoru $L^p$

$W^{k,p}(\om)$ ... Sobolev�v prostor funkc� s $p$-integrovatelnou
$k$-tou derivac�

$W^{1,p}_0(\om)$ ... Sobolev�v prostor s nulou na hranici

$u_{\om}=\aint_{\om}u$ ... integr�ln� pr�m�r $u$ p�es $\om$


$B(x,R)$ nebo $B_R$ ... koule o st�edu $x$ a polom�ru $R$

$\om(x,R)$ ... $B(x,R)\cap\om$

$BMO(\om)$ ... funkce s omezen�mi oscilacemi

$S(x,R)$ nebo  $S_R$ ... sf�ra o st�edu $x$ a polom�ru $R$

$\aint_{B(x,R)} u$ nebo $u_{B(x,R)}$ nebo $u_R$ ... integr�ln� pr�m�r
$u$ p�es  $B(x,R)$


$L^{p,\lambda}(\om)$ ... Morrey�v prostor

$\el^{p,\lambda}(\om)$ ... Campanat�v prostor

$C^{k,\pha}(\om)$ ... funkce s $\pha$-h\"olderovskou k-tou derivac�

$e_s$ ... b�zov� vektor

$\dif{s,h}{u}$ ... diference $h^{-1}[u(x+he_s)-u(x)]$

$\lambda_n$ ... Lebesgueova $n$-rozm�rn� m�ra

$H^{\pha}$ ... Hausdorffova $\pha$-rozm�rn� m�ra

$d_H$ ... Hausdorffova dimenze

$\supp\eta$ ... nosi� funkce

${\rm osc}\, v |_{M}$ ... oscilace $\sup_M v - \inf_M v$

$\sigma(A)$ ... spektrum matice

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% uvodni kapitola
%%% definice, prostory, rovnice
%%%%

\section{Prostory funkc�. Typy rovnic.} \label{uvod}

%%% kapitola 0 - definice, prostory, lemmata, rovnice

Obsah kapitoly: z�kladn� vlastnosti
funkc� ze Sobolevov�ch, Morreyov�ch a Campanatov�ch prostor�
(bez d�kazu); 
podm�nka elipticity;  typy rovnic.

\par\medskip
Nech� $\om\subset \er^n$ je omezen� a otev�en� mno�ina
s Lipschitzovskou hranic�.


\begin{theorem} {\rm (Vno�en� pro Sobolevy.)}
\ \par
1. Nech� $n>kp$. Potom $\exists c=c(n,k,p,\om)$ tak, �e
\[
\norm{u}{L^q(\om)}{}
\le
c\,
\norm{u}{W^{k,p}(\om)}{}
\,,
\]
kde $q=np/(n-p)$.
\par
2. Nech� $n<kp$. Potom $\exists c=c(n,k,p,\om)$ tak, �e
\[
\norm{u}{C^{h,\pha}(\overline{\om})}{}
\le
c\,
\norm{u}{W^{k,p}(\om)}{}
\,,
\]
kde $h=\lceil k-n/p \rceil$, $\pha=(k-n/p)-h$.
\end{theorem}

\begin{theorem} {\rm (Rellichova.)}
Nech� $n>kp$ a $q< np/(n-p)$. Potom vno�en� $W^{k,p}(\om)\subset
L^q(\om)$ je kompaktn�.
\end{theorem}

\begin{theorem} {\rm(Poincar�.)}
\label{th:poin}
Nech� $\om$ je souvisl�. 
\par
1. $\exists c = c(n,p,\om)$ tak, �e
\[
\intom |u-u_\om|^p \le c \, \normg{u}pp
\]
pro $\forall u\in W^{1,p}(\om)$.
Speci�ln� pro $\om=B(x,R)$ je $c=c_0\, R^p$. 
\par
2. $\exists c = c(n,p,\om)$ tak, �e
\[
\norm{u}p{p} \le c\, \normg{u}{p}{p}
\]
pro $\forall u\in W^{1,p}_0(\om)$. 
\end{theorem}



\par\noindent
{\bf Definice.} (Morreyovy a Campanatovy prostory.)
\par
1. Morrey�v prostor:
\[
L^{p,\lambda}(\om)=
\left\{ 
u\in L^p(\om):\ 
\sup_{x_0\in\om, \rho>0 }
\ 
\rho^{-\lambda}
\hskip-10pt
\int\limits_{\om(x_0,\rho)}
\hskip-10pt
|u(x)|^{p}\,dx
<\infty
\right\}
\,.
\]
2. Campanat�v prostor:
\[
\el^{p,\lambda}(\om)=
\left\{ 
u\in L^p(\om):\ 
\sup_{x_0 \in \om ,\, \rho>0}
\
\rho^{-\lambda}
\hskip-10pt
\int\limits_{\om(x_0,\rho)}
\hskip-10pt
|u(x)-u_{\om(x_0,\rho)}|^{p}\,dx
<\infty
\right\}
\,.
\]
%%%%
\par\noindent
{\bf Pozn�mka.} M�sto Lipschitzovsk�ch oblast� sta�� uva�ovat
obecn�j�� ``oblasti typu A'', kter� spl�uj�
\[
A |B(x_0,\rho)| \le |\om(x_0,\rho)| \le |B(x_0,\rho)|
\]
s vhodn�m $A>0$. Faktor $\rho^{-\lambda}$ lze pak nahradit
v�razem $|\om(x_0,\rho)|^{-\lambda/n}$.



\begin{theorem} {\rm(Vno�en� Morreyov�ch a Campanatov�ch prostor�.)}
  \label{th:mc}

1. Pro $0\le \lambda <n$ je $L^{p,\lambda}(\om)=\el^{p,\lambda}(\om)$.

2. Pro $\lambda = n$ je $L^{p,n}(\om)=L^{\infty}(\om)$, 
$\el^{p,n}(\om) = BMO(\om)$.

3. Je-li $u\in W^{1,p}(\om)$ a $\nabla u \in L^{p,\lambda}(\om)$, 
je $u\in \el^{p,p+\lambda}(\om)$.

4. Pro $n < \lambda < n+p$ je $\el^{p,\lambda}(\om) =
   C^{0,\pha}(\overline{\om})$, kde $\pha=(\lambda-n)/p$.

5. Pro $\lambda>n$ je $L^{p,\lambda}(\om)=\big\{0\big\}$;
pro $\lambda>n+p$ je
$\el^{p,\lambda}(\om)=\operatorname{lin}\big\{1\big\}$. 

\end{theorem}


\medskip
{\bf Definice.} [Rovnice s konstantn�mi koeficienty.]
\begin{equation}
-\diver\left(A \nabla u\right)=
-\frac{\partial}{\partial x_{\alpha}}
\left(
A_{ij}^{\alpha\beta} \frac{\partial u_j}{\partial x_\beta}
\right)
= 0
\,,
\qquad 
i=1,\dots,N
\,,
\label{konst}
\end{equation}
kde $\tilde{u}=(u_1,\dots,u_N):\er^n \to \er^N$ jsou nezn�m� funkce,
 $A_{ij}^{\alpha\beta}\in \er$ jsou konstanty.


\medskip
{\bf Definice.} [Podm�nka elipticity.]
\[
\lambda_0 |\xi|^2 
\le 
(A\xi,\xi)
\le
\lambda_1 |\xi|^2
\qquad
\forall \xi\in \er^{nN}
\]
kde
\[
(A\xi,\xi)=
A_{ij}^{\alpha\beta} 
\xi_i^{\alpha} \xi_j^{\beta}
\]

\medskip
{\bf Definice.} [Line�rn� rovnice.]
\begin{equation}
-\der{}{x_\pha}\left\{A_{ij}^{\pha\beta}(x)\der{u_j}{x_\beta}(x)\right\}
= 
-\der{f_i^{\pha}}{x_\pha}
\label{lin}
\end{equation}


\medskip
{\bf Definice.} [Kvaziline�rn� rovnice.]
\begin{equation}
-\der{}{x_\pha}\left\{A_{ij}^{\pha\beta}\big(u(x)\big)
\der{u_j}{x_\beta}(x)\right\}
= 
-\der{f_i^{\pha}}{x_\pha}
\label{kvazilin}
\end{equation}


\medskip
{\bf Definice.} [Neline�rn� rovnice.]
\begin{equation}
-\dd{x_{\alpha}}\Big\{ a_{i}^{\alpha}\big(\nabla u\big) \Big\}
= 0
\,,
\qquad
i=1,\dots,N
\,.
\label{nelin}
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% technika diferenci
%%%%

\section{Technika diferenc�.} \label{diference}

%% technika diferenci

Obsah kapitoly: lemma o odhadu derivace pomoc� diferenc�;
v�ta o regularit� neline�rn� rovnice; v�ta o regularit� kvaziline�rn�
rovnice za p�edpokladu spojitosti �e�en�.


\begin{lemma}  {\rm (O diferenc�ch.)}

1. Nech� $u\in W^{1,2}(\om)$, $\om_0\subset\subset \om$. Potom
\[
\int_{\om_0} |\dif{s,h}u|^2 \le K
\]
nez�visle na $0<|h|<\dist(\om_0,\partial\om)$.

2. Nech� $u\in L^2(\om)$, $\om_0\subset\subset \om$. Nech� 
\[
\int_{\om_0} |\dif{s,h}u|^2 \le K
\]
nez�visle na $0<|h|<\dist(\om_0,\partial\om)$.
Potom $\der{u}{x_s}\in L^2(\om_0)$ ve slab�m smyslu.
\end{lemma}

\dduk
1. Nejprve odhadujeme (posledn� krok je H�lderova nerovnost)
\begin{align*}
|\dif{s,h}u|^2 &= | \frac1h \int_0^1 \der{u}{x_s}(x+t\hes)\,h\,dt |^2
\\
&\le
(\int_0^1 |\nabla u|(x+t\hes)\,dt )^2
\\
&\le
\int_0^1 |\nabla u|^2(x+t\hes)\,dt
\,.
\end{align*}
Tedy pro dan� $h$ (Fubiniho v�ta)
\[
\int_{\om_0}|\dif{s,h}u|^2
\le
\int_{\om_0}\int_0^1 |\nabla u|^2(x+t\hes)\,dt\,dx
\le
\int_{\om}|\nabla u(y)|^2\,dy
\]
a odtud  z�v�r.

\par
2. Zvol $\varphi\in C^{\infty}_0(\om_0)$. Je
\[
\int_{\om} \big\{ \dif{s,h}u \big\} \,\varphi
=
\int_{\om} u\, \dif{s,-h}\varphi
\,.
\]
D�ky slab� kompaktnosti existuje $\chi\in L^2(\om_0)$ tak, �e
$\dif{s,h}u\rightharpoonup \chi$ v $L^2(\om)$ pro $h\to 0$.
\par
Na druh� stran� $\dif{s,-h}\varphi \to -\der{\varphi}{x_s}$
v (nej�m�n�) $L^2(\om_0)$. Tedy
\[
\int_{\om_0} \chi \, \varphi
=
-
\int_{\om_0} u \, \der{\varphi}{x_s}
\,,
\]
co� je hledan� z�v�r.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{theorem}  {\rm (Regularita neline�rn� rovnice.)}
Nech� $u\in W^{1,2}_{loc}$ je slab� �e�en� neline�rn� rovnice \eqref{nelin},
kde $A^{\alpha\beta}_{ij}=\frac{\partial a^{\alpha}_i}{\partial z^{\beta}_j}$
spl�uj� podm�nku elipticity.
Potom $u\in W^{2,2}_{loc}$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%
\dduk
Testujme rovnici funkc�
$\phi=\dif{s,-h}\left\{ \eta^2 \, \dif{s,h}u \right\}$,
kde $\eta$ je nez�porn� se�ez�vac� funkce.
M�me
\[
0=
\int_\om
a_i^{\alpha}(\nabla u)
\,
\der{\phi_i}{x_{\alpha}}
=
\int_\om
\underbrace{
\dif{s,h}
\big\{ a_i^{\alpha}(\nabla u) \big\}
}_{I_1}
\,
\underbrace{
\dd{x_{\alpha}} \left\{ \eta^2 \dif{s,h} u_i \right\}
}_{I_2}
\,.
\]

Zde
\begin{align*}
I_1(x)&=
\frac1h
\Big\{
\aia\big(\nabla u(x+\hes)\big)
-\aia\big(\nabla u(x) \big)
\Big\}
\\
&=
\frac1{h}
\int_0^1
\dd{t}
\Big\{
\aia
\big(
\nabla u(x) 
+ 
t\big[\nabla u(x+\hes)-\nabla u(x)\big]
\big)
\Big\}
\,dt
\\
&=
\der{}{x_\beta}
\big\{
\dif{s,h}{u_j}
\big\}
\underbrace{
\int_0^1
\der{\aia}{z_j^{\beta}}
\big(
\nabla u(x)+t\big[ \nabla u(x+\hes) -\nabla u(x)\big]
\big)
\,dt
}_{\tilde{A}_{ij}^{\alpha\beta}(x)}
\end{align*}
a
\[
I_2=\eta^2 \, 
\der{}{x_\pha}
\big\{
\dif{s,h}{u_i}
\big\}
+
2\eta\der{\eta}{x_\alpha}\,\dif{s,h}u_i
\,.
\]
Proto�e $\tilde{A}_{ij}^{\alpha\beta}$ spl�uj� podm�nku
elipticity, m�me
\begin{align*}
\lambda_0 
\int_\om
\left|
\nabla
\big\{ \dif{s,h}u \big\}
\right|^2
\eta^2
&\le
2\lambda_1
\int_\om
\eta 
|\nabla\{ \dif{s,h}u\}|
|\nabla \eta|
|\dif{s,h} u|
\\
&\le
\frac12[LS]
+
\frac{2\lambda_1^2}{\lambda_0}
\norm{\nabla\eta}{L^{\infty}}{2}
\int_{\supp \eta}|\dif{s,h} u|^2
\,.
\end{align*}
Tud��
\[
\int_\om
\left|
\nabla
\big\{ \dif{s,h}u \big\}
\right|^2
\eta^2
\le
c
\int_{\supp\eta}|\dif{s,h} u|^2
\,.
\]
Proto�e $u\in W^{1,2}_{loc}$, je d�ky prvn� ��sti lemmatu o
diferenc�ch prav� strana omezen� nez�visle na $h$. Odtud
d�ky druh� ��sti lemmatu z�v�r.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{theorem}  {\rm (Regularita kvaziline�rn� rovnice.)}
Nech� $v\in W^{1,2}_{loc}$ je slab� �e�en� kvaziline�rn� rovnice 
\eqref{kvazilin},
kde $A^{\alpha\beta}_{ij}\in C^1\cap L^{\infty}$
spl�uj� podm�nku elipticity nez�visle na $v$. 
Nav�c nech� $v$ je spojit�.
Potom $v\in W^{2,2}_{loc}$.
\end{theorem}
\dduk
Zvolme $x_0$, $R>0$.
Bu� $\eta$ nez�porn� se�er�zavac� funkce s nosi�em v $B(x_0,R)$,
rovn� $1$ na $B(x_0,R/2)$. Testujeme rovnici
  $\dif{s,-h}\left\{\eta^2\dif{s,h}v\right\}$.
M�me
\[
0=
\int_\om
A_{ij}^{\pha\beta}(v)\der{v_i}{x_\alpha}
\,
\der{\phi_j}{x_{\beta}}
=
\int_\om
\underbrace{
\dif{s,h}\left\{ A_{ij}^{\pha\beta}(v)\der{v_i}{x_\alpha} \right\}
}_{I_1}
\,
\underbrace{
\dd{x_{\beta}} \left\{ \eta^2 \dif{s,h} v_j \right\}
}_{I_2}
\,.
\]
Zde
\begin{align*}
I_1(x)
&=
\frac1h
\Big[
A_{ij}^{\pha\beta}\big(v(x+\hes)\big)\der{v_i}{x_\pha}(x+\hes)
-A_{ij}^{\pha\beta}\big(v(x)\big)\der{v_i}{x_\pha}(x)
\Big]
\\
&=
A_{ij}^{\pha\beta}\big(v(x+\hes)\big)
\,
\dif{s,h}\!
\left\{
\der{v_i}{x_\pha}(x)
\right\}
\\
&\hskip 1cm
+
\der{v_i}{x_\pha}(x)
\int_{0}^{1}\der{}{t}
\Big\{
A_{ij}^{\pha\beta}\big(v(x)+t[v(x+\hes)-v(x)]\big)
\Big\}
\,dt
\\
&=
A_{ij}^{\pha\beta}\big(v(x+\hes)\big)
\,
\dif{s,h}\!
\left\{
\der{v_i}{x_\pha}(x)
\right\}
\\
&\hskip1cm
+
\der{v_i}{x_\pha}(x)
\,
\dif{s,h}\big\{v_k(x)\big\}
\,
\int_0^1 \der{A_{ij}^{\pha\beta}}{v_k}\big(v(x)+t[v(x+\hes)-v(x)]\big)
\,dt
\\
&=I_{11}(x) + I_{12}(x)
\end{align*}
a %%%%aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
\[
I_2=\eta^2 \,
\dif{s,h}\!
\left\{
\der{v_j}{x_\beta}(x)
\right\}
+
2\eta\der{\eta}{x_\beta}\,\dif{s,h}v_j
=
I_{21} + I_{22}
\,.
\]
Odhadujeme
\begin{align*}
\int_{\om} 
I_{11} I_{21} &=
\int_{\om}
A_{ij}^{\pha\beta}
\big(v(x+\hes)\big)
\,
\dif{s,h}\!
\left\{\der{v_i}{x_\pha}(x)\right\}
\,
\dif{s,h}\!
\left\{
\der{v_j}{x_\beta}(x)
\right\}
\eta^2
\\
&\ge
\lambda_0 
\int_{\supp\eta}
\big| \dif{s,h}\big\{\nabla v\big\} \big|^2 \eta^2
\,,
\\
\int_\om I_{11} I_{22}
&=
2
\int_\om
A_{ij}^{\pha\beta}(\dots)
\dif{s,h}\!
\left\{\der{v_i}{x_\pha}(x)\right\}
\,
\dif{s,h}\big\{ v_i \big\}
\der{\eta}{x_\beta}
\eta
\\
&\le
\frac{\lambda_0}4
\int_{\supp\eta}
\big| \dif{s,h}\big\{\nabla v\big\} \big|^2 \eta^2
+
c\norm{\nabla \eta}{L^\infty}{2}
\int_{\supp \eta}
\big| \dif{s,h}v \big|^2
\,,
\\
\int_\om
I_{12} I_2
&=
\int_{\om}
\tilde{A}_{ij}^{\pha\beta}
\der{v_i}{x_\pha}
\dif{s,h}
\big\{v_k\big\}
\left[
\dif{s,h}\!
\left\{
\der{v_j}{x_\beta}
\right\}
\eta^2
+
2\eta
\der{\eta}{x_\beta}
\dif{s,h}\big\{ v_j \big\}
\right]
\\
&\le
c\int_{\supp\eta}
\big|\nabla v\big|\,
\big|\dif{s,h}\{ v \}\big|
\,
\Big\{
\big| \dif{s,h}\{ v \} \big|
\,\eta^2
+
\big| \dif{s,h}\{v\} \big|
\,
|\eta|
\,
\eta
\Big\}
\\
&\le
\frac{\lambda_0}4
\int_{\supp\eta}
\big| \dif{s,h}\big\{\nabla v\big\} \big|^2 \eta^2
+
c\norm{\nabla \eta}{L^\infty}{2}
\int_{\supp \eta}
\big| \dif{s,h}\{v\} \big|^2
\\
&\hskip 2cm
+
c
\int_{\supp\eta}
\big|\nabla v\big|^2
\big|\dif{s,h} \{ v \} \big|^2
\eta^2
\,.
\end{align*}
Celkem m�me
\begin{equation}
\int_{\supp\eta}
\big| \dif{s,h}\big\{\nabla v\big\} \big|^2 \eta^2
\le
c_1+
c_2
\int_{\supp \eta}
\big|\nabla v\big|^2
\big|\dif{s,h} \{ v \} \big|^2
\eta^2
\,.
\label{sl1}
\end{equation}
Zb�v� odhadnout �len vpravo, form�ln� odpov�daj�c� $|\nabla v|^4$.
Zvolme testovac� funkci $\psi=(v-v_R)|\dif{s,h}v|^2\eta^2$, kde
$v_R$ je pr�m�r $v$ p�es $B(x_0,R)$. M�me
\begin{align*}
0&=
\int_\om
A_{ij}^{\pha\beta}(v)\der{v_i}{x_\alpha}
\,
\der{\psi_j}{x_{\beta}}
\\
&=
\int_{\om}
A_{ij}^{\pha\beta}(v)\der{v_i}{x_\alpha}
\left[
\der{v_j}{x_\beta}
\big|\dif{s,h}\{v\}\big|^2
\eta^2
+
2(v-v_R)\dif{s,h}\{v_k\}
\dif{s,h}\!
\left\{\der{v_k}{x_\beta}\right\}
\eta^2
\right.
\\
&\hskip 4cm
+
\left.
2(v-v_R)
\big|\dif{s,h}\{v\}\big|^2
\der{\eta}{x_\beta}
\eta
\right]
\,.
\end{align*}
Odsud
\begin{align}
&
\lambda_0
\int_{\supp\eta}
\big|\nabla v\big|^2
\big|\dif{s,h} \{ v \} \big|^2
\eta^2
\label{sl2}
\\
&\le
c\,
\omega(R)
\int_{\supp\eta}
\big|\nabla v\big|
\big|\dif{s,h} \{ v \} \big|
\big| \dif{s,h}\big\{\nabla v\big\} \big|
\eta^2
+
\big| \nabla v \big|
\big|\dif{s,h} \{ v \} \big|^2
|\nabla \eta|
\eta
\nonumber
\\
&\le
c\, \omega(R)
\int_{\supp\eta}
\big|\nabla v\big|^2
\big|\dif{s,h} \{ v \} \big|^2
\eta^2
+
\big| \dif{s,h}\big\{\nabla v\big\} \big|^2 \eta^2
+\norm{\nabla\eta}{L^2(B_R)}{2}
\nonumber
\end{align}
D�ky spojitosti $v$ lze zmen�en�m $R$ u�init $\omega(R)=\oscvbr$ libovoln�
mal�. Potom z \eqref{sl1}, \eqref{sl2} plyne
\[
\int_{\supp\eta}
\big| \dif{s,h}\big\{\nabla v\big\} \big|^2 \eta^2
\le
c
\]
a odsud z�v�r.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\section{U�it� Morreyov�ch a Campanatov�ch prostor�}  \label{camp-morr}

%% cacciopoli a spol.

Obsah kapitoly: Caccioppoliho lemma; Campanatovo lemma; algebraick� lemma;
regularita gradientu pro line�rn� rovnici - v Campanatov� prostoru a v
prostoru H\"olderovsk�ch funkc�.

\begin{lemma} {\rm (Caccioppoli.)} \label{lm:cac}
Je-li $u\in W^{1,2}(B_R)$ slab� �e�en� rovnice s konstantn�mi
koeficienty \eqref{konst}, pak pro ka�d� $q\in \er^N$ plat�
\[
\int\limits_{B_{R/2}} |\nabla u|^2 \le \frac{c}{R^2} \int\limits_{B_R} |u-q|^2
\]
kde $c=c(\lambda_0,\lambda_1)$. Obecn�ji: vlevo integr�l p�es $B_\rho$,
vpravo faktor $(R-\rho)^{-2}$, a funguje vlastn�  pro ka�dou eliptickou rci.
\end{lemma}
\dduk
Bu� $\eta$ nez�porn� shlazovac� funkce rovn� jedn� na $B_{R/2}$
a nulov� vn� $B_R$. Lze ��dat $|\nabla\eta|\le c_1 / R$. Testovac� funkce
$\varphi=(u-q)\eta^2$ d�v�
\[
0=\int_{B_R}A\nabla u \cdot \nabla\varphi
=
\int_{B_R}A\nabla u \cdot \nabla u \, \eta^2
+
\int_{B_R}A\nabla u \cdot (u-q) \nabla \eta \, 2\eta
=
I_1 + I_2
\,.
\]
Nyn�
\begin{align*}
I_1 &\ge \lambda_0 \int_{B_R} |\nabla u|^2\, \eta^2
\ge \lambda_0 \int_{B_{R/2}} |\nabla u|^2
\\
|I_2| &\le \frac{2c_1\lambda_1}R  \int_{B_R} |\nabla u| \eta\, |u-q| 
\le
\frac{\lambda_0}2 \int_{B_R} |\nabla u|^2\, \eta^2
+\frac{2c_1^2\lambda_1^2}{R^2\lambda_0}\int_{B_R}|u-q|^2
\,.
\end{align*}
Odsud z�v�r s $c=4c_1^2\lambda_1^2 / \lambda_0^2$.
\eduk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} {\rm(Campanato.)} \label{lm:cam}
Je-li $u\in W^{1,2}(B_R)$ slab� �e�en� rovnice s konstantn�mi koeficienty
\eqref{konst}, pak pro
ka�d� $0<\rho<R$ plat�
\begin{align}
\int\limits_{B_{\rho}} |u|^2 
& \le 
c\left(\frac{\rho}{R}\right)^n
 \int\limits_{B_R} |u|^2
\label{campa1}
\\
\int\limits_{B_{\rho}} |u-u_\rho|^2 
& \le 
c\left(\frac{\rho}{R}\right)^{n+2}
 \int\limits_{B_R} |u-u_R|^2
\label{campa2}
\end{align}
kde $c=c(\lambda_0,\lambda_1,n)$.
\end{lemma}
\dduk
Ad \eqref{campa1}. Bez �jmy na obecnosti $\rho\le R/2$.
Zvol $k$ tak, aby $W^{k,2}\subset L^{\infty}$. Potom
\begin{equation}
\int_{B_\rho}|u|^2 
\le 
\kappa_n \rho^n \norm{u}{L^{\infty}(B_\rho)}{2}
\le
\kappa_n \rho^n \norm{u}{L^{\infty}(B_{R/2})}{2}
\,.
\label{cam1}
\end{equation}
Pomocn� v�po�et (konstanta vno�en� pro $B_R$): nech� 
\[
\norm{v}{L^{\infty}(B_1)}{2}
\le c_1 
\norm{v}{W^{k,2}(B_1)}{2}
\,,
\qquad
\norm{v}{W^{k,2}(B_1)}{2}
=
\norm{v}{L^2(B_1)}{2}
+
\norm{\nabla^k v}{L^2(B_1)}{2}
\,.
\]
Polo� $v(y)=u(Ry)$, $x\in B_1$, $u=u(x)\in W^{k,2}(B_R)$.
D�ky substituci $x=Ry$ ($dy=R^{-n}dx$) a vztahu 
$\nabla_y^k v(y) = R^k [\nabla_x^k u](Ry)$ m�me
\begin{align*}
\norm{v}{L^{\infty}(B_1)}2
&=
\norm{u}{L^{\infty}(B_R)}2
\\
\norm{v}{L^2(B_1)}2
&=
R^{-n}
\norm{u}{L^2(B_R)}2
\\
\norm{\nabla^k v}{L^2(B_1)}2
&=
R^{2k-n}
\norm{\nabla^k u}{L^2(B_R)}2
\end{align*}
Tedy (z�m�nou $R/2$ za $R$)
\begin{align}
\norm{u}{L^{\infty}(B_{R/2})}{2} 
&\le 
c_2 \left\{
R^{-n} \norm{u}{L^2(B_{R/2})}{2}
+
R^{2k-n} \norm{\nabla^k u}{L^2(B_{R/2})}{2}
\right\}
\,,
\label{cam2}
\\
c_2 &= c_1 (2^n + 2^{n-2k})
\,.
\nonumber
\end{align}
Druh� �len v \eqref{cam2} odhadneme pomoc� s�rie nerovnost�
($j=0,\dots,k-1$)
\[
\int_{B_{(1+j/k)R/2}}
|\nabla^{k-j} u|^2
\le
c_3 (2k)^2 R^{-2}
\int_{B_{(1+(j+1)/k)R/2}}
|\nabla^{k-(j+1)} u|^2
\]
kter� plynou z obecn�j��ho Caccioppoliho lemmatu
mezi koulemi o polom�rech $(1+j/k)R/2$ a $(1+(j+1)/k)R/2$
d�ky tomu, �e $\nabla^{k-j}u$ spl�uje tut� rovnici (konstantn�
koeficienty.) Celkem tedy
\[
\int_{L^2(B_{R/2})} |\nabla^k u|^2
\le
c_4 R^{-2k} \int_{B_R}|u|^2
\,,
\qquad
c_4 = c_3^k (2k)^{2k}
\,.
\]
Tato nerovnost spolu s \eqref{cam2}, trivi�ln�m vno�en�m
$L^2(B_{R/2})\subset L^2(B_R)$ a \eqref{cam1} d�vaj� 
\eqref{campa1} s $c=\kappa_n c_2 (1+c_4)$.
\par\medskip
Ad  \eqref{campa2}. Bez �jmy na obecnosti $\rho\le R/2$. 
Postupn�m u�it�m Poincar�ho nerovnosti,
\eqref{campa1} aplikovan� na $\nabla u$ (�e�� stejnou rovnici -
konstantn� koeficienty) a Cacciopoliho nerovnosti s $q=u_R$
m�me
\begin{align*}
\int_{B_\rho}|u-u_\rho|^2
&\le
c_1 \, \rho^2\int_{B_\rho}|\nabla u|^2 
\\
&\le
c_2 \, \rho^2 \left(\frac{\rho}{R}\right)^n \int_{B_{R/2}}|\nabla u|^2
\\
&\le
c_3 \, \left(\frac{\rho}{R}\right)^{n+2} \int_{B_{R}}|u-u_R|^2
\,.
\end{align*}
\eduk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} {\rm (Algebraick�.)}
\label{lm:alg}
Nech� $\varphi=\varphi(\rho)$ je nez�porn�, neklesaj�c� na $[0,R_0]$;
nech� $A$, $B$, $\alpha$, $\beta>0$, $\alpha>\beta$. Pak existuje 
$\epsilon_0>0$ a $C>0$  tak, �e je-li s vhodn�m $\epsilon\le \epsilon_0$
\[
\varphi(\rho)\le 
A \left[ \left(\frac{\rho}{R}\right)^{\alpha} + \epsilon \right]\varphi(R)
+ B R^{\beta}
\qquad
\forall
0<\rho\le R \le R_0
\,,
\]
je $\varphi(\rho)\le C \rho^{\beta}$ pro $\rho\in[0,R_0]$.
\end{lemma}
\dduk
Zvolme $\gamma\in(\beta,\alpha)$ a $\tau\in(0,1)$
tak, �e $2A\tau^{\alpha}=\tau^{\gamma}$. Polo� $\eps_0=\tau^{\alpha}$.
Potom
\begin{align*}
\varphi(\tau R) &\le A\left(\tau^{\alpha}+\eps\right)\varphi(R)+ BR^{\beta}
\le \tau^{\gamma}\varphi(R)+ BR^{\beta}
\\
\varphi(\tau^2 R) & \le \tau^{\gamma}\varphi(\tau R) + BR^{\beta}\tau^{\beta}
\le \tau^{2\gamma} \varphi(R) + 
BR^{\beta}\left[\tau^{\gamma}+\tau^{\beta}\right]
\\
\vdots
\\
\varphi(\tau^k R) & \le \tau^{k\gamma}\varphi(R) + BR^{\beta}
\left[ \tau^{(k-1)\beta} + \tau^{\gamma + (k-1)\beta} \dots 
\tau^{(k-2)\gamma + \beta} + \tau^{(k-1)\gamma}\right]
\end{align*}
V�raz v posledn� hranat� z�vorce je roven 
$(\tau^{k\beta}-\tau^{k\gamma})/(\tau^{\beta}-\tau^{\gamma})$.
Tud�� 
\[
\varphi(\tau^k R) \le c_1 [\tau^k]^{\beta}
\,,
\qquad
c_1=\varphi(R)+ BR^{\beta}/(\tau^{\beta}-\tau^{\gamma})
\,.
\]
\par
Nech� $0<\rho\le R$. Bu� $k\ge0$ cel� takov�, �e
 $\tau^{k+1}R<\rho\le \tau^{k}R$. Potom
\[
\varphi(\rho) \le \varphi(\tau^k R) \le c_1 [\tau^k]^{\beta}
\le \frac{c_1}{(\tau R)^{\beta}} [\tau^{k+1}R]^{\beta} \le c_2 \rho^{\beta}
\,,
\qquad c_2=\frac{c_1}{(\tau R)^{\beta}}
\,.
\]
\eduk%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} {\rm (Regularita line�rn� rovnice I.)}
Nech� $u\in W^{1,2}_{loc}$ je slab� �e�en� line�rn� rovnice \eqref{lin},
nech� $A_{ij}^{\pha\beta}(x)$ jsou spojit� a spl�uj� podm�nku elipticity
nez�visle na  $x$. Nech� $f\in L^{2,\lambda}_{loc}$,
kde $\lambda\in(0,n)$. 
\par
Potom $\nabla u \in L^{2,\lambda}_{loc}$.
\end{theorem}
\dduk
Nech� $\om_0\subset\subset\om$.
Ozna� $R_0=\dist(\om_0,\partial\om)$ a 
pro $x_0\in\om_0$ definuj
$\norm{u}{r}{}=\norm{u}{L^2(B(x_0,r))}{}$. 
C�lem je uk�zat
\begin{equation}
\normg{u}{\rho}{}\le C\rho^{\lambda/2}
\,,
\qquad
\forall \rho\le R_0
\,,
\label{cl1}
\end{equation}
kde $C=C(R_0,f)$ nez�vis� na $x_0\in\om_0$, z �eho� z�ejm�
plyne z�v�r. Nerovnost \eqref{cl1} se uk�e pomoc� algebraick�ho
lemmatu. Fixujme $\rho$, $R$, kde $\rho\le R \le R_0$.
\par
Lze ps�t $u=v+w$, kde $u$, $v$
�e�� rovnice
\begin{align}
-\diver\big(A(x_0)\nabla v\big)&=0
\qquad\textrm{na}
\ B(x_0,R)
\label{u1}
\\
v&=u
\qquad\textrm{na}
\ \partial B(x_0,R)
\nonumber
\\
-\diver\big(A(x_0)\nabla w\big)&=
-\diver\Big[
\big(A(x_0)-A(x)\big)\nabla u
+f
\Big]
\qquad\textrm{na}
\ B(x_0,R)
\label{u2}
\\
w&=0
\qquad\textrm{na}
\ \partial B(x_0,R)
\nonumber
\end{align}
Odhadujeme
\begin{align*}
\normg{u}{\rho}{}
&\le
\normg{v}{\rho}{}+\normg{w}{\rho}{}
\\
&\le
c_1\left(\frac{\rho}{R}\right)^{n/2}\normg{v}{R}{}
+\normg{w}{R}{}
\\
&\le
c_1\left(\frac{\rho}{R}\right)^{n/2}\normg{u}{R}{}
+(c_1+1)\normg{w}{R}{}
\,.
\end{align*}
U�ili jsme troj�heln�kovou nerovnost pro $u=v+w$,
Campanatovo lemma pro $v$ na $B(x_0,R)$, trivi�ln� odhad
$\normg{w}{\rho}{}\le \normg{w}{R}{}$ a
troj�heln�kovou nerovnost pro $v=u-w$.
\par
K odhadu posledn�ho �lenu testujeme \eqref{u2} funkc� $w$:
\begin{align*}
\int_{B(x_0,R)} A(x_0)\nabla w \cdot \nabla w
&=
\int_{B(x_0,R)}
\Big[
\big(A(x_0)-A(x)\big)\nabla u
+f
\Big]
\cdot \nabla w
\\
\lambda_0\normg{w}{R}{2}
&\le
\Big\{
\omega(R)
\normg{u}{R}{}
+
\norm{f}{R}{}
\Big\}
\normg{w}{R}{}
\\
\lambda_0\normg{w}{R}{}
&\le
\omega(R)\normg{u}{R}{}
+c_2 R^{\lambda/2}
\,.
\end{align*}
Zde $\omega(R)=\sup_{x\in B(x_0,R)}\big\{A(x)-A(x_0)\big\}$.
U�ili jsme elipticitu $A(x_0)$, h\"olderovu nerovnost
a p�edpoklad $f\in L^{2,\lambda}$.
\par
Celkem tedy
\[
\normg{u}{\rho}{}
\le
\left[
\left(
\frac{\rho}{R}
\right)^{n/2}
+
\lambda_0^{-1}\omega(R)
\right]
\normg{u}{R}{}
+
c_2\lambda_0^{-1}
R^{\lambda/2}
\,.
\]
D�ky spojitosti lze $\omega(R)$ u�init mal� dodate�n�m
zmen�en�m $R_0$ a to stejnom�rn� v��i $x_0\in\om_0$.
Tud�� \eqref{cl1} plyne z algebraick�ho 
lemmatu pro funkci $\phi(\rho)=\normg{u}{\rho}{}$,
$\pha=n/2$, $\beta=\lambda/2$.
\eduk%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} {\rm(Regularita line�rn� rovnice II.)}
Nech� $u\in W^{1,2}_{loc}$ je slab� �e�en� line�rn� rovnice \eqref{lin},
nech� $A(x)\in C^{0,\mu}_{loc}$
 a spl�uj� podm�nku elipticity nez�visle na $x$. 
Nech� $f\in C^{0,\nu}_{loc}$.
\par
Potom $\nabla u \in C^{0,\kappa}_{loc}$, kde
$\kappa=\min\{ \mu,\nu \}$.
\end{theorem}
\dduk
Nech� $\om_0\subset\subset\om$.
Ozna� $R_0=\dist(\om_0,\partial\om)$ a 
pro $x_0\in\om_0$ definuj
$\norm{u}{r}{}=\norm{u}{L^2(B(x_0,r))}{}$,
$u_r=u_{B(x_0,r)}$.
C�lem je uk�zat
\begin{equation}
\camp{u}{\rho}{}\le C\rho^{\frac{n}{2}+\kappa}
\,,
\qquad
\forall \rho\le R_0
\,,
\label{cl2}
\end{equation}
kde $C=C(R_0,f)$ nez�vis� na $x_0\in\om_0$. Odsud
plyne z�v�r d�ky vno�en� Campanatov�ch prostor�.
\par
Nerovnost \eqref{cl2} se uk�e pomoc� algebraick�ho
lemmatu. Fixujme $\rho$, $R$, kde $\rho\le R \le R_0$.
\par
Lze ps�t $u=v+w$, kde $u$, $v$
�e�� rovnice
\begin{align}
-\diver\big(A(x_0)\nabla v\big)&=0
\qquad\textrm{na}
\ B(x_0,R)
\label{u2-1}
\\
v&=u
\qquad\textrm{na}
\ \partial B(x_0,R)
\nonumber
\\
-\diver\big(A(x_0)\nabla w\big)&=
-\diver F
\qquad\textrm{na}
\ B(x_0,R)
\label{u2-2}
\\
w&=0
\qquad\textrm{na}
\ \partial B(x_0,R)
\nonumber
\end{align}
kde
\[
F=
\Big[
\big(A(x_0)-A(x)\big)
\big( \nabla u -(\nabla u)_R \big)
+
\big(A(x_0)-A(x)\big)(\nabla u)_R
+\big( f-(f)_R \big)
\Big]
\,.
\]
Odhadujeme
\begin{align*}
\camp{u}{\rho}
&\le
\camp{v}{\rho}
+
\camp{w}{\rho}
\\
&\le
c_1\left(\frac{\rho}{R}\right)^{\frac{n}2+1}
\camp{v}{R}
+
2\normg{w}{R}{}
\\
&\le
c_1\left(\frac{\rho}{R}\right)^{\frac{n}{2}+1}
\camp{u}{R}
+
(c_1+2)
\normg{w}{R}{}
\,.
\end{align*}
U�ili jsme postupn� troj�heln�kovou nerovnost pro $u=v+w$,
Campanatovo lemma pro $\nabla v$ na $B(x_0,R)$, snadn� odhad
(H�lderova nerovnost) 
$\norm{(\nabla w)_\rho}{\rho}{}\le\normg{w}{\rho}{}$,
trivi�ln� odhad $\normg{w}{\rho}{}\le\normg{w}{R}{}$
a kone�n� troj�heln�kovou nerovnost pro $v=u-w$.
\par
Testov�n�m \eqref{u2-2} funkc� $w$ dostaneme d�ky elipticit�
\[
\normg{w}{R}{}
\le
c_2
\norm{F}{R}{}
\le
c_2 R^{\mu}\norm{\nabla u - (\nabla u)_R}{R}{}
+
c_3 R^{\mu}\normg{u}{R}{}
+
c_4 R^{\frac{n}{2}+\nu}
\]
- u�ili jsme $A\in C^{0,\mu}$ a $f\in C^{0,\nu}\subset L^{2,n+2\nu}$,
odkud
\[
\norm{f-f_R}{R}{}
\le c_4
R^{\frac{n}2+\nu}
\,.
\]
Celkem tedy (zna��me $\phi(\rho)=\norm{\nabla u - (\nabla u)_\rho}{\rho}{}$)
\begin{equation}
\phi(\rho)
\le
\left[
c_1\left( \frac{\rho}{R} \right)^{\frac{n}{2}+1} + c_1 R^{\mu}
\right]
\phi(R)
+
c_2 
R^{\frac{n}2+\nu}
+
c_2
R^{\mu}\normg{u}{R}{}
\,.
\label{u2-m0}
\end{equation}
Zb�v� o�et�it posledn� �len. Zvol $\eps\in(0,n/2)$.
Proto�e $f\in \el^{2,n-2\eps}\subset L^{2,n-2\eps}$
a $A(x)$ jsou spojit�, je podle p�edchoz� v�ty $\nabla u \in L^{2,n-2\eps}$.
Tud�� $\normg{ u}{R}{}\le c_3 R^{n/2-\eps}$ a tedy
\begin{equation}
\phi(\rho)
\le
\left[
c_1\left( \frac{\rho}{R} \right)^{\frac{n}{2}+1} + c_1 R^{\mu}
\right]
\phi(R)
+
c_2 
R^{\frac{n}2+\nu}
+
c_2
R^{\frac{n}2+\mu-\eps}
\,.
\label{u2-m}
\end{equation}
Nyn� rozli�me dva p��pady. Pokud $\mu>\nu(=\kappa)$, sta�� zvolit $\eps=\mu-\nu$
a dostaneme
\[
\phi(\rho)
\le
\left[
c_1\left( \frac{\rho}{R} \right)^{\frac{n}{2}+1} + c_1 R^{\mu}
\right]
\phi(R)
+
c_2 
R^{\frac{n}2+\kappa}
\,.
\]
Odsud z�v�r d�ky algebraick�mu lemmatu pro $\pha=n/2+1$, $\beta=n/2+\kappa$
po p��padn�m zmen�en� $R_0(\ge R)$.
\par
Pokud $\nu\ge\mu$, polo�me $\eps=\mu/2$.
Z \eqref{u2-m} plyne
\[
\phi(\rho)
\le
\left[
c_1\left( \frac{\rho}{R} \right)^{\frac{n}{2}+1} + c_1 R^{\mu}
\right]
\phi(R)
+
c_2 
R^{\frac{n}2+\frac{\mu}2}
\]
a tedy d�ky algegraick�mu lemmatu $\phi(\rho)\le c \rho^{n/2+\mu/2}$.
Tud�� $\nabla u \in \el^{2,n+\mu}\subset L^{\infty}$, odkud
$\normg{u}{R}{}\le c R^{n/2}$.
\par
Dosazen�m do \eqref{u2-m0} m�me
\[
\phi(\rho)
\le
\left[
c_1\left( \frac{\rho}{R} \right)^{\frac{n}{2}+1} + c_1 R^{\mu}
\right]
\phi(R)
+
c_2 
R^{\frac{n}2+\kappa}
\]
nebo� $\kappa=\mu\le\nu$. Z�v�r op�t pou�it�m algebraick�ho lemmatu.
\eduk

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{��ste�n� regularita.} \label{partial}

%% partial regularity

Obsah kapitoly: pokr�vac� lemma (Vitali); lemma o odhadu Hausdorffovy dimenze;
��ste�n� regularita kvaziline�rn� rovnice; Giustiho lemma; 'nep��m�'
d�kaz ��ste�n� regularity.

\begin{lemma} {\rm (Pokr�vac�.)}
Nech� $A\subset R^n$ je omezen� a $r(x):A\to(0,1)$. Pak
existuje  posloupnost $\{x_j\}\subset A$ tak,
�e  jsou $B(x_j,r(x_j))$ po dvou disjunktn�
a
\[
A \subset \bigcup_{j} B(x_j,3r(x_j))
\,.
\]
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%
\dduk
V prvn�m kroku uva�uji pouze koule $B(x,r(x))$,
spl�uj�c� $r(x)\in[1/2,1)$. D�ky omezenosti $A$ mezi nimi najdu
kone�n� maxim�ln� disjuktn� podsyst�m.
V $n$-t�m kroku p�id�v�m disjunktn� z koul� 
spl�uj�c�ch $r(x)\in[2^{-n-1},2^{-n})$. V�dy jich p�ibude
nejv��e kone�n� mnoho. Vybran� koule zna��m $B(x_j,r(x_j))$.
\par
Nyn� nech� $y\in A$ je libovoln� a $y\neq x_j$ pro $\forall j$.
Existuje $n$, �e $y\in[2^{-n-1},2^{-n})$. P�itom koule $B(y,r(y))$
nebyla vybr�na v $n$-t�m kroku. Tedy existuje $z\in B(y,r(y))\cap B(x_j,r(x_j))$,
kde $x_j$ bylo vybr�no nejpozd�ji v $n$-t�m kroku.
\par
Kl��ov� pozorov�n�: $2r(x_j)\ge 2^{-n} \ge r(y)$. Odsud
\[
|y-x_j|\le |y - z| + |z-x_j| < r(y) + r(x_j) \le 3 r(x_j)
\]
a
tedy
$y\in
B(x_j,3r(x_j))$.
\eduk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} {\rm (Odhad dimenze.)}
\label{lm:dim}
Nech� $f\in L^1_{loc}(\om)$, $\pha\in[0,n)$. Polo�
\[
E=\big\{ x\in \om:\ \lim\sup_{\rho\to0+} \rho^{-\pha}\int_{B(x,\rho)}|f|>0 \big\}
\,.
\]
Potom
$H^{\pha}(E)=0$, a tud�� $d_H(A)\le\pha$.
\end{lemma}
\dduk
Vzhledem k $\sigma$-subaditivit� $H^{\pha}$ je
\begin{align*}
H^{\pha}(E)&=\sup_{K} H^{\pha}(E\cap K),
\qquad
K\subset\subset\om
\ 
\textrm{kompaktn�}
\\
H^{\pha}(E\cap K)&=\sup_{s>0} H^{\pha}(F_s),
\\
F_s&=\big\{x\in E\cap K:\ \lim\sup_{\rho\to0+} \rho^{-\pha}\int_{B(x,\rho)}|f|>s \big\}
\end{align*}
Dok�eme $H^{\pha}(F_s)=0$ pro $F_s$ pevn�. 
Zvolme $\eps\in(0,1)$.
Proto�e
\[
\lim_{\rho\to0+} \rho^{-n} \int_{B(x,\rho)} |f| = \kappa_n |f(x)| < \infty 
\]
pro skoro v�echna $x\in\om$,
je $\lambda_n(E)=0$. 
\par
Tedy existuje $Q\supset F_s$ otev�en�, $\lambda_n(Q)<\eps$.
\par
Nyn� pro ka�d� $x\in F_s$ najdu $r(x)\in(0,\eps)$ tak,
�e $B(x,r(x))\subset Q$ a 
\[
\big(r(x)\big)^{-\pha} \int_{B(x,r(x))} |f| > s 
\,.
\]
Dle pokr�vac�ho lemmatu vyberu disjunktn� $B(x_i,r_i)$
(zna��m $r_i=r(x_i)$)
tak, �e
$F_s \subset \cup_{i} B(x_i,3r_i)$. Tedy
\[
H^{\pha,\eps}(F_s)
\le
\sum_i (3r_i)^\pha
=
3^{\pha}
\sum_i r_i^{\pha}
=
\frac{3^{\pha}}{s}
\int_{\cup_i B(x_i,r_i)} |f|
\le
\frac{3^{\pha}}{s} \int_{Q} |f|
\,.
\]
Pro $\eps\to0$ jde lev� strana k $H^{\pha}(F_s)$, prav� k $0$
(nebo� $\lambda_n(Q)<\eps$)
d�ky absolutn� spojitosti integr�lu.
\eduk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \label{th:pr1}
{\rm (��ste�n� regularita kvaziline�rn� rovnice.)}
Nech� $u\in W^{1,2}_{loc}(\om)$ je �e�en� rovnice
\begin{equation} \label{eq3}
-\diver \Big\{ A(u)\nabla u \Big\} = f
\,.
\end{equation}
Nech� $A$ je stejnom�rn� spojit� a eliptick�,
nech� $f\in L^{p}_{loc}$, kde $p>n$. 
Potom existuje $\Sigma\subset\om$ takov�, �e
\begin{itemize}
\item[(1)]
$u\in C^{0,\alpha}_{loc}(\om\setminus\Sigma)$, kde $\alpha=1-n/p$;
\item[(2)]
$d_H(\Sigma) < n-2$.
\end{itemize}
\end{theorem}
\dduk
Zvolme $x_0\in\om$.
Pro $R\le R_0=\dist(x_0,\om)$ ozna�me $\tilde{A}=A(u_R)$.
Na $B(x_0,R)$ lze ps�t $u=v+w$, kde $u$, $v$ �e��
\begin{align}
-\diver\big(\tilde{A}\nabla v\big)&=0
\qquad\textrm{na}
\ B(x_0,R)
\label{cr1}
\\
v&=u
\qquad\textrm{na}
\ \partial B(x_0,R)
\nonumber
\\
-\diver\big(\tilde{A}\nabla w\big)&=
-\diver \Big( \big( \tilde{A}-A(u) \big) \Big)\nabla u + f \Big)
\qquad\textrm{na}
\ B(x_0,R)
\label{cr2}
\\
w&=0
\qquad\textrm{na}
\ \partial B(x_0,R)
\nonumber
\end{align}
Ozna�me $\norm{u}{\rho}{}=\norm{u}{L^2(B(x_0,\rho))}{}$.
Podobn� jako ve v�t� o regularit� line�rn� rovnice odvod�me
pro $\rho\le R$
\begin{align*}
\norm{\nabla u}{\rho}{}
&\le
c_1 \left( \frac{\rho}{R} \right)^{\frac{n}2} 
\norm{\nabla u}{R}{}
+
c_2 \norm{\nabla w}{R}{}
\\
\norm{\nabla w}{R}{}
&\le
\norm{\big(\tilde{A}-A(u)\big)\nabla u}{R}{}
+
\norm{f}{R}{}
\end{align*}
Odhadujeme
\[
\norm{f}{R}{2}
=
\int_{B_R} |f|^2 
\le
|B_R|^{1-\frac2p}\left( \int_{B_R} |f|^p \right)^{2/p}
\le
c_3 R^{n-\frac{2n}p}
\,.
\]
V d�kazu budeme d�le pot�ebovat n�sleduj�c� hlubok� v�sledek: z toho, �e
$u\in W^{1,2}_{loc}$ �e�� rovnici \eqref{eq3}, lze odvodit, 
pro vhodn� $q>2$ lok�ln� plat� odhad
\begin{equation}\label{rh}
  \left( \aint_{B_R} |\nabla u|^q \right)^{1/q}
     \le
  \left( \aint_{B_{2R}} |\nabla u|^2 \right)^{1/2}
\end{equation}
- tzv. reverzn� H�lderova nerovnost.
\par
Ozna�me $\omega$ modul spojitosti $A$, tj.
\[
|A(u)-A(v)| \le \omega(|u-v|^2)
\,.
\]
B�NO $\omega$ je spojit�, neklesaj�c�, konk�vn� a 
$\omega(0+)=0$, $\omega(+\infty-)=\omega_{\infty}<\infty$.
\par
Odhadujeme
\begin{align*}
\norm{\big(\tilde{A}-A(u)\big)\nabla u}{R}{2}
&=
\int_{B_R}  |\nabla u|^2 |A(u)-A(u_R)|^2
\\
&\le
\left( \intbr |\nabla u|^q \right)^{2/q}
\left( \intbr \omega^{\frac{2q}{q-2}}(|u-u_R|^2) \right)^{1-2/q}
\\
&\le
c\,
R^{2n(\frac1q-\frac12)}
\int_{B_{2R}} |\nabla u|^2
\ 
\omega_{\infty}^{\frac{2q}{q-2}-1}
\left( \intbr \omega(|u-u_R|^2) \right)^{1-2/q}
\\
&=
c\, 
\int_{B_{2R}} |\nabla u|^2
\left( \aint_{B_R} \omega(|u-u_R|^2) \right)^{1-2/q}
\\
&\le
c\, 
\int_{B_{2R}} |\nabla u|^2
\left\{
\omega\left(
\aint_{B_R} |u-u_R|^2
\right)
\right\}^{1-2/q}
\end{align*}
Pou�ili jsme oby�ejnou a reverzn� \eqref{rh} H�lderovu nerovnost,
odhad $\omega\le \omega_{\infty}$, $|B_R|=cR^n$ a 
nakonec konk�vnost $\omega$.
\par
P�i zna�en� $\phi(\rho)=\norm{\nabla u}{\rho}{}$, $\beta=1/2-1/q$
celkem dostaneme
\[
\phi(\rho)
\le
c_1
\left[
\left( \frac{\rho}{R} \right)^{n/2} 
+
\omega^{\beta}\left(
\aint_{B_R} |u-u_R|^2
\right)
\right]
\phi(2R)
+
c_2 R^{\frac{n}2(1-\frac2p)}
\,.
\]
Chceme pou��t algebraick� lemma. To ale p�jde pouze tam, kde je 
$u-u_R$ mal�. Za t�mto ��elem definujeme
\[
\Sigma :=
\Big\{
x_0 \in \om:\ 
\liminf_{R\to0+} 
\aint_{B(x_0,R)} |u-u_R|^2 > 0
\Big\}
\]
Pokud $x_0 \notin \Sigma$, je s  mal�m $R$ mal� t�
$\aint_{B_R} |u-u_R|^2$ - d�ky spojitosti dokonce na 
okol� $x_0$. Tedy dle Lemmatu~\ref{lm:alg} je
\[
\phi(\rho) = \normg{u}{\rho}{} \le c \rho^{\frac{n}2(1-\frac2p)}
\,.
\]
Tj. na okol� $x_0$ je $\nabla u\in L^{n-2n/p}$ a podle
V�ty~\ref{th:mc}, bodu 3 je $u\in \el^{n+2(1-n/p)}$; podle
bodu 4 je tedy $u\in C^{0,\pha}$, kde $\pha=1-n/p$, c.b.d.
\par
Zb�v� odhadnout velikost singul�rn� mno�iny $\Sigma$. 
Z Poincar�ho  (V�ta~\ref{th:poin}) a H\"olderovy nerovnosti
\begin{align*}
\aint_{B_R} |u-u_R|^2
&\le
c\, R^{2-n}
\intbr |\nabla u|^2 
\\
&\le
c\, R^{2-n}
\left( 
\intbr |\nabla u|^q
\right)^{2/q}
|B_R|^{1-2/q}
\\
&=
c\,
R^{2-2n/q}
\left( 
\intbr |\nabla u|^q
\right)^{2/q}
\,.
\end{align*}
Odtud
\[
\Sigma \subset 
\Big\{
x_0\in \om:\ \liminf_{R\to 0+} R^{q-n} 
\int_{B(x_0,R)} |\nabla u|^q > 0
\Big\}
\,.
\]
Proto�e dle \eqref{rh} je $\nabla u\in L^q_{loc}$ s n�jak�m $q>2$, 
je podle Lemmatu~\ref{lm:dim} $d_H(\Sigma)\le n-q$.
\eduk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} {\rm (Giusti.)} \label{lm:giusti}
Nech� $u\in W^{1,2}_{loc}(\om)$ je slab� �e�en� \eqref{eq3},
nech� $\tilde{\om}\subset\subset \om$.
Potom pro ka�d� $\tau\in(0,1)$ a $M\ge0$ existuj� $\eps_0>0$ a
$R_0>0$ s n�sleduj�c� vlastnost�: pokud pro $x\in\tilde{\om}$ 
existuje $R<\min\{R_0,\dist(x,\partial\om)\}$ tak, �e
\par
(a)
\[
\uu(x,R):=\aint_{B(x,R)} |u-u_{B(x,R)}|^2 \le \eps_0^2
\,,
\]
\par
(b)
\[
u_{B(x,R)} \le M
\,,
\]
je $\uu(x,\tau R) \le 2 c_* \tau^2  \uu(x,R) $, kde
konstanta $c_*$ z�vis� pouze na elipticit� $A$.
\end{lemma}
\dduk
Sporem: existuj� $\tau\in(0,1)$, $M\ge0$ tak, �e lze nal�zt
posloupnosti $\eps_n\to0$, $R_n\to0$ a body $x_n\in\tilde{\om}$
tak, �e (a) $\uu(x_n,R_n)=\eps_0^2$ a (b) $u_{x_n,R_n}\le M$,
av�ak $\uu(x_n,\tau R_n) > 2 c_* \tau^2 \uu(x_n,R_n)$.
\par
Jak zvolit $c_*$, aby nastal spor, uvid�me za chv�li.
\par
Zavedeme pomocn� funkce
\[
v_n(y) = \frac1{\eps_n} \big[ u(x_n+R_n y) - u_{B(x,R)} ]
\,,
\qquad
y\in B(0,1)
\]
Pozorujeme, �e
\begin{align}
  (v_n)_{B(0,1)} &= 0 \nonumber
\\
  \vv_n(0,1) &= \int_{B(0,1)} |v_n|^2 = 1 \label{p1}
\\
  \vv_n(0,\tau) &> 2 c_* \tau^2 \label{p2}
\end{align}
kde $\vv_n(0,\rho)=\aint_{B(0,\rho)} |v_n - (v_n)_{B(0,\rho)} |^2$.
Funkce $v_n$ �e��
\[
-\diver\big\{ A_n(y) \nabla v_n \big\} = 0
\,,
\qquad
y\in B(0,1)
\]
kde $A_n(y)=A(R_n y + x_n, \eps_n v_n(y) + u_{B(x_n,R_n)} )$. Z
Caccioppoliho lemmatu \ref{lm:cac} plyne pro $\sigma \in(0,1)$
\[
\int_{B(0,\sigma)} |\nabla v_n |^2 
\le 
\frac{c_1}{(1-\sigma)^2}
\int_{B(0,1)} |v_n|^2
\]
kde $c_1$ z�vis� jen na elipticit� $A_n$. Tedy z omezenosti $v_n$
v $L^2(B(0,1))$, viz \eqref{p1}, plyne i omezenost 
$\nabla v_n$ v $L^2_{loc}(B(0,1))$.
\par
P�echodem k podposloupnosti m�me $x_n\to x_0 \in
\textrm{cl}\,(\tilde{\om})$, $u_{B(x_n,R_n)}\to u_0$, $\nabla v_n
\rightharpoonup \nabla v_0$ v $L^2_{loc}$ a $v_n\to v_n$ v $L^2_{loc}$
a tak� s.v. v $B(0,1)$.
\par
Odsud $A_n(y) \to A_0=A(x_0,u_n)$ s.v. a $v_0(y)$ �e�� v $B(0,1)$
\[
-\diver\big\{ A_0(y) \nabla v_0 \big\} = 0
\]
Tedy podle Campanatova lemmatu \ref{lm:cam} 
\[
\vv_0(0,\tau) \le c_* \tau^2 \vv_0(0,1)
\]
kde $c_*$ je hledan� konstanta, z�visej�c� jen na elipticit� 
$A_0=A(x_0,u_0)$. Av�ak limitn�m p�echodem
plyne z \eqref{p1}, \eqref{p2}
\begin{align*}
  \vv_0(0,1)&=\int_{B(0,1)} |v_0|^2 \le 1
\\
  \vv_0(0,\tau)& \ge 2 c_* \tau^2
\end{align*}
co� je spor.
\eduk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} {\rm (\uv{Nep��m�} d�kaz ��ste�n� regularity.)}
\label{th:pr2}
Nech� $u\in W^{1,2}_{loc}(\om)$ je �e�en� rovnice
\[
-\diver \Big\{ A(x,u)\nabla u \Big\} = 0
\,.
\]
Nech� $A$ je spojit� a eliptick�.
Pak existuje $\Sigma\subset \om$ takov�, �e
\begin{itemize} \itemsep 0pt
\item[(1)] $u\in C^{0,\pha}_{loc}(\om\setminus\Sigma)$ pro libovoln�
$\pha\in(0,1)$;
\item[(2)] $d_H(\Sigma)<n-2$.
\end{itemize}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\dduk
Polo�me
\begin{align*}
  \Sigma &= \Sigma_1 \cup \Sigma_2
\\
  \Sigma_1 &:= 
\Big\{ x\in \om: \sup_{R<\delta} u_{B(x,R)} = +\infty \Big\}
\\
  \Sigma_2 &:= 
\Big\{ x \in \om:\ 
\liminf_{R\to0+} 
\uu(x,R) > 0
\Big\}
\end{align*}
kde $\delta>0$ je pevn�, libovoln� a 
$\uu(x,\rho)=\aint_{B(x,\rho)}|u-u_{B(x,\rho)}|^2$.
Stejn� jako v d�kaze V�ty~\ref{th:pr1} pomoc� reverzn�ho H�ldera
odvod�me odhad $d_H(\Sigma_2)<n-2$. 
Pro $\Sigma_1$ to v se�it� nem�m !!!???
\par
Zb�v� regularita $u$ mimo $\Sigma$.
Dok�eme, �e pro ka�d� $x\notin\Sigma$ existuje okol� $U(x)$ tak, �e
$u\in C^{0,\pha}(U(x))$ pro libovoln� $\pha\in(0,1)$. Hlavn�m
n�strojem je p�itom Giustiho lemma~\ref{lm:giusti}.
\par
Nech� $x$, $\pha$ jsou d�na. Proto�e $x\notin\Sigma_1$, existuje $M>0$
tak, �e $\sup_{R<\delta} u_R < M$. Zvolme $\tau\in(0,1)$ tak, aby
$2c_{*}\tau^2 < \tau^{2\pha}$, kde $c_*$ poch�z� z Giustiho
lemmatu~\ref{lm:giusti} (a jak v�me, z�vis� jen na elipticit� $A$.)
\par
Giustiho lemma n�m zaru�uje existenci jist�ch $\eps_0$ a $R_0>0$. 
Proto�e $x\notin\Sigma_2$, lze naj�t $R<R_0$ tak, �e
$\uu(x,R)<\eps_0^2$. B�NO $R<\delta$, a tedy z�v�r Giustiho lemmatu
d�v�
\[
\uu(x,\tau R) \le 2c_* \tau^2 \uu(x,R) < \tau^{2\pha} \uu(x,R)
\]
Speci�ln� $\uu(x,\tau R)\le \eps_0^2$ a tedy opakov�m u�it�m z�v�ru
Giustiho lemmatu
$\uu(x,\tau^k R) < \tau^{2k\pha} \uu(x,R)$.
Kone�n� ze spojitosti existuje okol� $U(x)$ tak, �e
\begin{equation} \label{dis1}
\uu(y,\tau^k R) < \tau^{2k\pha} \uu(y,R) 
\qquad
\forall y\in U(x)
\end{equation}
Nech� nyn� $\rho\in(0,R)$. C�lem je uk�zat
\[
\uu(y,\rho) \le c\, \rho^{2\pha}
\qquad
\forall y\in U(x)
\]
kde $c$ nez�vis� na $y$, $\rho$. Odtud hledan� z�v�r $u\in
\el^{2,2\pha+n}=C^{0,\pha}$ na $U(x)$.
\par
Zvolme $k\ge0$ cel� tak, �e $\tau^{k+1}R < \rho \le \tau^k R$.
Po��tejme
\begin{align*}
  \uu(y,\rho) &= \frac{1}{|B(y,\rho)|} \int_{B(y,\rho)} |u-u_{B(y,\rho)}|^2
\\
  &\le \frac{1}{|B(y,\rho)|} \int_{B(y,\rho)} |u-u_{B(y,\tau^k R)}|^2
\\
  &\le \frac{1}{|B(y,\tau^{k+1}R)|} \int_{B(y,\tau^k R)}
|u-u_{B(y,\tau^k R)}|^2
\\
  &= \tau^{-n} \uu(y,\tau^k R)
 \le \underbrace{\tau^{-n-2\pha}R^{-2\pha}\uu(y,R)}_c \rho^{2\pha}
\end{align*}
U�ili jsme toho, �e $c=u_B$ je minimum funkce $c\mapsto
\int_B|u-c|^2$ a nerovnost� \eqref{dis1} a $\tau^k \le \rho/(\tau R)$.
D�kaz je hotov. 
\eduk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Syst�my bl�zk� $-\Delta$.} \label{blizke}

%%% systemy 'blizke' laplacianu

Obsah kapitoly: dv� v�ty o H\"olderovskosti gradientu pro line�rn� rovnici
'bl�zkou' laplaci�nu.

\begin{theorem}
  Nech�  $u\in W^{1,2}_{loc}$ �e�� rovnici
\[
-\diver \big( A(x)\nabla u)=0
\]
kde $A(x)$ je symetrick�. Jestli�e
\begin{equation}\label{p:51}
\frac{\lambda_0}{\lambda_1} > \frac{n-2}{n-1}
\,,
\end{equation}
kde $\lambda_i$ jsou konstanty elipticity a $n$ je dimenze,
je $u$ lok�ln� H\"olderovsk�.
\end{theorem}
\dduk
Uk�eme, �e funkce
\[
\phi(R)= R^{-\lam} \int_{B(x_0,R)} |\nabla u|^2
\]
je s vhodn�m $\lam$ omezen� (nez�visle na $x_0\in
\om_0\subset\subset\om$ a 
$R<\dist(\overline{\om_0},\partial \om)$.)
\par
Proto�e $\phi(R)$ je nez�porn� a (lok�ln�) absolutn� spojit�, 
sta�� zaru�it, �e%
\footnote{N�sleduj�c� �vahy jsou v po��dku pro skoro v�echna $R$.}
\[
\phi'(R) = 
-\lam R^{-\lam-1} 
\int_{B_R}|\nabla u|^2 
+
R^{-\lam} \int_{S_R} |\nabla u|^2 
\ge0
\,.
\]
Hlavn� bod d�kazu spo��v� v tom, �e $\int_{B_R}|\nabla u|^2$
odhadneme pomoc� $\int_{S_R}|\nabla u|^2$. Za t�mto ��elem definujeme
pomocnou funkci $v$, kde
\begin{equation}
\begin{aligned}
  -\Delta v &=0 \qquad x\in B_R
\\
  v&=u \qquad x\in S_R
\end{aligned}
\label{rv}
\end{equation}
Plat�
\begin{equation}
\lam_0 \int_{B_R} |\nabla u|^2
\le
\intbr \big( A \nabla u,\nabla u\big)
\le
\intbr \big( A \nabla v, \nabla v)
\le
\lam_1 \intbr |\nabla v|^2
\label{o1}
\end{equation}
Prvn� a t�et� nerovnost jsou podm�nky elipticity. Druh� nerovnost
plyne z toho, �e $u$ �e�� Euler-Lagrangeovu rovnici 
funkcion�lu $\psi\mapsto \int_{B_R}(A\nabla\psi,\nabla\psi)$,
$\psi|_{S_R}=u$ a je zde tedy (jedin�m d�ky elipticit�) minimiz�rem.
\par
D�le m�me
\[
\intbr |\nabla v|^2 
\le 
\frac{R}{n-1} \intsr |\nabla_T v|^2
=
\frac{R}{n-1} \intsr |\nabla_T u|^2
\le
\frac{R}{n-1} \intsr |\nabla u|^2
\,.
\]
Zde $\nabla_T$ zna�� derivace pouze ve sm�ru te�n�m k $S_R$.
Prvn� nerovnost plyne z teorie potenci�lu ($v$ je harmonick�),
n�sleduj�c� rovnost z toho, �e $u=v$ na $S_R$.
\par
Celkem m�me slibovan� odhad
\[
\intbr |\nabla u|^2 
\le
 \frac{\lam_1}{\lam_0} \cdot  \frac{R}{n-1}
\intsr |\nabla u|^2
\]
a tedy
\[
\phi'(R) \ge R^{-\lam} \intsr |\nabla u|^2 
\left(
1 -   \frac{\lam_1}{\lam_0} \cdot  \frac{\lam}{n-1}
\right)
\,.
\]
D�ky \eqref{p:51} lze zvolit $\lam$ tak, �e
\[
 \frac{\lam_0}{\lam_1}
\ge
\frac{\lam}{n-1}
>
\frac{n-2}{n-1}
\]
Z prvn� nerovnosti plyne, �e $\phi'(R)\ge0$, tedy $\phi$ je
(lok�ln�) omezen�, neboli $\nabla u \in L^{2,\lam}_{loc}$.
\par
Druh� nerovnost zaru�uje $\lam>n-2$, tj. $\lam=n-2+\eps$, $\eps>0$.
Podle V�ty~\ref{th:mc} bodu 3 je $u\in\el^{n+\eps}$; podle bodu
5 je $u\in C^{0,\eps/2}$.
\eduk

\begin{theorem}
  Nech�  $u\in W^{1,2}_{loc}$ �e�� rovnici
\[
-\diver \big( A(x)\nabla u)=0
\]
kde $A(x)$ je symetrick�. Jestli�e
\begin{equation}\label{p:52}
\frac{\lambda_0}{\lambda_1} > \frac{\sqrt{n-1}-1}{\sqrt{n-1}+1}
\,,
\end{equation}
kde $\lambda_i$ jsou konstanty elipticity a $n$ je dimenze,
je $u$ lok�ln� H\"olderovsk�.
\end{theorem}
\dduk
D�kaz je p��buzn� d�kazu p�edchoz� v�ty. Rozd�l spo��v� v tom,
�e zlep��me odhad \eqref{o1}.
\par
Rovnici pro $u$ p�ep��eme jako
\begin{equation}
-\Delta u = \diver \Big\{ \big(I-\Lambda A\big)\nabla u \Big\}
\,.
\label{ru}
\end{equation}
Konstantu $\Lambda$ zvol�me tak, aby norma $B=I-\Lambda A$ byla
minim�ln�. Matice $B$ je symetrick�, tedy 
$\norm{B}{}{}=\max\{|\lambda|;\ \lambda\in\sigma(B)\}$. Proto�e
$1-\Lambda \lam_1$ a $1-\Lambda \lam_0$ jsou zjevn� nejmen��
a nejv�t�� prvek $\sigma(B)$, je optim�ln� volbou
$\Lambda=2/(\lam_1+\lam_0)$ d�vaj�c�
$\norm{B}{}{}=(\lam_1-\lam_0)/(\lam_1+\lam_0)$.
\par
Funkce $v$ je op�t ur�ena rovnic� \eqref{rv}. 
Rovnici \eqref{ru} testujeme $w=u-v\in W^{1,2}_0(B_R)$:
\begin{align*}
\intbr \nabla u \cdot \nabla w
&\le
\intbr
\left|
\big(I-\Lambda A)
\nabla u \cdot \nabla w
\right|
\\
&\le
\frac{\lam_1-\lam_0}{\lam_1+\lam_0}
\intbr |\nabla u| |\nabla w|
\\
&\le
\frac12 \left( \frac{\lam_1-\lam_0}{\lam_1+\lam_0} \right)^2
\intbr |\nabla u|^2 
+ 
\frac12 \intbr |\nabla w|^2
\end{align*}
D�le testujeme funkc� $w$ rovnici \eqref{rv} a dostaneme $\intbr
\nabla v \cdot \nabla w=0$. Odtud
\begin{align*}
  \intbr \nabla u \cdot \nabla w &= \intbr |\nabla w|^2
\\
  \intbr |\nabla u|^2 &= \intbr |\nabla v|^2 + |\nabla w|^2
\end{align*}
Dohromady m�me
\[
\left\{
1- 
\left( \frac{\lam_1-\lam_0}{\lam_1+\lam_0} \right)^2
\right\}
\intbr |\nabla u|^2 \le \intbr |\nabla v|^2
\,.
\]
To je hledan� zlep�en� odhadu \eqref{o1}, nebo� $\lam_0/\lam_1$
je obecn� men�� ne� v�raz ve slo�en� z�vorce vlevo.
\par
Nyn� podobn� jako v d�kazu p�edchoz� v�ty dostaneme
\[
\phi'(R) 
\ge 
R^{-\lam} \intsr |\nabla u|^2 
\left(
1 -  \left\{
1- 
\left( \frac{\lam_1-\lam_0}{\lam_1+\lam_0} \right)^2
\right\}^{-1}
 \frac{\lam}{n-1}
\right)
\,.
\]
K zavr�en� d�kazu je tedy t�eba naj�t $\lam$ tak, �e
\[
1- 
\left( \frac{\lam_1-\lam_0}{\lam_1+\lam_0} \right)^2
\ge
\frac{\lam}{n-1}
>
\frac{n-2}{n-1}
\,,
\]
co� lze pr�v� kdy� plat� \eqref{p:52}.
\eduk%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par\medskip
\noindent
{\bf Pozn�mky.}
\bod Podm�nky \eqref{p:51}, \eqref{p:52} 
vyjad�uj�, �e $\lam_0/\lam_1$ je bl�zko $1$,
tj. matice $A$ je t�m�� diagon�ln� (laplaci�n).
\bod Podm�nka \eqref{p:52} je slab�� ne� \eqref{p:51}
\bod Ko�elev 78' dok�zal analogickou v�tu s je�t� slab�� podm�nkou
\[
\frac{\lambda_0}{\lambda_1} 
>
\frac{ \sqrt{ 1 + \frac{(n-2)^2}{n-1} } - 1}%
{ \sqrt{ 1 + \frac{(n-2)^2}{n-1} } + 1}
\,,
\]
kter� je z�rove� optim�ln� - jej� libovoln� poru�en� umo��uje 
existenci nespojit�ho �e�en�.


\end{document}
