\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 16, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/16\hfil On the behaviour of solutions]
{Estimates for solutions to nonlinear boundary-value problems
in conic domains}

\author[T. S. Gadjiev, S. Y. Aliev \hfil EJDE-2005/16\hfilneg]
{Tahir S. Gadjiev, Sardar Y. Aliev} 

\address{Tahir S. Gadjiev \hfill\break
Institute of Mathematics \& Mechanics of NAS Azerbaijan,
Department of Nonlinear Analysis, 9, F. Agayev str., AZ1141, Baku,
Azerbaijan} \email{tgadjiev@mail.az}

\address{Sardar Y. Aliev \hfill\break
Baku State University, Department of Mathematics, 23 Z. Khalilov
str., AZ1148, Baku, Azerbaijan} \email{ibvag@yahoo.com}

\date{}
\thanks{Submitted January 27, 2004. Published February 1, 2005.}
\subjclass[2000]{35J20, 35D10} 
\keywords{Nonlinear equation; behavior of solutions; nonsmooth domain}

\begin{abstract}
 We obtain sharp estimates on the solution and its
 derivative near the conic points. In particular,
 we show that the solution satisfies
 $|u(x)|\leq C|x|^\lambda$
 where lambda is an eigenvalue of the Sturm-Liouville problem.
 Also we prove that the solution  has square summable
 weighted second generalized derivatives.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction and preliminaries}

We consider mixed boundary-value problems in a bounded domain
$\Omega \subset \mathbb{R}^n$, $n\geq 2$  for the equation
\begin{equation} \label{e1}
\sum_{i=1}^n \frac{d}{dx_i}a_i( x,u,u_{x}) +a( x,u,u_{x})
=0,\quad x\in \Omega
\end{equation}
This study includes equations such as
$-\mathop{\rm div}(k+|\nabla u|^{p-2})+\mu_1|u|^\beta+u^2\phi(x)$, where
$p>1$ and $k\geq 0$.

The domain $\Omega $ is assumed to satisfy  the isoperimetric inequalities
defined in \cite{m1}. The boundary of the domain is decomposed as  $\partial
\Omega =\Gamma_{1}\cup \Gamma_{2}$. Then Dirichlet conditions are given on
$\Gamma_{1}$, and  Neumann conditions on $\Gamma_{2}$.

Our aim is to obtain sharp estimates on the solution and its derivative
near the conic points. Also to obtain estimates for $|u|$ and $|\nabla u( x)|$
which correspond to $\varepsilon =0$ in \cite{g1}, but not obtained there.
For the Dirichlet problem, these equations were
considered in \cite{k1}. For the Dirichlet problem with linear equations,
estimates on conical domains were considered in \cite{k2}.
The mixed boundary-value problem for linear equations on
conical domains was considered in \cite{w1}. Here we study a non-linear case.

Let us set some notation. $B_{d}( 0) $ is ball of
radius $d$ with the center at the point $0$.
$\Omega_0 ^d =\Omega \cap B_{d}( 0)$ is cone in
$\mathbb{R}^n$; i.e., for sufficiently small $d$
\[
\Omega_0 ^d =\{ ( r,\omega ) : 0<r<d,\;\omega =(
\omega_{1},\omega_{2},\dots ,\omega_{n-1}) \in G\},
\]
where $( r,\omega ) $ are spherical coordinates. $G$ is a
domain on a unit sphere $S^{n-1}$ with infinitely differentiable boundary
$\partial G$,
\[
\Gamma_0 ^d =\{ ( r,\omega ) : 0<r<d;,\,\omega \in \partial G\}
=\Gamma_{0,1}^d \cup \Gamma_{0,2}^d \subset
\partial \Omega
\]
is the lateral surface of the cone $\Omega_0 ^d $,
$G_{\rho }=\Omega_0 ^d \cap \{ |x| =\rho \}
$, ${0<\rho<d}$. $dx=r^{n-1}drd\omega $,
$d\Omega_{\rho }=\rho^{n-1}d\omega ,\,d\omega $ is an element of area
of the unit sphere, $|\nabla u| ^{2}=( \frac{\partial u}{\partial r}
) ^{2}+\frac{1}{r^{2}}| \nabla_{\omega }u| ^{2}$,
where $|\nabla_{\omega }u|$ is projection of vector $\nabla u$ on
tangent plane to the sphere $S^{n-1}$ at the point $\omega $,
\[
\Delta u=\frac{\partial ^{2}u}{\partial
r^{2}}+\frac{n-1}{n}\,\frac{\partial u}{\partial
r}+\frac{1}{r^{2}}\Delta_{\omega }u.
\]
Here $\Delta_{\omega }u$ is the Laplace-Beltrami operator on a unit sphere.

Denote by $W_{\alpha ,0}^{m}( \Omega ) $ the space of functions
having generalized derivatives up to order $m$ in $\Omega $ with norm
\[
\| u\|_{W_{\alpha ,0}^{m}( \Omega ) }^{2}
=\sum_{| k|^m =0}\int_{\Omega } r^{\alpha
-2( m-k) }\Big| \frac{\partial ^{| k| }u}{\partial
x_{1}^{k_{1}}\dots \partial x_{n}^{k_{n}}}\Big| ^{2}dx\,.
\]
The function that are continuously differentiable in
$\overline{\Omega }$ and vanishing on $\Gamma_{1}$ form a dense subset.
In particular
\[
\| u\|_{W_{\alpha ,0}^{2}( \Omega ) }^{2}
=\in_{\Omega }\big( r^{\alpha }u_{xx}^{2}+r^{\alpha -2}| \nabla u| ^{2}
+r^{\alpha -4}u^{2}\big) dx\,.
\]
By $W_{2,0}^{1}( \Omega ) $ we denote  the subset of the Sobolev space
$W_{2}^{1}( \Omega ) $ consisting of  continuously differentiable
functions in $\overline{\Omega }$ vanishing on $\Gamma_{1}$. (This is
as dense subset of functions).

We shall use Hardy inequalities and some of its implications.
For any function $u\in W_{2,0}^{1}( \Omega_0 ^d ) $, we have
\begin{equation} \label{e2}
\int_{\Omega_0 ^d } r^{\alpha -4}u^{2}dx\leq
\frac{4}{(4-n-\alpha ) ^{2}}\int_{\Omega_0 ^d }
r^{\alpha -2}u_{r}^{2}dx\,,\quad \alpha <4-n,
\end{equation}
which follows by integration with respect to $\omega \in G$
the correspondent Hardy inequality \cite{h1}.

Allowing  isoperimetricity for the domain $\Omega $, we consider the
eigenvalue problem
\begin{equation} \label{e3}
\begin{gathered}
\Delta_{\omega }u+\lambda ( \lambda +n-2) u=0\,,\quad \omega \in G,\\
u\big|_{\gamma_0 }=0,\quad \frac{\partial u}{\partial u}\big|_{\gamma_{1}}=0,
\end{gathered}
\end{equation}
where $\partial G\in \gamma_0 \cup \gamma_{1}$. In \cite{f1}, it was shown
that this problem has at least one positive eigenvalue
$\lambda=\lambda (G)$. Then by the variational principle
for all $u\in W_{2,0}^{1}( G) $,
\begin{equation} \label{e4}
\underset{G}{\int }u^{2}d\omega \leq \frac{1}{\lambda ^{2}+\lambda (
n-2) }\underset{G}{\int }|\nabla_{\omega }u|
^{2}d\omega.
\end{equation}
Note that constants in inequalities \eqref{e2} and \eqref{e4} are the best
possible.

When we multiply inequality \eqref{e4} by $1/r$ and integrate with respect to
$r\in ( 0,d) $, we have that  for any function
\[
u\in V=\big\{ v\in W_{2}^{1}( \Omega ) :
v( x)=0,\; x\in \Gamma_{0,1}^d ,\; \frac{\partial v}{\partial n}
=0,\; x\in \Gamma_{0,2}^d \big\},
\]
\begin{equation} \label{e5}
\int_{\Omega_0 ^d } r^{-n}u^{2}dx\leq \frac{1}{\lambda
^{2}+\lambda ( n-2) }\int_{\Omega_0 ^d }
r^{2-n}|\nabla u|^{2}dx\,.
\end{equation}
For any function $u\in V$,
\begin{equation} \label{e6}
\int_{\Omega_0 ^d } r^{\alpha -4}u^{2}dx\leq
\big[\big( 2-\frac{n+\alpha }{2}\big) ^{2}+\lambda ( \lambda +n-2) \big]
^{-1}\int_{\Omega_0 ^d } r^{\alpha -2}|\nabla u|^{2}dx,
\end{equation}
whenever the integral in the right-hand side is finite.
Here $\alpha \leq 4-n$.
To obtain this inequality we  multiply inequality \eqref{e4} by
$1/r$ and integrate with respect to $r\in ( 0,d) $. Then
\begin{equation} \label{e7}
\int_{\Omega_0 ^d } r^{\alpha -4}u^{2}dx\leq \frac{1}{\lambda
^{2}+\lambda ( n-2) }\int_{\Omega_0 ^d } r^{\alpha-4}|\nabla_{\omega }u|^{2}dx.
\end{equation}
If $\alpha <4-n$  inequality \eqref{e6} is obtained by adding
\eqref{e2} and \eqref{e7}. If $\alpha =4-n$   inequality
\eqref{e6} coincides with \eqref{e5}.

By a generalized solution of the mixed boundary-value problem
for equation \eqref{e1}, we mean a function $u( x)$ in $W_{2,0}^{1}( \Omega ) $
such that
\begin{equation} \label{e8}
\int_{\Omega } [ a_i( x,u,u_{x}) \eta
_{x_i}+a( x,u,u_{x}) \eta ( x)] dx=0,\quad
\forall \eta ( x) \in W_{2,0}^{1}( \Omega )\,.
\end{equation}
In this paper, we use the repeated index convention;
this is, the summation of terms with repeated indices.

 On the coefficient  we require the following conditions:
 The functions $a_i( x,u,p) $ are measurable at any $x\in\Omega $,
 $u\in \mathbb{R}$,  $p\in \mathbb{R}^n$; differentiable with respect
 to $p_{j}$ ($j=1,\dots ,n$); and satisfy
\begin{gather}
\upsilon ( |u|) \xi ^{2}\leq \frac{\partial
a_i( x,u,p) }{\partial p_{j}}\xi_i\xi_{j}
\leq \mu (|u|) \xi ^{2},\quad \forall \xi \in \mathbb{R}^{n}, \label{e9}\\
\frac{\partial a_i( 0,0,p) }{\partial p_{j}}=\delta
_i^{j}\,,\quad i,j=\overline{1,n}, \label{e10}\\
\Big[ \sum_{i=1}^n a_i^{2}( x,u,p) \Big]^{1/2}
\leq \mu_{1}(|u|)
( |p| +g( x) ) \,,\quad 0\leq g( x) \in L_{q}( \Omega ), \label{e11}
\end{gather}
where $\delta_i^{j}$ is the Kronecker symbol, $q>n$, $g( 0)<\infty $.

The function $a( x,u,p) $ is measurable at $x\in \Omega$, $u\in \mathbb{R}$,
$p\in \mathbb{R}^n$ satisfies
\begin{equation} \label{e12}
| a( x,u,p) | \leq \mu_{2}(| u| ) ( |p|^{2}+f( x)),
\end{equation}
where $0\leq f( x)$, $f \in L_{q/2}( \Omega )$,
$q>n$,  $v( t)[ \mu( t) ,\mu_{1}( t) ,\mu_{2}( t)]$ is positive
nondecreasing function (positive non-increasing) at $t\geq 0$, $\mu, v>0$,
$\mu_{1},\mu_{2}\geq 0$.

In \cite{g2} the boundedness and H\"older continuity of
generalized solution of \eqref{e8} was proved under the conditions
\eqref{e9}--\eqref{e12}. Assuming that the $\mathop{\rm vrai\,max}$
$M$ of $|u( x)|$ is known,
there exists $\gamma >0$, $C_0 >0$ dependent only
on $M,n,q,\mu ,\mu_{1},\mu_{2},v,\Omega $  such that
\[
| u( x) | =| u( x) -u( 0)| \leq C_0 | x| ^{\gamma },\quad | x| <d.
\]
For continuous functions $\mathop{\rm vrai\,max}$ is the same as the
$\max$ over the domain on which the function is defined. 

\section{Main results}

\begin{theorem} \label{thm1}
Let $u( x) $ be a generalized solution of \eqref{e8}.
Assume \eqref{e9}--\eqref{e12}
and that for any $k>0$ there exists $d_0 >0$ such that for
$p\in \mathbb{R}^n$, $|x|+|u|<d_0 $, $0\leq h( x) \in L_{q}$, and $q>n$
we have
\begin{equation} \label{e13}
\Big( \sum_{i=1}^n [ a_i( x,u,p)-a_i( 0,0,p)] ^{2}\Big) ^{1/2}
\leq K|p| +h( x)\,.
\end{equation}
Also assume that $g( x) \in {W}_{\alpha-2}^{0}( \Omega )$,
$h( x) \in W_{\alpha-2,0}^{0}( \Omega )$,
$f( x) \in W_{\alpha,0}^{0}( \Omega )$, $\alpha \leq 4-n$, and
\begin{equation} \label{e14}
\lambda >2-( n+\alpha )/2\,.
\end{equation}
Then
\begin{equation} \label{e15}
\begin{aligned}
&\int_{\Omega } r^{\alpha -2}| \nabla u|^{2}dx
&\leq C(1+\| g\|_{W_{\alpha -2}^{0}( \Omega) }
+\| f\|_{q/2,\Omega }+\| h\|_{W_{\alpha
-2,0}^{0}( \Omega ) }+\| f\|_{W_{\alpha,0}^{0}( \Omega ) }^{2}),
\end{aligned}
\end{equation}
where $C$ is constant depending on
$M,v,\mu_{1},\mu_{2},\mu ,\alpha ,n,\lambda ,q,\mathop{\rm meas}\Omega ,
\mathop{\rm meas}G$.
\end{theorem}

\begin{proof}
For any $\delta \in ( 0,d) $ if $r$ is the radius vector of the point
$x\in \overline{\Omega }$ then  $r_{\delta }=| r-\delta l| \neq 0$,
for all $x\in \overline{\Omega}$, where for the fixed point
$z\in S^{n-1}\backslash \overline{G}$ and unit
radius vector $l=\overrightarrow{0z}=( l_{1},\dots ,l_{n}) $, the vector
$\delta l$ does not  belong to $\Omega_0 ^d $. Therefore, the function
$\eta( x) =r_{\delta }^{\alpha -2}u( x) $ is admissible in
identity \eqref{e8}. We obtain
\begin{equation} \label{e16}
\begin{aligned}
&\int_{\Omega} r_{\delta }^{\alpha -2} a_i( x,u,u_{x})u_{x_i}dx
 +\int_{\Omega} r_{\delta }^{\alpha -2} u( x) a( x,u,u_{x}) dx\\
&\quad+ \int_{\Omega } (\alpha -2) u( x) r_{\delta}^{\alpha -4}
 a_i(x,u,u_{x}) ( x_i-\delta l_i)dx=0.
\end{aligned}
\end{equation}
Since $a_i(x,u,p)=p_j \int_0^1 {\frac{\partial a_i(x,u,\tau
p)}{\partial (\tau p_j)}d\tau} +a_i(x,u,0)$, by \eqref{e10} we have
\begin{equation} \label{e17}
\begin{gathered}
a_i( 0,0,p) =p_i+a_i^{0}\,,\quad
a_i^{0}\equiv a_i(0,0,0) ,\quad i=\overline{1,n} \\
a_i( x,u,p) p_i=|p| ^{2}+a_i^{0}p_i+
[ a_i( x,u,p) -a_i( 0,0,p)] p_i.
\end{gathered}
\end{equation}
Taking this into account, choosing some small number $d$ and
dividing the domain $\Omega $ into two subdomains $\Omega_0 ^d $
and $\Omega \backslash \Omega_0 ^d $ we estimate the obtained
integrals in each of subdomains separately. Then  we apply
inequality \eqref{e6}, use estimates from \cite{l1} and the fact that
$u( x)$ is H\"older continuous. Finally, using
conditions of the theorem passing to the limit as $\delta
\to +0$ we obtain the required estimate.
\end{proof}

\noindent\textbf{Remark.}
Let $n=2$, $0\in \partial \Omega $ be a corner point,
$G=( 0,\omega_0 ), \omega_0 $ is size of the angle in the
neighbourhood of $0$, $\Omega_0 ^d =( 0,d) \times (0,\omega_0 ) $.
In this case eigenvalues problem \eqref{e3}  has the form
\begin{equation} \label{e18}
\begin{gathered}
u''+\lambda ^{2}u=0,\quad  u=u( \omega ),\quad \omega \in G,\\
u( \omega ) \Big|_{\omega =0}=0, \quad
 \frac{\partial u}{\partial n}\Big|_{\omega =\omega_0 }=0\,.
\end{gathered}
\end{equation}
Here, the least positive eigenvalue of this problem is
$\lambda =\pi / (2\omega_0 )$ and condition \eqref{e14} takes the form
\[
\frac{\pi }{\omega_0 }>2-\alpha ,\quad \alpha \leq 2.
\]

Before,  estimating  $|u( x)|$, we prove the following lemma.

\begin{lemma} \label{lem0}
Let $u( x) $ be a generalized solution of \eqref{e1} and let
conditions \eqref{e9}--\eqref{e12} be satisfied.
Then for any function
\[
v( x) \in V=\{ v\in W_{2}^{1}( \Omega_0 ^{\rho }) : v( x) =0,\;
x\in \Gamma_{0,1}^{\rho };\;\frac{\partial v}{\partial n}=0,\;
x\in \Gamma_{0,2}^{\rho }\}
\]
and almost all $\rho \in ( 0,d)$ the following equality holds
\begin{equation} \label{e19}
\int_{\Omega_0 ^{\rho }} [ a_i( x,u,u_{x})
v_{x_i}+a( x,u,u_{x}) v( x) ] dx
=\int_{G_{\rho }}a_i( x,u,u_{x}) v( x) \cos (r,x_i) dG_{\rho }
\end{equation}
\end{lemma}

To prove it we substitute $\eta ( x) =v( x)( \chi_{\rho })_{h}(
x)$, for $v\in W_{2,0}^{1}( \Omega ) $ into the integral identity
\eqref{e8}, where $\chi_{\rho }( x) $ is characteristic function
of the set $\Omega_0 ^{\rho }$ and $( \chi_{\rho })_{h}$ is its
Sobolev averaging. Such $\eta $ is admissible by virtue of Theorem
\ref{thm1}. Passing to the limit as $h\to 0$ we obtain
\eqref{e19}. Passage to the limit is justified by the use of
properties of mean functions \cite[theorem 3.10, p.113]{s1} and
Theorem \ref{thm1}.

\begin{theorem} \label{thm2}
Let $u( x) $ be a generalized solution of \eqref{e1}. Assume
conditions \eqref{e9}--\eqref{e12} and that
\begin{equation}
\Big( \sum_{i=1}^n [ a_i( x,u,p) -a_i( 0,0,p)] ^{2}\Big) ^{1/2}
\leq \delta (|x| ) |p| +h( x),
\end{equation}
for any $x\in \Omega_0 ^d $, $u\in R,p\in \mathbb{R}^{n}$,
where $\delta ( r) $ is a nondecreasing positive function satisfying
the Diny condition $\int_0^d \frac{\delta ( r) }{r}dr<\infty $.
In addition we assume that
\begin{equation} \label{e21}
\begin{gathered}
a_i( x,u,p) p_i\geq v_0 |p|^{2}-\mu
_{3}|u| ^{\beta }-u^{2}\varphi ( x) ;\\
a( x,u,p) u\leq \mu_0 |p|^{2}+\mu
_{3}|u| ^{\beta }+u^{2}\varphi ( x),
\end{gathered}
\end{equation}
where $2n/( n-2) >\beta >2$, $0\leq \varphi( x) \in L_{q/2}( \Omega )$,
$q>n$, $v_0 >0$, $\mu_0 ,\mu_{3}\geq 0$;
$ g( x) \in W_{2-n}^{0}( \Omega )$, $h( x) \in W_{2-n,0}^{0}( \Omega )$,
$f( x) \in W_{4-n,0}^{0}( \Omega)$, and
\[
 \rho ^{2}\int_{G} g^{2}( \rho ,\omega ) d\omega +\rho
^{2}\int_{G} h^{2}( \rho ,\omega ) d\omega
+\int_{\Omega_0 ^{\rho }} r^{4-n}f^{2}( x) dx\leq
k\rho ^{s},
\]
with $s>2\lambda ( G)$, $0<\rho <d$.
Then
\begin{equation} \label{e23}
 | u( x) | \leq C|x|^{\lambda ( G)}\,,
\end{equation}
where $\lambda (G) $ is the least positive
eigenvalue of \eqref{e3} and the constant $C$ depends only on the
known quantities of the problem.
\end{theorem}

\begin{proof} Substitute $v( x) =r^{2-n}u( x) $
in identity \eqref{e19}. Such a function is admissible by virtue of
\eqref{e5} and Theorem \ref{thm1}. Taking into account \eqref{e17} and estimating
integrals with multipliers $a_i^{0}$ and expression $u\ u_{x_0 }$ we obtain
\begin{align*}
&\int_{\Omega_0 ^{\rho }} r^{2-n}|\nabla u|^{2}dx\\
&\leq \frac{n-2}{2}\int_{G} u^{2}d\omega \\
&\quad +\int_{\Omega_0 ^{\rho }}  [ a_i( x,u,u_{x})-a_i( 0,0,u_{x}) ] |
 \big[ r^{2-n}| u_{x_i}| +( 2-n)
 r^{-n}| x_i| | u( x) | \big] dx\\
&\quad +\int_{\Omega_0 ^{\rho }} r^{2-n}|u|\, | a( x,u,u_{x}) | dx\\
&\quad +\rho \int_{G} | u( x) |
 [a_i( x,u,u_{x}) -a_i( 0,0,u_{x})]| \cos ( r,x_i)|_{r=\rho }d\omega \\
&\quad +C_{9}\rho ^{-\varepsilon }\| g\|_{W_{2-n}^{0}( \Omega ) }
+\rho ^{2-\varepsilon }\int_{G} g^{2}( \rho ,\omega ) d\omega
+\rho \int_{G}u u_{\rho }d\omega
\end{align*} % 24
Denoting $v( \rho ) =\int_0^{\rho } dr\int_{G}( ru_{r}^{2}
+\frac{1}{r}|\nabla_{\omega}u| ^{2}) d\omega $ and estimating integrals in the
right-hand side by means of inequalities \eqref{e4}, \eqref{e5}, Cauchy inequality
with $\varepsilon >0$, and H\"older property of $u( x) $, we
obtain
\begin{equation} \label{e25}
v( \rho ) \leq c\rho \,^{2\lambda }\ ,\ \ \ 0<\rho <d
\end{equation}
where constant $C$ depends on $M,d,v,\mu_{1},\mu_{2},\mu ,n,\lambda
,q,\mathop{\rm meas} G, \mathop{\rm meas} \Omega ,\| g\|_{q,\Omega}$,
\[
\| h\|_{W_{2-n,0}^{0}( \Omega) },\| g\|_{W_{2-n}^{0}(\Omega ) },
\| f\|_{W_{4-n,0}^{0}( \Omega ) },\| f\|_{q/2,\Omega },
\int_0^d \frac{\delta ( r) }{r}dr,k,s
\]
Consider the function
\begin{equation} \label{e26}
z( x') =\rho ^{-\lambda ( G) }u( \rho x') ,\quad  0<\rho <d
\end{equation}
in layer $Q'=\{ x': 1/2<|x'| <1\}$, $u\equiv 0$ out of $\Omega $,
and use inequalities from \cite[ch.2, inequality (2.22)]{l1}.
Taking into account estimate \eqref{e25}, we obtain
\begin{equation} \label{e287}
\int_{\rho /2<|x| <\rho } |u|^{q}dx\leq C\rho ^{n+q\lambda },\quad
2\leq q\leq 2n/( n-2) ,\; n>2.
\end{equation}
Then taking into consideration  results from \cite[ch.4, theorem
7.6]{l1}, by the assumption of this theorem, we obtain
\begin{equation} \label{e28}
| u( x) | \leq M_2\rho ^{\lambda ( G)}
\end{equation}
where $x\in \Omega_0 ^d \cap \{ \rho /2<|x| <\rho <d \} $ and
$M_2$ is a constant depending on the known quantities. Taking that
$|x| = 2\rho/3 $ we obtain the required estimate \eqref{e23} and
the proof is complete.
\end{proof}


\begin{theorem} \label{thm3}
Let $u( x) $ be a generalized solution of \eqref{e1} and
assumptions of theorem \ref{thm1} be satisfied.
Assume that for $x\in \overline{\Omega }$ and
$u,p\in \mathbb{R}^n$ the functions $a_i(x,u,p)$, $i=\overline{1,n}$
and $a( x,u,p) $ be differentiable with respect to their arguments
and the following inequalities hold:
\begin{equation} \label{e29}
\begin{gathered}
a_i( x,u,p) p_i\geq v_0 |p| ^{2}-\varphi_0 ( x) \\
\Big[ \sum_{i=1}^n \big( \big| \frac{\partial a_i}{\partial u}\big| ^{2}
+\big| \frac{\partial a}{\partial x_i}\big|^{2}\big) \Big] ^{1/2}
+\Big(\sum_{i,j=1}^n  \big| \frac{\partial a_i}{\partial x_{j}}\big|
^{2}\Big) ^{1/2} \leq \mu_{4}(|u|) ( |p| +\varphi_{1}( x) )
\\
\Big(\big| \frac{\partial a}{\partial u}\big| ^{2}
+\sum_{i=1}^n \big| \frac{\partial a}{\partial x_i}\big|^{2}\Big) ^{1/2}
\leq \mu_{5}( |u| ) \big(|p| ^{2}+\varphi_{2}( x) \big) ,
\end{gathered}
\end{equation}
 where $\varphi_i( x)$, $i=0,1,2$ are nonnegative functions.
Also assume that $\varphi_0 (x)$,
$\varphi_{2}( x) \in L_{q/2}( \Omega )$,
$\varphi_{1}( x) \in L_{q}( \Omega )$, $q>n$.
Then $u( x) \in W_{\alpha ,0}^{2}( \Omega ) $ and
\begin{align*} % 30
&\| u\|_{W_{\alpha ,0}^{2}( \Omega ) }^{2}\\
&\leq c_{1}(1+\| f\|_{q,\Omega }+\| f\|_{q/2,\Omega }
 +\| \varphi_0 \|_{q/2,\Omega}+\| \varphi_{2}\|_{q/2,\Omega }
 +\| \varphi_{1}\|_{q,\Omega }\\
&\quad +\| h\|_{W_{\alpha-2,0}^{2}( \Omega ) }^{2}
+\|g\|_{W_{\alpha -2}^{0}( \Omega )}^{2}
+\| f\|_{W_{\alpha,0}^{0}( \Omega ) }^{2}\\
&\quad +c_{2}\Big\{\int_{\Omega} r^{( \alpha +h) q/4-n}[\varphi_0 ^{q/2}(
x) +\varphi_{1}^{q}( x)
+\varphi_{2}^{q/2}( x) +f^{q/2}( x) +g^{q}(x) ]\Big\}^{4/q}\,,
\end{align*}
where $\alpha \leq 4-n$. Provided that the last
integral is finite, the constat $c_{1},c_{2}>0$ depends on the known parameters.
\end{theorem}

To proof this theorem we considered a sequence
of domains $\Omega_{k,\rho }$, which are intersections of $\Omega_0 ^d $
 and some layers. Making some transformations and using an estimate from
\cite{l1} and summing all the obtained inequalities over $k=1,2,\dots$.
Using Theorem \ref{thm1} we obtain the following corollary.

\begin{corollary} \label{coro1}
Let the conditions of Theorem \ref{thm3}, except for \eqref{e14},
be fulfilled. Then generalized solution $u( x)$ of problem \eqref{e1}
is in $W^{2}( \Omega )$, for the following cases:
\begin{enumerate}
\item  $n\geq 4$;
\item  $n=2$  and $0<\omega_0 <\frac{\pi }{2}$;
\item  $n=3$ and $G\subset G_0 =\{ \omega =( \theta ;\varphi ):
0<| \theta | <\omega_0 <\pi ,\;  0<\varphi <2\pi \} $,
where $\omega_0 $ is solution of equation
$p_{1/2}( \cos \omega_0 ) =0$  for Legendre functions.
\end{enumerate}
\end{corollary}

\begin{proof} (1) According to theorem \ref{thm3}
$u( x) \in W_{4-n,0}^{2}( \Omega ) $. Condition \eqref{e14} is
trivial  if $\alpha =4-n$ because $\lambda =\lambda ( G) >0$. Now the
statement follows from inequality
\[
\int_{\Omega_0^d} u_{xx}^{2}dx
\leq d^{n-4}\int_{\Omega_0 ^d } r^{4-n}u_{xx}^{2}dx\leq \mbox{const.}
\]
(2) Suppose $\alpha =0$ in Theorem \ref{thm3} then condition \eqref{e13} is trivial.
If $n=2$ the statement follows from the remark.

\noindent(3) Condition \eqref{e14} becomes $\lambda ( G) >1/2$. Let
$\Omega_0 \subset S^{2}$  be a domain in which the eigenvalue
problem \eqref{e3} is solvable for $\lambda ( G) =1/2$ and
$\partial \Omega_0 =\partial ^{1}\Omega_0 \cup \partial^{2}\Omega_0 $:
\begin{equation} \label{e31}
\begin{aligned}
\Delta_{\omega }u+( 1/2) ( 1+1/2) u=0,\quad  \omega \in \Omega_0  \\
u\Big|_{\partial ^{1}\Omega_0 }=0,\quad
 \frac{\partial u}{\partial u}\Big|_{\partial ^{2}\Omega_0} =0
\end{aligned}
\end{equation}
The condition $\lambda>1/2 $ implies  $\Omega \subset \Omega_0 $; see \cite{g2}.
We are seeking of solution problem \eqref{e31} of the form
$u= v( \theta ) $. Then for $v( \theta )$ we obtain
\begin{equation} \label{e32}
\begin{gathered}
\frac{1}{\sin \theta } \frac{d}{d\theta }\big( \sin \theta \frac{dv}{d\theta }\big)
+\frac{1}{2}\big( 1+\frac{1}{2}\big)v=0,\quad
0<| \theta | <\omega_0 ,\\
v( -\omega_0 ) =0\,\quad \frac{\partial v}{\partial n}(\omega_0 ) =0\,.
\end{gathered}
\end{equation}
The solution to this equation is a Legendre function of the first genus
$v(\theta ) =p_{1/2}( \cos \theta )$, which has exactly one
zero in the interval $0<\theta <\pi $ which we denote by $\omega_0 $
(see \cite{l1}).
Therefore, the corollary is proved.
\end{proof}

\begin{theorem} \label{thm4}
Let $u( x)$ be a generalized solution of \eqref{e1}. Let functions
$a_i( x,u,p)$, $a( x,u,p)$ be differentiable with respect to their
arguments and conditions \eqref{e9}--\eqref{e12}, \eqref{e29}
 with $q=\infty $  be satisfied.
Under the assumptions in Theorem \ref{thm2},
\begin{equation}
| \nabla u( x) | \leq c|x|^{\lambda ( G) -1}\,
\end{equation}
where $\lambda ( G)$ is the least positive eigenvalue of
\eqref{e3}, and constant $c$  depends only on the
known quantities.
\end{theorem}

\begin{proof} As in the proof of Theorem \ref{thm2} consider function
$z(x') =\rho ^{-\lambda ( G) }u(\rho x') $, $0<\rho <d$ in
the layer $Q'=\{x':  1/2<| x'| <1\} $
assuming that $u\equiv 0$ outside of $\Omega $. Under our
conditions, the theorem from \cite{h1} on boundedness of
modulus of gradient of solution inside of domain and near smooth
pieces of boundary is valid:
\begin{equation}
\mathop{\rm vrai\, max}{}_{Q'} |\nabla'z | \leq M_{3}
\end{equation}
where $M_{3}>0$ depends on $v,v_0,\mu,\mu_1,\mu_2$
$\mathop{\rm vrai\,max}_{Q'} | {z(x')} |$. Then for the
function $u( x) $ we obtain
\begin{equation}
|\nabla u( x)| \leq M_{1}\rho ^{\lambda( G) -1}\,,\quad
x\in \Omega_0 ^d \cap \{ \rho :2<|x| <\rho <d\}.
\end{equation}
Taking $|x| =2\rho/3 $, we obtain the required estimate.
\end{proof}

\begin{thebibliography}{00}

\bibitem{f1} A. F. Filippov;
\emph{Smoothness of generalized solutions near the vertex of a
polyhedral angle} (Rusian). Differentsial'nye Uravnenija 9 (1973),
1889--1903, 1927.


\bibitem{g1} T. S. Gadjiev;
\emph{Some sharp estimates of solutions of mixed boundary
problems for nonlinear equations in nonsmooth domains}.
Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 14 (2001),
42--47, 207.

\bibitem{g2} T. S. Gadjiev;
\emph{Mathematical physics and nonlinear mechanics} (Rusian),
 Kiev, publisher (Naukovo Dumka), 1986, v. 5 (39).

\bibitem{h1} G. H. Hardy, J. E. Littlewood;
\emph{Inequalities} (Russian). Gosudarstv. Izdat. Inostr. Lit.,
 Moscow, 1948.

\bibitem{k1}
V. A. Kondrat'ev, M. V. Borsuk;
\emph{The behavior of the solution of the Dirichlet problem
for a second-order quasilinear elliptic equation near a
corner point}. (Russian)  Differentsial'nye Uravneniya
 24  (1988), no. 10, 1778--1784, 1838;
translation in  Differential Equations  24  (1988),
 no. 10, 1185--1190 (1989)

\bibitem{k2} V. A. Kondrat'ev;
\emph{On the boundary problem for elliptic equation on
nonsmooth domains},
Proceeding Moscow Mathematical Society, 1967, v. 16.

\bibitem{l1} O. A. Ladyzhenskaya, N. N. Uraltseva;
\emph{Linear and quasilinear equations of elliptic type} (Rusina),
Second edition, revised. Izdat. ``Nauka'', Moscow, 1973.

\bibitem{m1} V. G. Maz'ya;
\emph{Sobolev spaces} (Russian) Leningrad. Univ., Leningrad, 1985.

\bibitem{s1} S. L. Sobolev;
\emph{Introduction to the theory of cubature formulae} (Rusina),
Izdat. ``Nauka'', Moscow, 1974

\bibitem{t1} P. Tolksdorf;
\emph{On the Dirichlet problem for quasilinear equations
in domains with conical boundary points}.
Comm. Partial Differential Equations  8  (1983),
no. 7, 773--817.

\bibitem{w1} N. M. Wigley;
\emph{Mixed boundary value problems in plane domains
with corners}.  Math. Z.  115  1970 33--52.

\bibitem{w2} E. T. Whittaker, G. N. Watson;
\emph{A course of modern analysis. An introduction to
the general theory of infinite processes and of analytic
functions; with an account of the principal transcendental
functions}. Reprint of the fourth (1927) edition.
Cambridge Mathematical Library. Cambridge University Press,
Cambridge, 1996.

\end{thebibliography}


\section*{March 18, 2005. Addendum}
In response to the editor's request, we want to add a
reference that should have been included in the original
 bibliography.
\begin{itemize}
\item[{[13]}] Borsuk, M. V.; Behavior of generalized solutions of the  Dirichlet
problem for second-order quasilinear elliptic equations of divergence type
near a conical point. (Russian)
  Sibirsk. Mat. Zh.  31  (1990), no. 6, 25--38;  translation
 in Siberian Math. J.  31  (1990), no. 6, 891--904 (1991)
\end{itemize}
Also we want to compare  this reference with our article.
The two articles have the same structure and visual appearance.
Both articles follow the ideas presented by Condratyev \cite{k2},
and have the same components: Weight inequalities, investigation
of a corresponding spectral problem, and study of Holder
continuity of solutions.

However, these two articles are different: [13] studies
a Dirichlet boundary problem, while our article studies
a mixed boundary problem.

1. The weight inequalities \eqref{e4}-\eqref{e7} require isoperimetric
conditions on the domain, which are not needed for the
Dirichlet problem.

2. The study of the spectrum for problem \eqref{e3} in our
article follows the method in \cite{f1}. In the Dirichlet case,
the study follows the work by Mikhlen (see [13]). 
The mixed boundary problem has  smallest eigenvalue 
$\lambda/(2\omega_0)$ and critical point $\pi/(2\omega_0)$, 
while the Dirichlet problem has  smallest
eigenvalue $\lambda/\omega_0$ and critical point $\pi/\omega_0$.

3. The study of Holder continuity of solutions for the
mixed problem follows ideas in \cite{g2}. Meanwhile
for the Dirichlet problem, the study follows ideas in \cite{l1}.

\end{document}