\documentclass[reqno]{amsart}
\begin{document}
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.~2000(2000), No.~39, pp.~1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 2000 Southwest Texas State University  and
University of North Texas.}
\vspace{1cm}

\title[\hfilneg EJDE--2000/39\hfil Regular oblique derivative problem]
{ Regular oblique derivative problem in Morrey spaces }

\author[D.K. Palagachev, M.A. Ragusa, \& L.G. Softova\hfil EJDE--2000/39\hfilneg]
{Dian K. Palagachev, Maria Alessandra Ragusa, \& Lubomira G. Softova}

\address{Dian K. Palagachev \hfill\break\indent
Dipartimento Interuniversitario di  Matematica,
    Politecnico di Bari, \hfill\break\indent
    Via E. Orabona, 4, 70125 Bari, Italy}
\email{dian@@pascal.dm.uniba.it}

\address{Maria Alessandra Ragusa \hfill\break\indent
Dipartimento di Matematica, Universit\`{a} di Catania, \hfill\break\indent
   Viale A. Doria, 6, 95125 Catania, Italy}
\email{maragusa@@dipmat.unict.it}

\address{Lubomira G. Softova \hfill\break\indent
Bulgarian Academy of Sciences,
   Institute of Mathematics and Informatics, \hfill\break\indent
   Dept. Math. Physics, \hfill\break\indent
   ``Acad. G. Bonchev'' Str., bl. 8, 1113 Sofia, Bulgaria}
\email{luba@@dipmat.unict.it}

\date{}
\thanks{Submitted December 17, 1999. Published May 23, 2000.}
\subjclass{35J25, 35B65, 35R05}
\keywords{Uniformly elliptic operator, regular oblique derivative
problem, Morrey spaces}

\begin{abstract}
 This article presents a study of the regular oblique derivative problem
 $$ \displaylines{
  \sum_{i,j=1}^n a^{ij}(x) \frac{\partial^2 u }{\partial x_i\partial x_j} =f(x) \cr
  \frac{\partial u }{\partial \ell(x)}+ \sigma(x) u = \varphi(x)\,.
 }$$
 Assuming that the coefficients $a^{ij}$  belong to the Sarason's class
 of functions with vanishing mean oscillation, we show existence and
 global regularity of strong solutions in Morrey spaces.
\end{abstract}

\maketitle

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{crlr}[thm]{Corollary}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{rem}[thm]{Remark}
\makeatletter
\def\theequation{\thesection.\@arabic\c@equation}
\makeatother

\section{Introduction}

The goal of the present paper is to study the global regularity in
Morrey spaces for strong solutions to the  {\it non-degenerate\/}
oblique derivative problem
\begin{equation}\label{1.1}
\begin{gathered}
\sum_{i,j=1}^n a^{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}
   =f(x)\quad \text{for almost all}\ x\in \Omega,\\[3pt]
 \frac{\partial u}{\partial \ell(x)}+ \sigma(x)u=\varphi(x)
    \quad \text{in the trace sense on}\ \partial \Omega\,.
\end{gathered} \end{equation}
Here the coefficients of the uniformly elliptic operator may be
discontinuous and the first order boundary operator, prescribed in terms of
directional derivative with respect to a unit vector field $\ell(x)$,
may be {\it nowhere\/} tangential to the boundary of $\Omega$.
More precisely, we assume  that $a^{ij}$'s belong to the Sarason class,
VMO, of functions with vanishing mean oscillation \cite{S}.

The interests in the study of boundary-value problems for elliptic operators
with principal coefficients  in VMO increased significantly in the last
ten years.  This is mainly due to the fact that VMO contains as a proper
subspace $C^0(\overline\Omega)$ that  ensures the extension of the
$L^p$-theory of operators with {\it continuous\/} coefficients	to
{\it discontinuous\/} coefficients \cite[Chapter~9]{GT}, \cite{LU}.
On the other hand, the Sobolev spaces
$W^{1,n}(\Omega)$ and $W^{\theta,\theta/n}(\Omega)$, $0<\theta<1$, are also
contained in VMO, whence the discontinuities of $a^{ij}$'s expressed in
terms of belonging to VMO become more general than those studied
before (cf.\cite{Miranda}, \cite{CFL1}).
We refer the reader to the survey \cite{C}, where an excellent presentation of the
state-of-the-art and relations with another similar results can be found
concerning the regularizing properties of these operators in the framework of
Sobolev spaces. The Dirichlet problem for such kind of	equations has been
well studied both in the linear (\cite{CFL1}, \cite{CFL2}) and in the
quasilinear (\cite{Pa}) cases.
Concerning the regular oblique derivative problems for
elliptic operators with VMO principal coefficients, we should mention
the articles \cite{DP1} in the linear and \cite{DP2} in the quasilinear case,
respectively. The results of \cite{DP1} have been extended also to elliptic
operators with lower order terms and general boundary operators (\cite{MP}).
Recently, the $W^{2,p}$-theory
developed in \cite{MP} has been applied in the study of {\it degenerate\/}
oblique derivative problem in Sobolev spaces (see \cite{MPV}). The degeneracy
means that the field $\ell$ can be tangential to the boundary of $\Omega$
at the points  of some non-empty subset.

In the present paper we derive global regularizing property  in Morrey spaces
of elliptic operators with VMO coefficients.
Precisely, it is proved that any strong solution $(u\in W^{2,p}(\Omega))$
of
\eqref{1.1} with $f\in L^{p,\lambda}(\Omega)$ and $\varphi\in
W^{(p,\lambda)}(\partial \Omega)$, $1<p<+\infty$, $0<\lambda<n$, admits second
derivatives lying in the Morrey space $L^{p,\lambda}(\Omega)$
(Theorem~\ref{thm2.1}).
As consequence of that regularizing property we derive
also strong solvability in $W^{2,p,\lambda}(\Omega)$ of \eqref{1.1}
(Theorem~\ref{thm2.2}) for any
$f\in L^{p,\lambda}(\Omega)$ and $\varphi\in
W^{(p,\lambda)}(\partial \Omega)$. (See the next Section for the definition of
the spaces used.) Finally, the known relations between the Morrey
and the H\"older spaces permit us to obtain {\it finer\/} H\"older continuity
of the gradient
$Du$ of the strong solutions to \eqref{1.1} for suitable values of $p$ and
$\lambda$.

The crucial point of our investigations is the local boundary Morrey
regularity
of the strong solutions to \eqref{1.1} (Lemma~\ref{lem4.1}). The approach is
based on an explicit representation of solution's second derivatives (near the
boundary) in terms of singular integral operators with Calder\'on--Zygmund
kernels and their commutators and operators with positive kernels. This method
has been already used in the study of Dirichlet problem (\cite{CFL2}, see
also \cite{DP1}). Since the representation formula derived in \cite{DP1}
concerns constant coefficients elliptic and boundary operators, we apply here,
in contrast to \cite{DP1}, a new approach in order to deal with non-homogeneous
boundary conditions described by variable oblique derivative operator. This is
reached by introducing a special auxiliary function which, roughly speaking,
absorbs the right-hand side of the boundary condition. Thus, a new
representation formula for the second derivatives occurs, which involves
densities depending on the same second derivatives, {\it but also\/} on the
strong solution and its gradient. To estimate effectively the Morrey norms of
the second derivatives, we make use of a special non-dimensional norms. Indeed,
that approach seems to be more natural when one studies the oblique derivative
problem and this is due to the first order operator defined on the boundary
$\partial \Omega$.

The rest of the paper is organized as follows. In Section~2 we state the
problem, the assumptions on the data and the main results. Section~3 is
devoted
to some auxiliary results. Special emphasize is given on a construction and
properties of the auxiliary function (Lemma~\ref{lem3.1}) mentioned above, by
the aid
of which we are able to represent (locally near the boundary) the solution of
\eqref{1.1}. In Section~4 the local boundary Morrey regularity of the  strong
solutions to \eqref{1.1} is derived. Finally, a combination of that result
with
the interior regularity enables us to prove the main results of the paper
(Section~5).

Results similar to the present here were derived for Dirichlet
problem in \cite{DPR0} and \cite{DPR}.


\section{The Problem and Assumptions}
\setcounter{equation}{0}


Let $\Omega\subset{\mathbb R}^n$, $n\geq3$, be a bounded domain with sufficiently
smooth
boundary $\partial \Omega$. Consider the unit vector field
$\ell(x)=(\ell_1(x),\ldots,\ell_n(x))$ prescribed on $\partial \Omega$ and the
first-order boundary operator
$$
{\mathcal B}\equiv \frac{\partial}{\partial \ell(x)} +\sigma(x)\quad x\in\partial
\Omega.
$$
In $\Omega$ we will consider the second order uniformly elliptic operator
$$
{\mathcal L}\equiv a^{ij}(x)D_{ij}
$$
where the usual summation convention on repeated indices is accepted and
$D_{ij}\equiv \frac{\partial^2}{\partial x_i\partial x_j}$.

Our goal will be to study global regularity and strong solvability in
the framework of Morrey spaces of the next oblique derivative problem
\begin{equation}\label{2.1}
\begin{gathered}
{\mathcal L}u = f(x)\quad \text{for almost all}\ x\in \Omega,\\
{\mathcal B}u = \varphi(x)\quad \text{in the trace sense on}\ \partial \Omega\,.
\end{gathered}
\end{equation}

Before giving the list of assumptions concerning the data of \eqref{2.1}, let
us
recall some definitions and state useful notations. As usual,
the classical Sobolev space of functions having weak derivatives up to order
$k$ which belong to $L^p(\Omega)$ will be denoted by $W^{k,p}(\Omega)$.

Let $p\in(1,+\infty)$ and $\lambda\in(0,n)$. The function $f\in
L^1_{\text{loc}}(\Omega)$ is said to belong to the
 Morrey space
$L^{p,\lambda}(\Omega)$ if
$$
\|f\|_{L^{p,\lambda}(\Omega)} \equiv \bigg(
\sup_{\underset{x\in \Omega}{\rho>0}}
\rho^{-\lambda}
\int\limits_{B_\rho (x) \cap \Omega} |f(y)|^p dy \bigg)^{1/p} < +\infty\,,
$$
where, hereafter $B_\rho(x)$ denotes an $n$-dimensional ball of radius $\rho$
and centered at the point $x$.

We will consider also subspaces of $W^{k,p}(\Omega)$ formed by functions
having
their $k$-th order derivatives in $L^{p,\lambda}(\Omega)$. The symbol
$W^{k,p,\lambda}(\Omega)$ stands for these subspaces. Precisely,
$$
W^{k,p,\lambda}(\Omega)=\big\{ u\in W^{k,p}(\Omega)\colon\quad
D^\alpha u\in L^{p,\lambda}(\Omega),\quad |\alpha|=k\big\}.
$$
The norm in that space is naturally defined by
$$
\|u\|_{W^{k,p,\lambda}(\Omega)} =
\|u\|_{L^{p,\lambda}(\Omega)} + \|D^k u\|_{L^{p,\lambda}(\Omega)}.
$$
By means of the interpolation inequality, it is clear that also the
lower-order
derivatives $D^\alpha u\in L^{p,\lambda}(\Omega)$ for $0<|\alpha|<k$.
We shall make use also of the non-dimensional norms
$$
\|u\|^*_{W^{k,p,\lambda}(\Omega)} =
\|u\|_{L^{p,\lambda}(\Omega)} + d^{k/2}\|D^k
u\|_{L^{p,\lambda}(\Omega)},\quad
d=\text{diam\,}\Omega.
$$

To interpret the boundary condition in \eqref{2.1} in the trace sense on
$\partial \Omega$, we will use the space of functions defined on
$\partial \Omega$ which are traces of functions lying in
$W^{1,p,\lambda}(\Omega)$. That functional class is well studied by
Campanato (cf. \cite{Ca1}). Thus, define
$W^{(p,\lambda)}(\partial \Omega)$ to be the Banach space formed by functions
$\varphi$ defined on $\partial \Omega$ and having the finite norm
\begin{align*}
\|\varphi\|_{W^{(p,\lambda)}(\partial \Omega)}=&\bigg(
\sup_{\underset{z'\in\partial \Omega}{
   \rho>0}}{\rho^{-\bar\lambda}} \int\limits_{B_{\rho}(z')\cap \partial
\Omega} |\varphi(x')|^p d \sigma_{x'}\bigg)^{1/p}\\
&+\bigg(\sup_{\underset{z',\, \bar z'\in \partial \Omega}{
   \rho>0}} \rho^{-\lambda}
   \int\limits_{B_{\rho}(z')\cap \partial \Omega}
   \int\limits_{B_{\rho}(\bar z')\cap \partial \Omega}
   \frac{|\varphi(x')-\varphi(\bar x')|^p}{|x'-\bar x'|^{p+n-2}} d\sigma_{x'}
d\sigma_{\bar x'}    \bigg)^{1/p},
\end{align*}
with $\bar \lambda=\max\{\lambda-1,0\}$.

In order to formulate the regularity assumptions
on the coefficients of the  operator ${\mathcal L}$,
we need also to recall the definitions of the John--Nirenberg
space (\cite{JN}) of functions with bounded mean oscillation (BMO) and the
Sarason class VMO of the functions with vanishing mean oscillation
(\cite{S}).
A locally integrable function $f(x)$ is said to belong to BMO if
$$
\| f \|_* \equiv \sup_{B \subset {\mathbb R}^n} \frac{1}{|B|}
\int\limits_B |f(x)-f_B| dx < +\infty
$$
with $f_B$ being the integral average
$\frac{1}{|B|} \int_B f(x)dx$ of the function $f(x)$ over the set $B$,
and $B$ ranges in the class of balls of ${\mathbb R}^n$.
If $f(x)\in BMO$ denote
$$
\gamma (r) = \sup_{{\rho \leq r,}\ {x \in {\mathbb R}^n}}
\frac{1}{|B_\rho|} \int\limits_{B_\rho} |f(x)-f_{B_\rho}| dx.
$$
Then,  $f(x)\in VMO$ if $\gamma(r) =o(1)$ as $r\to 0^+$ and refer to
$\gamma(r)$ as the VMO-modulus of $f(x)$.

It should be noted that replacing the ball $B$ above by the intersection
$B\cap
\Omega$, one obtains the definitions of $BMO(\Omega)$ and $VMO(\Omega)$.
Later on, having a function defined on $\Omega$ that belongs to $BMO(\Omega)$
($VMO(\Omega)$) it is possible to extend it to all ${\mathbb R}^n$
preserving its $BMO$ (VMO) character (see \cite[Proposition~1.3]{Acq}).

We are in a position now to list our assumptions. Concerning the operator
${\mathcal L}$, we suppose that it is uniformly elliptic one with VMO
coefficients.
That is,
\begin{equation}\label{2.2}
\begin{gathered}
\exists\ \kappa >0: \  \kappa^{-1} |\xi|^2 \leq a^{ij}(x) \xi_i\xi_j \leq
\kappa |\xi|^2 \quad \forall \xi \in {\mathbb R}^n,\  \text{a.a.}\ x \in \Omega,\\
a^{ij}(x) \in VMO(\Omega),\quad a^{ij}(x)=a^{ji}(x).
\end{gathered}
\end{equation}
We set also $\gamma_{ij}(r)$ for the VMO-modulus of the function $a^{ij}(x)$
and let $\gamma(r) = \left( \sum_{i,j=1}^n \gamma^2_{ij}(r)\right)^{1/2}$.
An immediate consequence of \eqref{2.2}
is the essential
boundedness of $a^{ij}$'s.

As it concerns the boundary operator ${\mathcal B}$, we assume
\begin{equation}\label{2.3}
\begin{gathered}
\ell_i(x),\ \sigma(x)\in C^{0,1}(\partial\Omega),\quad
	\partial \Omega\in C^{1,1},\\
\ell(x)\cdot\nu(x)=\ell_i(x)\nu_i(x)>0,\quad \sigma(x)<0\quad \text{for each}\
	x\in\partial \Omega,
\end{gathered}
\end{equation}
with $\nu(x)=(\nu_1(x),\ldots,\nu_n(x))$ being the unit {\it inward\/} normal
to
$\partial \Omega$. The simple geometric meaning of \eqref{2.3} is that the
field $\ell(x)$ is nowhere tangential to $\partial \Omega$, that is,
\eqref{2.1}
is a {\it regular oblique derivative problem\/} (see \cite{PP}).


The main results of the paper are contained in the following theorems.

\begin{thm}\label{thm2.1}
Let $\eqref{2.2}$ and $\eqref{2.3}$ be true, $p\in(1,+\infty)$ and
$\lambda\in(0,n)$.
Assume further that $u\in W^{2,p}(\Omega)$ solves the problem $\eqref{2.1}$
with $f\in L^{p,\lambda} (\Omega)$ and $\varphi\in
W^{(p,\lambda)}(\partial \Omega)$.

Then $D_{ij}u \in L^{p,\lambda} (\Omega) $ and there is
a constant $C=C(n,p,\lambda,\kappa,\gamma,\ell,\sigma,\partial \Omega)$
such that
\begin{equation}\label{2.4}
\|u\|_{W^{2,p,\lambda} (\Omega)} \leq
C \left(\| u \|_{L^{p,\lambda} (\Omega)}  +
\|f \|_{L^{p,\lambda} (\Omega)} + \|\varphi\|_{W^{(p,\lambda)} (\partial
\Omega)} \right).
\end{equation}
\end{thm}

The regularizing property of the couple $({\mathcal L},{\mathcal B})$ implies
well-posedness of the oblique derivative problem \eqref{2.1}
in  the Morrey space $W^{2,p,\lambda} (\Omega)$.
\begin{thm}\label{thm2.2}
Let $\eqref{2.2}$ and $\eqref{2.3}$ be satisfied, $p\in(1,+\infty)$ and
$\lambda\in(0,n)$.

Then, for every $f\in L^{p,\lambda}(\Omega)$ and $\varphi\in
W^{(p,\lambda)}(\partial \Omega)$ there exists a unique solution of the
oblique derivative problem $\eqref{2.1}$.
Moreover,
\begin{equation}\label{2.5}
   \| u \|_{W^{2,p,\lambda}(\Omega)}
   \leq C \left(  \| f \|_{L^{p,\lambda}(\Omega)}+
	   \|\varphi\|_{W^{(p,\lambda)}(\partial \Omega)}\right)
\end{equation}
with a constant
$C=C(n,p,\lambda,\kappa,\gamma,\ell,\sigma,\partial \Omega)$.
\end{thm}

An immediate consequence of Theorem~\ref{thm2.1}
and the imbedding properties of the Morrey spaces for suitable values of $p$
and
$\lambda$
(cf. \cite{Ca2}) is the next global H\"older regularity result for the
gradient
$Du$ of the strong solutions to  \eqref{2.1}.
\begin{crlr}\label{corol2.3}
Let $u\in W^{2,p}(\Omega)$ be a strong solution to $\eqref{2.1}$
with $f\in L^{p,\lambda}(\Omega)$ and
$\varphi\in W^{(p,\lambda)}(\partial \Omega)$.

Then, if $n-p<\lambda<n$,
the gradient $Du$ is a H\"older continuous function on $\overline\Omega$
with exponent $\alpha=1-(n-\lambda)/p$ and
$$
   \| Du \|_{C^{0,\alpha}(\overline\Omega)}= \sup_{x,y\in \Omega}
   \frac{|Du(x)-Du(y)|}{|x-y|^\alpha}
   \leq C \left(  \| f \|_{L^{p,\lambda}(\Omega)}+
	   \|\varphi\|_{W^{(p,\lambda)}(\partial \Omega)}\right).
$$
\end{crlr}

Let us point out that the solely assumptions $f\in L^p(\Omega)$
and $\varphi\in W^{1-1/p,p}(\partial \Omega)$ imply $u\in W^{2,p}(\Omega)$
(see \cite{DP1}). Thus, if $p>n$ the Sobolev imbedding theorem yields $Du\in
C^{\beta}(\overline\Omega)$ with $\beta=1-n/p$. On the other hand,
Corollary~\ref{corol2.3} ensures H\"older continuity of the gradient {\it
also\/} for $p\in(1,n]$, assuming {\it finer\/} regularity of the data
expressed
in terms of their belonging to the Morrey space $L^{p,\lambda}(\Omega)$
with $\lambda\in(n-p,n)$.


\begin{rem}\em
The results presented here can be applied in studying Morrey regularity
of the strong solutions to \eqref{2.1} for general elliptic
operators
$$
{\mathcal L}\equiv a^{ij}(x)D_{ij} + b^i(x)D_i + c(x)
$$
with $a^{ij}\in VMO(\Omega)$ and the lower order coefficients
$b^i(x)$ and $c(x)$ owning suitable Lebesgue integrability. We refer the
reader
to \cite{MP} for details concerning the case of Sobolev spaces.
\end{rem}


\section{Auxiliary Results}
\setcounter{equation}{0}
\setcounter{thm}{0}


Let $\tilde\Gamma$ be a portion of the hyperplane $\{x_n=0\}$,
$x=(x_1,\ldots,x_{n-1},x_n)\equiv (x',x_n)$, and let $\tilde\varphi(x')$ be a function
defined on $\tilde\Gamma$ which belongs to $W^{(p,\lambda)}(\tilde\Gamma)$.
 The Banach space $W^{(p,\lambda)}(\tilde\Gamma)$
is equipped now with the non-dimensional norm
\begin{align*}
\|\tilde\varphi\|^*_{W^{(p,\lambda)}(\tilde\Gamma)}=&\bigg(
\sup_{\underset{z'\in\tilde\Gamma}{
   \rho\in (0,d]}} {\rho^{-\bar\lambda}}
\int\limits_{B'_{\rho}(z')\cap \tilde\Gamma}
   |\tilde\varphi (x')|^p dx'\bigg)^{1/p} \\
&+d^{1/2}\bigg( \sup_{\underset{z',\, \bar z'\in\tilde\Gamma}{
   \rho\in (0,d]}} {\rho^{-\lambda}}
   \int\limits_{B'_{\rho}(z')\cap \tilde\Gamma}
   \int\limits_{B'_{\rho}(\bar z')\cap \tilde\Gamma}
   \frac{|\tilde\varphi(x')-\tilde\varphi(\bar x')|^p}{|x'-\bar x'|^{p+n-2}} dx' d\bar x'
   \bigg)^{1/p},
\end{align*}
and $B'_{\rho}(z')$ is an $(n-1)$-dimensional ball of radius $\rho$ and
centered at $z'\in\{x_n=0\}$, $\bar \lambda=\max\{\lambda-1,0\}$,
$d=\text{diam\,}\tilde\Gamma$.

Now, following \cite{GT} (see the proof of Theorem~6.26 therein), we take a
function $\eta(y')\in C_0^2({\mathbb R}^{n-1})$ such that
$\int_{{\mathbb R}^{n-1}} \eta(y')dy'=1$. Fixing arbitrary $x_0=(x'_0,0)$ and
$R>0$, and
denoting $B^+_R=B_R(x_0)\cap \{x_n>0\}$, $\Gamma_R=B_R(x_0)\cap \{x_n=0\}$,
without loss of
generality we may take	$\Gamma_R$ instead of $\tilde\Gamma$ at the above
definition of the norm $\|\tilde\varphi\|^*_{W^{(p,\lambda)}(\tilde\Gamma)}$
and set $d=R$.	Later, having
$\tilde\varphi\in W^{(p,\lambda)}(\Gamma_R)$ we suppose that $\tilde\varphi$ is extended to all
${\mathbb R}^{n-1}$ as a function with a compact support, preserving its
$W^{(p,\lambda)}$-norm.

Supposing that the boundary $\partial \Omega$ is locally flatten near the
point
$x_0$ such that $\Omega\subset \{x_n>0\}$, we recall that  the {\it regular
obliqueness condition\/} \eqref{2.3} ensures $\ell_n(x_0)\neq0$. Consider now
the function
\begin{equation}\label{3.1}
\phi(x)=\phi(x',x_n)= \frac{x_n}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(x'-x_n y') \eta(y') dy'.
\end{equation}
Essential step in our further considerations is ensured by the next
\begin{lem}\label{lem3.1}
The function $\phi(x)$ belongs to $W^{2,p,\lambda}(B^+_R)$ and	satisfies
\begin{equation}\label{3.2}
\phi(x',0)=0, \quad \frac{\partial \phi}{\partial x_n}(x',0)=
	\frac{\tilde\varphi(x')}{\ell_n(x_0)}\quad \text{for}\quad x'\in\Gamma_R.
\end{equation}
Moreover,
\begin{equation}\label{3.3}
\|\phi\|^*_{W^{2,p,\lambda}(B^+_R)}  =
   \|\phi\|_{L^{p,\lambda}(B^+_R)} + R\|D^2\phi\|_{L^{p,\lambda}(B^+_R)}
	\leq  C R^{1/2} \|\tilde\varphi\|^*_{W^{(p,\lambda)}(\Gamma_R)}
\end{equation}
with $C=C(n,p,\lambda,\ell,\eta)$.
\end{lem}

\paragraph{Proof.} We will prove Lemma~\ref{lem3.1} in two steps.

{\bf Step 1: A bound of $\boldsymbol{\|\phi\|_{L^{p,\lambda}(B^+_R)}.}$} Let
$\rho\in(0,R]$,  ${\bar x}\in B^+_R$
and $B^+_\rho(\bar x)=B_\rho(\bar x)\cap \{x_n>0\}$.
Then, making use of the Jensen integral inequality as well as
of  Fubini's theorem, we obtain
\begin{align*}
&\rho^{-\lambda} \int\limits_{B^+_\rho(\bar x)\cap B^+_R} |\phi(x)|^pdx
=
	 \frac{1}{[\ell_n(x_0)]^p} \rho^{-\lambda}
\int\limits_{B^+_\rho(\bar x)\cap B^+_R} \big| x_n \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(x'-x_n y')\eta(y')dy'\big|^p dx\\
&\quad \leq  C(n,p,\ell,\text{supp\,}\eta) \rho^{-\lambda}
	\int\limits_{\text{supp\,}\eta} |\eta(y')|^p \big(
\int\limits_{B^+_\rho(\bar x)\cap B^+_R}  x_n^p
	|\tilde\varphi(x'-x_n y')|^p dx \big) dy'.
\end{align*}
Now, setting $ I_{B^+_\rho(\bar x)\cap B^+_R}(y')=\rho^{-\lambda}
\int_{B^+_\rho(\bar x)\cap B^+_R} x_n^p |\tilde\varphi(x'-x_ny')|^p dx$ and
$Q_\rho(\bar x)$ for the cube $\big\{x\in {\mathbb R}^n\colon$ $|x_i-\bar x_i|\leq
\rho$
for $i\leq n-1;$ $\max\{0,-\rho+\bar x_n\}\leq x_n\leq \rho+\bar x_n\big\}$,
we
have
\begin{align*}
I_{B^+_\rho(\bar x)\cap B^+_R}(y')\leq &\ I_{Q_\rho(\bar x)}(y')=
	\rho^{-\lambda} \int\limits_{Q_\rho(\bar x)} x_n^p
	|\tilde\varphi(x'-x_ny')|^p dx' dx_n\\
\leq &\  \rho^{-\lambda} \int\limits_{\max\{0,-\rho+\bar
x_n\}}^{\rho+\bar
   x_n} x_n^p \int\limits_{Q'_\rho(\bar x)}  |\tilde\varphi(z')|^p dz' dx_n
\end{align*}
with $Q'_\rho(\bar x)=\big\{z'\in {\mathbb R}^{n-1}\colon$ $-\rho+\bar x_i-x_ny_i\leq
z_i\leq \rho+\bar x_i-x_ny_i,$\quad $i\leq n-1\big\}$.

Since, $\int_{Q'_\rho(\bar x)}|\tilde\varphi(z')|^p dz'\leq \rho^{\bar\lambda}
\left(\|\tilde\varphi\|^*_{W^{(p,\lambda)}(\Gamma_R)}\right)^p$,
$\bar\lambda=\max\{\lambda-1,0\}$, using $\bar x_n\leq R$, $\rho\leq R$, one
has
\begin{align*}
& I_{B^+_\rho(\bar x)\cap B^+_R}(y')\leq \rho^{\bar \lambda-\lambda}
\left(\|\tilde\varphi\|^*_{W^{(p,\lambda)}(\Gamma_R)}\right)^p
\int\limits_{\max\{0,-\rho+\bar x_n\}}^{\rho+\bar
   x_n} x_n^p dx_n \\
&\quad\quad\quad \leq  C(n,p,\ell) R^{p+\max\{1-\lambda,0\}}
   \left(\|\tilde\varphi\|^*_{W^{(p,\lambda)}(\Gamma_R)}\right)^p.
\end{align*}
The last bound and the fact that $y'\in \text{supp\,}\eta$ show that
\begin{equation}\label{3.4}
\|\phi\|_{L^{p,\lambda}(B_R^+)}\leq
C(n,p,\ell,\text{supp\,}\eta) R^{1+\max\{1-\lambda,0\}/p}
   \|\tilde\varphi\|^*_{W^{(p,\lambda)}(\Gamma_R)}.
\end{equation}


{\bf Step 2: An estimate for  $\boldsymbol{\|D^2\phi\|_{L^{p,\lambda}(B_R^+)}}$.}
We will calculate now the first and second derivatives of the function $\phi$
given by \eqref{3.1}. For, after
the change $z'=x'-x_ny'$ of the variables in \eqref{3.1}, one has
$$
\phi(x',x_n)= \frac{x^{2-n}_n}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(z') \eta\left(\frac{x'-z'}{x_n}\right) dz',
$$
whence
\begin{align*}
\frac{\partial \phi}{\partial x_i}(x',x_n)=&\
	      \frac{x^{1-n}_n}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(z') \frac{\partial \eta}{\partial x_i}\left(\frac{x'-
      z'}{x_n}\right) dz'       \quad \text{for}\quad i<n,\\[2pt]
\frac{\partial \phi}{\partial x_n}(x',x_n)=&\
	      \frac{(2-n)x^{1-n}_n}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(z') \eta\left(\frac{x'-z'}{x_n}\right) dz' \\
 &\ -	 \frac{x^{-n}_n}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(z') D'\eta\left(\frac{x'-z'}{x_n}\right)\cdot (x'-z') dz'
\end{align*}
with $D'=(\partial /\partial
y_1,\ldots,\partial /\partial y_{n-1})$, $x'\cdot y'=\sum_{j=1}^{n-1} x_j
y_j$.

Returning to the original variables, we obtain
\begin{align}\label{3.5}
\frac{\partial \phi}{\partial x_i}(x',x_n)=&\
	      \frac{1}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(x'-x_ny') D_i \eta (y') dy'
      \quad \text{for}\quad i<n,\\[2pt]
\label{3.6}
\nonumber
\frac{\partial \phi}{\partial x_n}(x',x_n)=&\
	      \frac{2-n}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(x'-x_ny') \eta (y') dy' \\
 &\ -	 \frac{1}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(x'-x_ny') D'\eta(y')\cdot y' dy'.
\end{align}
Now,  remembering $\eta\in C^2_0({\mathbb R}^{n-1})$ and
$\int_{{\mathbb R}^{n-1}}\eta(y')dy'=1$, the divergence theorem implies
\begin{align}\label{3.7}
\nonumber
& \int\limits_{{\mathbb R}^{n-1}}\!\!\!D_i\eta (y')dy'=\!\!\!
\int\limits_{{\mathbb R}^{n-1}}\!\!\!D_{ij}\eta (y')dy'=\!\!\!
\int\limits_{{\mathbb R}^{n-1}}\!\!\!D'(D_{i}\eta) (y')\cdot y'dy'=0\ \forall i,j\leq
n-1;\\[-8pt]
\\[-2pt]
\nonumber &
\int\limits_{{\mathbb R}^{n-1}}\!\!\!\eta (y') dy'=1,\quad
\int\limits_{{\mathbb R}^{n-1}}\!\!\!D'\eta (y')\cdot y' dy'=1-n,\quad
\int\limits_{{\mathbb R}^{n-1}}\!\!\!D'(D'\eta\cdot y')\cdot y'  dy'=(1-n)^2.
\end{align}
Therefore, \eqref{3.2} follows from \eqref{3.1} and \eqref{3.6} putting
$x_n=0$
therein.

Since $\eta\in C_0^2({\mathbb R}^{n-1})$, we can differentiate \eqref{3.5} and
\eqref{3.6} once again. Thus, straightforward calculations yield
\begin{align*}
D_{ij}\phi(x',x_n) =&\ \frac{1}{\ell_n(x_0)} \frac{1}{x_n}
\int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(x'-x_ny')D_{ij}\eta(y')dy'\quad  i,j\leq n-1,\\[2pt]
D_{in}\phi(x',x_n) =&\ \frac{1-n}{\ell_n(x_0)} \frac{1}{x_n}
      \int\limits_{{\mathbb R}^{n-1}}\tilde\varphi(x'-x_ny')D_{i}\eta(y')dy'\\
	  &  - \frac{1}{\ell_n(x_0)} \frac{1}{x_n}
      \int\limits_{{\mathbb R}^{n-1}}\tilde\varphi(x'-x_ny')D'(D_i\eta)(y')\cdot y'dy'
  \quad  i\leq n-1,\\[2pt]
D_{nn}\phi(x',x_n) =&\ \frac{(2-n)(1-n)}{\ell_n(x_0)} \frac{1}{x_n}
      \int\limits_{{\mathbb R}^{n-1}}\tilde\varphi(x'-x_ny')\eta(y')dy'\\
	  &  + \frac{2n-3}{\ell_n(x_0)} \frac{1}{x_n}
      \int\limits_{{\mathbb R}^{n-1}}\tilde\varphi(x'-x_ny')D'\eta(y')\cdot y'dy'\\
	  &  + \frac{1}{\ell_n(x_0)} \frac{1}{x_n}
  \int\limits_{{\mathbb R}^{n-1}}\tilde\varphi(x'-x_ny')D'(D'\eta\cdot y')\cdot y'dy'.
\end{align*}
These formulae and \eqref{3.7} lead to
\begin{align}\label{3.8}
\nonumber
D_{ij}\phi(x',x_n) =&\ \frac{1}{\ell_n(x_0)} \frac{1}{x_n}
\int\limits_{{\mathbb R}^{n-1}}
 [\tilde\varphi(x'-x_ny')-\tilde\varphi(x')]D_{ij}\eta(y')dy',\quad  i,j\leq n-1,\\[2pt]
\nonumber
D_{in}\phi(x',x_n) =&\ \frac{1-n}{\ell_n(x_0)} \frac{1}{x_n}
      \int\limits_{{\mathbb R}^{n-1}}[\tilde\varphi(x'-x_ny')-\tilde\varphi(x')]D_{i}\eta(y')dy'\\
\nonumber
	  &  - \frac{1}{\ell_n(x_0)} \frac{1}{x_n}\!\!
      \int\limits_{{\mathbb R}^{n-1}}\!\![\tilde\varphi(x'\!-\!x_ny')\!-\!\tilde\varphi(x')]D'(D_i\eta)(y')\!\cdot\!
y'dy',\quad   i\!\leq\! n\!-\!1,\\
\\[-12pt]
\nonumber
D_{nn}\phi(x',x_n) =&\ \frac{(2-n)(1-n)}{\ell_n(x_0)} \frac{1}{x_n}
      \int\limits_{{\mathbb R}^{n-1}}[\tilde\varphi(x'-x_ny')-\tilde\varphi(x')]\eta(y')dy'\\
\nonumber
	  &  + \frac{2n-3}{\ell_n(x_0)} \frac{1}{x_n}
      \int\limits_{{\mathbb R}^{n-1}}[\tilde\varphi(x'-x_ny')-\tilde\varphi(x')]D'\eta(y')\cdot y'dy'\\
\nonumber
	  &  + \frac{1}{\ell_n(x_0)} \frac{1}{x_n}
  \int\limits_{{\mathbb R}^{n-1}}[\tilde\varphi(x'-x_ny')-\tilde\varphi(x')]D'(D'\eta\cdot y')\cdot
y'dy'.
\end{align}


Now, getting a look on the formulae \eqref{3.8}, it is
clear that the integrals appearing there are all of the type
(modulo a constant multiplier)
$$
\psi(x)=\psi(x',x_n)= \frac{1}{x_n}
\int\limits_{{\mathbb R}^{n-1}} [\tilde\varphi(x'-x_ny')-\tilde\varphi(x')]\mu(y')dy'
$$
with $\mu(y')$ being
$\eta(y')$, $D_i\eta(y')$,  $y'\cdot D'\eta(y')$,
$D_{ij}\eta(y')$ or $D'(D'\eta\cdot y')\cdot y'$.

Proceeding as in Step~1 with $\bar x\in B_R^+$, $\rho\in(0,R]$, we obtain
$$
\rho^{-\lambda} \int\limits_{B_\rho^+(\bar x)\cap B_R^+} |\psi(x)|^p dx\leq
C(n,p,\ell,\text{supp\,}\mu) \int\limits_{\text{supp\,}\mu} |\mu(y')|^p
J(y')dy'
$$
with
$$
J(y')=\rho^{-\lambda} \int\limits_{B_\rho^+(\bar x)\cap B_R^+}
\frac{1}{x_n^p} |\tilde\varphi(x'-x_ny')-\tilde\varphi(x')|^p dx'dx_n.
$$

Since
$$
J(y')\leq C\sum_{j=1}^{n-1}
\rho^{-\lambda}\!\!\!\! \int\limits_{B_\rho^+(\bar x)\cap
B_R^+}\!\!\!\! \frac{1}{x_n^p}
|\tilde\varphi(x_1,\ldots,x_{j-1},x_j-x_ny_j,x_{j+1},\ldots,x_{n-1})-\tilde\varphi(x')|^p dx,
$$
replacing the balls at the last integrals by sets of the type
$T_j=\{x\in{\mathbb R}^n\colon$ $-\rho+\bar x_i\leq x_i\leq \rho+\bar x_i$ $(i\neq j)$,
$-\rho+\bar x_j\leq x_j\leq \bar x_j-x_n\}$, it is easily seen that
\begin{align*}
&\rho^{-\lambda} \int\limits_{T_j}  \frac{1}{x_n^p}
|\tilde\varphi(x_1,\ldots,x_{j-1},x_j-x_ny_j,x_{j+1},\ldots,x_{n-1})-\tilde\varphi(x')|^p dx\\
&\quad\quad
\leq
C \sup_{\underset{z',\, \bar z'\in \Gamma_R}{
   \rho>0}} \rho^{-\lambda}
   \int\limits_{B_{\rho}(z')\cap \Gamma _R}
   \int\limits_{B_{\rho}(\bar z')\cap \Gamma _R}
   \frac{|\tilde\varphi(x')-\tilde\varphi(\bar x')|^p}{|x'-\bar x'|^{p+n-2}} d\sigma_{x'}
d\sigma_{\bar x'}
\end{align*}
(see \cite{Ad}, \cite{Ca1}, \cite{Mi} for details). This implies
\begin{equation}\label{3.9}
\|D^2\phi\|_{L^{p,\lambda}(B_R^+)}\leq C
\|\psi\|_{L^{p,\lambda}(B_R^+)}\leq C
\|\tilde\varphi\|_{W^{(p,\lambda)}(\Gamma _R)}.
\end{equation}
The estimates \eqref{3.4} and \eqref{3.9} yield \eqref{3.3}.
\hfill $\diamondsuit$ \smallskip

In our further considerations
we will need some precise results on the
boundedness in Morrey spaces of suitable integral operators. We refer the
readers to the corresponding theorems and proofs given in \cite{DPR}
and \cite{DR2}.

\begin{prp}\label{prp3.2} {\em \cite[Theorem~2.3]{DR2}}
Let $U$ be an open subset of ${\mathbb R}^n$, $f \in L^{p,\lambda}(U)$,
$p\in(1,+\infty)$, $\lambda\in (0,n)$, $a \in VMO \cap L^\infty ({\mathbb R}^n)$.
Let $k(x,z)$ be a Calder\'on--Zygmund kernel (see \cite{CFL2}) in the
$z$ variable for almost all $x \in U$ such that
$$
\max_{|\alpha|\leq 2n} \left\| \frac{\partial^\alpha}{\partial z^\alpha}
k(x,z) \right\|_{L^\infty (D \times \Sigma)} = M < +\infty,
$$
with $\Sigma=\{x\in{\mathbb R}^n\colon\ |x|=1\}$.
For an arbitrary $\varepsilon > 0$ set
\begin{align*}
K_\varepsilon f (x) =&\
 \int\limits_{\underset{x\in U}{|x-y|>\varepsilon}}
k(x,x-y) f(y) dy,\\
C_\varepsilon (a,f) (x) =&\  \int\limits_{\underset{x\in U}{|x-y|>\varepsilon}}
k(x,x-y) (a(x)-a(y)) f(y) dy.
\end{align*}
There exist $Kf$, $C(a,f) \in L^{p,\lambda}(U)$ such that
$$
\lim_{\varepsilon \to 0}
\| K_\varepsilon f - K f \|_{L^{p,\lambda} (U)}
=
\lim_{\varepsilon \to 0}
\| C_\varepsilon (a,f) - C(a,f) \|_{L^{p,\lambda} (U)}=0.
$$
Moreover,
$$
\| K f \|_{L^{p,\lambda} (U)}
\leq C
\| f \|_{L^{p,\lambda} (U)},
\quad
\| C(a,f) \|_{L^{p,\lambda} (U)}
\leq C
\| a \|_* \| f \|_{L^{p,\lambda} (U)}
$$
for some positive constant $ C = C(n,p,\lambda,M)$.
\end{prp}

\begin{prp}\label{prp3.3} {\em \cite[Theorem~2.5]{DPR}}
Let $x\in {\mathbb R}^n_+$ and define
$$
\tilde Kf (x) = \int\limits_{{\mathbb R}^n_+}
\frac{f(y)}{|\tilde x - y|^n} dy, \quad
\tilde x \equiv (x_1,\dots,x_{n-1},-x_n).
$$
There exists a constant $C$ independent of $f(x)$, such that
$$
\| \tilde Kf \|_{L^{p,\lambda}({\mathbb R}^n_+)}
\leq
C  \| f \|_{L^{p,\lambda}({\mathbb R}^n_+)}.
$$
\end{prp}

\begin{prp}\label{prp3.4} {\em \cite[Theorem~2.6]{DPR}}
Let $f \in L^{p,\lambda} ({\mathbb R}^n_+)$, $p\in(1,+\infty)$,
$\lambda\in(0,n)$, $a \in VMO \cap L^\infty ({\mathbb R}^n_+)$.
Then, for any $x \in {\mathbb R}^n_+$	the commutator
$$
   \tilde C (a,f)(x) =
   \int\limits_{{\mathbb R}^n_+} \frac{|a(x) - a(y)|}{|\tilde x - y|^n}
   f(y)  dy
$$
is bounded from $L^{p,\lambda}({\mathbb R}^n_+)$ into itself. There exists
a constant $C$ independent of $a(x)$ and $f(x)$ such that
$$
\| \tilde C (a,f) \|_{L^{p,\lambda}({\mathbb R}^n_+)} \leq
C \|a\|_* \| f \|_{L^{p,\lambda}({\mathbb R}^n_+)}.
$$
\end{prp}



\section{Boundary Morrey Regularity}
\setcounter{equation}{0}
\setcounter{thm}{0}

As in the previous section, we suppose that the boundary $\partial \Omega$
is locally flatten near an arbitrary point $x_0\in \partial \Omega$ such that
$\Omega\subset \{x_n>0\}$. The following result implies boundary regularizing
property of the couple $({\mathcal L},{\mathcal B})$ in Morrey spaces:

\begin{lem}\label{lem4.1}
Let $\eqref{2.2}$ and $\eqref{2.3}$ be satisfied and $p\in(1,+\infty)$,
$1<q\leq p<+\infty$,
$\lambda\in (0,n)$. Suppose $r>0$ and let
$u\in W^{2,q}(B^+_r)$
be a solution to the equation ${\mathcal L}u=f\in L^{p,\lambda}(B^+_r)$ such that
${\mathcal B}u=\varphi$ on $B_r\cap \{x_n=0\}$
with $\varphi\in W^{(p,\lambda)}(B_r\cap \{x_n=0\})$.

Then there exists $R\in (0,r)$ small enough such that $D_{ij}u\in
L^{p,\lambda}(B^+_R)$. Moreover, there is a constant
$C=C(n,\kappa,p,\lambda,\ell,\sigma,\partial \Omega)$ such that
\begin{equation}\label{4.1}
 \| D_{ij}u \|_{L^{p,\lambda}(B^+_R)} \leq
C  \left( \|u\|_{L^{p,\lambda}(B_R^+)}+
	  \|f\|_{L^{p,\lambda}(B_R^+)}+
	  \|\varphi\|_{W^{(p,\lambda)}(B_R\cap \{x_n=0\})}\right).
\end{equation}
\end{lem}

\paragraph{Proof.}
We will utilize the explicit representation formula of the second derivatives
$D^2u$ derived in \cite[Lemma~4.2]{DP1}. However, as that formula concerns
oblique derivative problem for constant coefficients elliptic operator and
homogeneous boundary condition with constant coefficients boundary operator,
first of all we shall reduce the original problem to a homogeneous one.

Without loss of generality we may suppose that the ball $B_r$ is centered at
the origin. Let $x_0=(x'_0,x_{0n})$, $x'_0=(x_{01},\ldots, x_{0\,n-1})$.
Obviously, we have
$$
\begin{gathered}
a^{ij}(x_0)D_{ij}u(x)=\big[a^{ij}(x_0)-a^{ij}(x)\big]D_{ij}u(x)+f(x)
	\quad \text{a.e. in}\ B_r^+,\\
\ell_i(x'_0)D_iu(x')+\sigma(x'_0)u(x')=\big[\ell_i(x'_0)-\ell_i(x')
     \big]D_iu(x')\\
\quad\quad
\quad\quad
   +\big[\sigma(x'_0)-\sigma(x')\big]u(x') +\varphi(x')\quad
	x'\in B_r\cap \{x_n=0\}.
\end{gathered}
$$
Consider now the right-hand side of the boundary condition above and denote it
by $\tilde\varphi$. That is,
\begin{equation}\label{4.2}
\tilde\varphi(x',u)=\big[\ell_i(x'_0)-\ell_i(x')
     \big]D_iu(x') +\big[\sigma(x'_0)-\sigma(x')\big]u(x') +\varphi(x').
\end{equation}
Define $\phi(x)=\phi(x,u)$ by \eqref{3.1} with $\tilde\varphi$ given by
\eqref{4.2}. Since $\tilde\varphi (x',u)$ depends {\it affinely\/} on $u$, it is clear
that also the dependence of $\phi$ on $u$ will be affine one. Later,
remembering
the properties of $\phi$ established in Lemma~\ref{lem3.1} (see \eqref{3.2}),
it
is obvious that
$$
\frac{\partial \phi}{\partial \ell(x'_0)}(x)+
\sigma(x'_0)\phi(x)=\tilde\varphi(x',u)\quad         \text{for}\ x_n=0.
$$
That is why, the function $u(x)-\phi(x)$ satisfies
$$
\begin{gathered}
a^{ij}(x_0)D_{ij}(u(x)-\phi(x))=\big[a^{ij}(x_0)-a^{ij}(x)\big]D_{ij}u(x)\\
\quad\quad
\quad\quad
 \quad\quad
+f(x) -a^{ij}(x_0)D_{ij}\phi(x)   \quad \text{a.e. in}\ B_r^+,\\[4pt]
\frac{\partial (u-\phi)}{\partial \ell(x'_0)}+
   \sigma(x'_0)(u(x')-\phi(x'))=0 \quad     x'\in B_r\cap \{x_n=0\}.
\end{gathered}
$$

Therefore, \cite[Lemma~3.1]{DP1} implies
$$
u(x)=\phi(x)+\!\int\limits_{B^+_r}\! G(x_0,x,y)\left\{
\left(a^{ij}(x_0)\!-\!a^{ij}(y)\right)D_{ij} u(y) \!+\!f(y)\!-\!a^{ij}(x_0)D_{ij}\phi(y)
\right\} dy,
$$
where
$$
G(x_0,x,y)= \Gamma(x_0,x-y)-\Gamma(x_0,T(x,x_0)-y)+\theta(x_0,T(x,x_0)-y);
$$
$\Gamma(x_0,\xi)$ is the normalized fundamental solution of the operator
$a^{ij}(x_0)D_{ij}:$
$$
\Gamma(x_0,\xi)=  \frac{1}{n(2-n)\omega_n\sqrt{\text{det\,}\{a^{ij}(x_0)\}}}
	\left(A^{ij}(x_0)\xi_i\xi_j\right)^{(2-n)/2}
$$
with $\omega_n$ and $A^{ij}(x_0)$ being the measure of the unit ball in ${\mathbb R}^n$
and the inverse matrix of $\{a^{ij}(x_0)\}$, respectively;
$$
T(x,y)=x-\frac{2x_n}{a^{nn}(y)}\boldsymbol{a}^n(y),\quad T(x)=T(x,x),\quad
\boldsymbol{a}^n(y)=(a^{1n}(y),\ldots,a^{nn}(y));
$$
\begin{align*}
\theta(x_0,\xi)=&\ \frac{2}{n\omega_n\sqrt{\text{det\,}\{a^{ij}(x_0)\}}}
	\frac{\ell_n(x'_0)}{a^{nn}(x_0)}\\
&\ \times \int\limits_0^\infty
\frac{e^{\sigma(x'_0)s}(\xi+s T(\ell(x'_0) ))_n}{ \left(A^{ij}(x_0)
(\xi+s T(\ell(x'_0) ))_i(\xi+s T(\ell(x'_0) ))_j\right)^{n/2}} ds
\end{align*}
with $(\xi+s T(\ell(x'_0) ))_i$ being the $i$-th component of the vector
$\xi+s T(\ell(x'_0))\in{\mathbb R}^n$.

Now, similar arguments as these used in the proof of \cite[Lemma~4.2]{DP1}
lead to
\begin{align}\label{4.3}
\nonumber
D_{ij}u(x)=&\ D_{ij}\phi(x)\\
\nonumber
&\  + \text{P.V.}\!\! \int\limits_{B^+_r}\! \Gamma_{ij}(x,x\!-\!y)
      \left\{\!\left(a^{ij}(x)\!-\!a^{ij}(y)\right)D_{ij} u(y) \!+\!
       f(y)\!-\!{\mathcal L}(x)\phi(y)\!\right\} dy\\
  &\ + c_{ij}(x)\left(f(x)-{\mathcal L}(x)\phi(x)\right)
     + I_{ij}(x,x) + J_{ij}(x,x)
\end{align}
for almost all $x\in B^+_r$,
where ${\mathcal L}(x)\phi(y)=a^{ij}(x)D_{ij}\phi(y)$ and
$\Gamma_i(x,\xi) = D_{\xi_i} \Gamma(x,\xi)$,
$\Gamma_{ij}(x,\xi) = D_{\xi_i\xi_j} \Gamma(x,\xi)$,
$\theta_{i}(x,\xi) = D_{\xi_i} \theta(x,\xi)$,
$\theta_{ij}(x,\xi) = D_{\xi_i\xi_j} \theta(x,\xi)$,
$$
c_{ij}(x) = \int\limits_{|\xi|=1} \Gamma_i (x,\xi) \xi_j d \sigma_\xi;
$$
$$
I_{ij} (x,z)  =
\int\limits_{B_r^+} \Gamma_{ij} (z,T(x,z)-y)
\left\{ \left(a^{hk} (z) - a^{hk} (y) \right)
D_{hk}u(y) + f(y) -{\mathcal L}(x)\phi(y) \right\} dy
$$
for $i,j<n$;

\noindent
$I_{in} (x,z)$

\noindent
\hfill $
=\int\limits_{B_r^+} \Gamma_{ij} (z,T(x,z)-y)
\left\{ \left(a^{hk} (z) - a^{hk} (y) \right) D_{hk}u(y) +
f(y) -{\mathcal L}(x)\phi(y) \right\}
B_j (z) dy$

\noindent
for $i<n$;


\noindent
$I_{nn} (x,z)$

\noindent
\hfill $
=\!\!\int\limits_{B_r^+}\!  \Gamma_{ij} (z,T(x,z)\! -\! y)
\left\{\!\left(a^{hk}(z)\! -\! a^{hk}(y) \right) D_{hk}u(y)\!
+\! f(y)\!-\!{\mathcal L}(x)\phi(y)\!\right\} B_i(z)B_j(z)dy$;
$$
J_{ij} (x,z) = \int\limits_{B_r^+} \theta_{ij} (z,T(x,z)-y)
\left\{ \left(a^{hk} (z) - a^{hk} (y) \right) D_{hk}u(y)
+ f(y)-{\mathcal L}(x)\phi(y) \right\} dy
$$
for $i,j<n$;


\noindent
$J_{in} (x,z)$


\noindent
\hfill $
  = \int\limits_{B_r^+} \theta_{ij} (z,T(x,z)-y)
\left\{ \left(a^{hk} (z) - a^{hk} (y) \right) D_{hk}u(y)
+ f(y)-{\mathcal L}(x)\phi(y) \right\}
B_j (z) dy$

\noindent
for $i<n$;


\noindent
$J_{nn} (x,z)$

\noindent
\hfill $
  = \!\!\int\limits_{B_r^+}\!\!      \theta_{ij} (z,T(x,z)\! -\! y)
 \left\{\!\left(a^{hk}(z)\! -\! a^{hk}(y) \right)
D_{hk}u(y) \!+\! f(y)\!-\!{\mathcal L}(x)\phi(y)\! \right\} B_i(z)B_j(z)dy$.

The vector $B(z)=(B_1(z),\ldots,B_n(z))$ above is given by the formula
$$
B(z)=\frac{\partial }{\partial x_n}T(x,z),\quad\text{that is}\quad
B(z)=\left(-2\frac{a^{1n}(z)}{a^{nn}(z)},\ldots,
-2\frac{a^{n-1,\,n}(z)}{a^{nn}(z)},-1\right).
$$

Suppose now $q<p$ and let $s\in[q,p]$. Take an arbitrary $w\in
W^{2,s,\lambda}(B^+_r)$ and define
$$
{\mathcal S}w =
\phi(x)+\!\int\limits_{B^+_r}\! G(x_0,x,y)\left\{\!
\left(a^{ij}(x_0)\!-\!a^{ij}(y)\right)D_{ij} w(y)
\!+\!f(y)\!-\!a^{ij}(x_0)D_{ij}\phi(y)\! \right\} dy,
$$
with $\tilde\varphi=\tilde\varphi(x',w)$ given by \eqref{4.2} and $\phi(x)=\phi(x,w)$ defined
by \eqref{3.1}.

The idea in proving Lemma~\ref{lem4.1} will be to show that ${\mathcal S}$ is a
{\it contraction mapping\/} from $W^{2,s,\lambda}(B^+_R)$ into itself for
small
enough $R\in (0,r)$. (The fact that ${\mathcal S}$ maps $W^{2,s}(B_r^+)$ into
itself
will follow from the calculations below.) Then, having in mind that $u\in
W^{2,q}(B^+_R)$ is a fixed point
of the map ${\mathcal S}$ it will follow easily the statement of
Lemma~\ref{lem4.1}
and the estimate \eqref{4.1}.

Take now two arbitrary functions $w_1,\ w_2\in W^{2,s,\lambda}(B^+_r)$.
Denoting $w=w_1-w_2$, one has
\begin{align*}
{\mathcal S}w_1- {\mathcal S}w_2 =&\
\phi(x,w)\\
&\ +\!\!\int\limits_{B^+_r}\!\! G(x_0,x,y)\left\{\!
\left(a^{ij}(x_0)\!-\!a^{ij}(y)\right)D_{ij} w(y) \!-\!a^{ij}(x_0)D_{ij}\phi(y,w)\!
\right\} dy
\end{align*}
with
\begin{equation}\label{4.4}
\tilde\varphi(x',w)=\big[\ell_i(x'_0)-\ell_i(x')
     \big]D_iw(x') +\big[\sigma(x'_0)-\sigma(x')\big]w(x'),
\end{equation}
and
$$
\phi(x)=\phi(x',x_n)= \frac{x_n}{\ell_n(x_0)} \int\limits_{{\mathbb R}^{n-1}}
	\tilde\varphi(x'-x_n y') \eta(y') dy'\quad \text{(see \eqref{3.1})}.
$$
Taking into account the
assumptions \eqref{2.2},
we have
\begin{align*}
&\|{\mathcal S}w_1- {\mathcal S}w_2\|_{L^{s,\lambda}(B^+_r)} \leq C(n,\kappa)  \Bigg(
\|\phi(x,w)\|_{L^{s,\lambda}(B^+_r)}\\
&\quad\quad+
\Big\|\!\! \int\limits_{B^+_r}\! G(x_0,x,y) \left((a^{ij}(x_0)\!-\!a^{ij}(y))
D_{ij} w(y)\! +\! D_{ij}\phi(y,w) \right) dy\Big\|_{L^{s,\lambda}(B^+_r)}\!\!\Bigg).
\end{align*}
Since  $G(x_0,x,y)=O(|x-y|^{2-n})$ as $|x-y|\to0$ (see \cite[Lemma~3.1,
Remark~3.1]{DP1}) and $a^{ij}\in L^\infty(\Omega)$,
the integral $\int_{B^+_r}\! G(x_0,x,y) (
(a^{ij}(x_0)\!-\!a^{ij}(y))D_{ij} w(y) \!+\! D_{ij}\phi(y,w)) dy$ is a Riesz
potential. Thus, the
classical theory (cf. \cite[Lemma~7.12]{GT}, \cite[Lemma~I.1]{Ca2}) implies
\begin{align*}
&\left\|\ \int\limits_{B^+_r} G(x_0,x,y) \left(
(a^{ij}(x_0)-a^{ij}(y))D_{ij} w(y) + D_{ij}\phi(y,w) \right) dy
\right\|_{L^{s,\lambda}(B^+_r)} \\
&\quad\quad \leq C(n,s) r^2 \left(
\left\|D^{2} w\right\|_{L^{s,\lambda}(B^+_r)}+
\left\|D^{2} \phi(\cdot,w)\right\|_{L^{s,\lambda}(B^+_r)}\right).
\end{align*}

Further, according to \eqref{4.3} one has
\begin{align*}
D_{ij}({\mathcal S}w_1&\ - {\mathcal S}w_2)(x)= D_{ij}\phi(x,w)\\
  &\   + \text{P.V.} \int\limits_{B^+_r} \Gamma_{ij}(x,x-y)
      \left\{\left(a^{ij}(x)-a^{ij}(y)\right)D_{ij} w(y)
       -{\mathcal L}(x)\phi(y,w)\right\} dy\\
  &\ - c_{ij}(x){\mathcal L}(x)\phi(x,w)
     + I_{ij}(x,x,w) + J_{ij}(x,x,w)\quad \text{for a.a.}\ x\in B^+_r,
\end{align*}
with $c_{ij}$, $I_{ij}(x,x,w)$ and $J_{ij}(x,x,w)$ being as above with $u$
replaced by $w$ and missing term $f(y)$ at the integrands.

Since $\Gamma_{ij}(x,\xi)$
are Calder\'on--Zygmund kernels in the $\xi$ variable,
 Proposition~\ref{prp3.2} implies
\begin{align*}
&\left\|\text{P.V.} \int\limits_{B^+_r} \Gamma_{ij}(x,x-y)
      \left\{\left(a^{ij}(x)-a^{ij}(y)\right)D_{ij} w(y)
       -{\mathcal L}(x)\phi(y,w)\right\} dy \right\|_{L^{s,\lambda}(B^+_r)}\\
&\quad\quad \leq C(n,s,\kappa,\gamma_{ij},M,\partial \Omega) \left(
	\gamma(r)\|D^2w\|_{L^{s,\lambda}(B^+_r)} +
	\|D^2\phi(\cdot,w)\|_{L^{s,\lambda}(B^+_r)} \right)
\end{align*}
with $M = \max_{i,j=1,\ldots,n} \max_{|\alpha|\leq 2n}
\left\| \frac{\partial^\alpha \Gamma_{ij} (x,\xi)}{\partial \xi^\alpha}
\right\|_{L^\infty(\Omega \times \Sigma)}$.

Further, the geometric properties of the mapping $T$ ensure $c_1|\tilde x
-y|\leq |T(x)-y|\leq c_2 |\tilde x-y|$ (cf. \cite{CFL2}) for some positive
constants $c_1$  and $c_2$. Thus, Propositions~\ref{prp3.3} and \ref{prp3.4}
yield
\begin{align*}
&\|I_{ij}(\cdot,\cdot,w)\|_{L^{s,\lambda}(B^+_r)},\
    \|I_{ij}(\cdot,\cdot,w)\|_{L^{s,\lambda}(B^+_r)}\\
&\quad \leq C(n,s,\kappa,\gamma_{ij},M,\partial \Omega)\left\|
	\sum_{h,k=1}^n	\tilde C(a^{hk},D_{hk}w)+\tilde K({\mathcal
L}\phi(\cdot,w))
	       \right\|_{L^{s,\lambda}(B^+_r)}\\
&\quad \leq C\left(
	\gamma(r)\|D^2w\|_{L^{s,\lambda}(B^+_r)}+
	\|D^2\phi(\cdot,w)\|_{L^{s,\lambda}(B^+_r)}\right).
\end{align*}
Finally,
$$
\|c_{ij}(x) {\mathcal L}(x)\phi(x,w)\|_{L^{s,\lambda}(B^+_r)} \leq
C  \|D^2\phi(\cdot,w)\|_{L^{s,\lambda}(B^+_r)}.
$$

Therefore,
\begin{align}\label{4.5}
\nonumber
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\|{\mathcal S}w_1- {\mathcal S}w_2\|^*_{W^{2,s,\lambda}(B^+_r)} \leq  &\    C
     \left(    r\gamma(r)\|D^2w\|_{L^{s,\lambda}(B^+_r)}+
     r^2\|D^2w\|_{L^{s,\lambda}(B^+_r)}\right.\\
	& \left.  + r\|D^2\phi(\cdot,w)\|_{L^{s,\lambda}(B^+_r)}+
    \left\|\phi(\cdot,w)\right\|_{L^{s,\lambda}(B^+_r)}
\right)
\end{align}
with $C=C(n,s,\kappa,\gamma_{ij},M,\partial \Omega)$.

To express the last two norms above in terms of
$\|w\|^*_{W^{2,s,\lambda}(B^+_r)}$, we use Lemma~\ref{lem3.1}. Thus,
\begin{align*}
\|\phi\|^*_{W^{2,s,\lambda}(B^+_r)} = &\
   \|\phi(\cdot,w)\|_{L^{s,\lambda}(B^+_r)}+ r
     \|D^2\phi(\cdot,w)\|_{L^{s,\lambda}(B^+_r)}\\
 \leq &\ c r^{1/2}\|\tilde\varphi\|^*_{W^{(s,\lambda)}(B^+_r\cap\{x_n=0\})}.
\end{align*}
On the other hand,  \eqref{4.4} implies
\begin{align*}
r^{1/2}\|\tilde\varphi(\cdot,w)\|^*_{W^{(s,\lambda)}(B^+_r\cap\{x_n=0\})}
 \leq & C r^{1/2}\left(
\|(\ell_i(x'_0)-\ell_i(x'))D_iw\|^*_{W^{(s,\lambda)}(B^+_r\cap\{x_n=0\})}\right.
\\
&\quad \left. +
\|(\sigma(x'_0)-\sigma(x'))w\|^*_{W^{(s,\lambda)}(B^+_r\cap\{x_n=0\})}
\right).
\end{align*}
Remembering the Lipschitz regularity of the coefficients of the
boundary
operator \eqref{2.3}, the Ra\-de\-ma\-cher theorem and
\cite[Theorem~1.2]{Ca1} yield
\begin{align*}
& r^{1/2}\|(\ell_i(x'_0)-
  \ell_i(x'))D_iw\|^*_{W^{(s,\lambda)}(B^+_r\cap\{x_n=0\})} \\
&\quad\quad +
    r^{1/2}\|(\sigma(x'_0)-\sigma(x'))w\|^*_{W^{(s,
    \lambda)}(B^+_r\cap\{x_n=0\})}
 \leq  C r^{1/2} \|w\|^*_{W^{2,s,\lambda}(B^+_r)}.
\end{align*}
Therefore, \eqref{3.3} implies
$$
\|\phi(\cdot,w)\|^*_{W^{2,s,\lambda}(B^+_r)}
\leq C r^{1/2} \|w\|^*_{W^{2,s,\lambda}(B^+_r)}.
$$
Taking into account \eqref{2.2},
\eqref{4.5} reads
$$
\|{\mathcal S}w_1-{\mathcal S}w_2\|^*_{W^{2,s,\lambda}(B^+_r)} \leq C(r)
\|w_1-w_2\|^*_{W^{2,s,\lambda}(B^+_r)}, \quad C(r)=o(1)\ \text{as}\ r\to0,
$$
where $C(r)=C(\gamma(r)+r+r^{1/2})$.
Taking $r=R$ to be sufficiently small above we have $C(R)<1$, that is, ${\mathcal
S}$ is a contraction mapping from $W^{2,s,\lambda}(B^+_R)$ (equipped with the
norm $\|\cdot\|^*_{W^{2,s,\lambda}(B_R^+)}$)
 into itself
for each $s\in[q,p]$. Now, remembering that $u\in W^{2,q}(B^+_R)$
is a fixed point of ${\mathcal S}$, and using the imbedding
$W^{2,p,\lambda}(B^+_R)\subset W^{2,q,\lambda}(B^+_R)\subset
W^{2,q}(B^+_R)$, as well as the fact that the fixed point of ${\mathcal S}$
should be {\it unique\/} one,
we obtain $D^2u \in L^{p,\lambda}(B^+_R)$.

To get the estimate \eqref{4.1}, we have to take the $L^p$-norm of the both
sides of \eqref{4.3}. The calculations are similar to these already carried out
in obtaining \eqref{4.5}. Precisely, taking $w_1=u$ and $w_2=0$ we have
\begin{align*}
\|u\|^*_{W^{2,p,\lambda}(B^+_r)} =  &\
\|{\mathcal S}u\|^*_{W^{2,p,\lambda}(B^+_r)} \\
       \leq &\
\|{\mathcal S}u - {\mathcal S}0\|^*_{W^{2,p,\lambda}(B^+_r)} +
\|{\mathcal S}0\|^*_{W^{2,p,\lambda}(B^+_r)}.
\end{align*}
The first norm above is estimated exactly as in \eqref{4.5}, while the second
one gives
$\|f\|_{L^{p,\lambda}(B_R^+)}$ and $\|\varphi\|_{W^{(p,\lambda)}(B_R\cap
\{x_n=0\})}$.

This completes the proof of
Lemma~\ref{lem4.1}.  \hfill $\diamondsuit$

\section{Global Morrey Regularity and Solvability of the Problem \eqref{2.1}}
\setcounter{equation}{0}

\paragraph{Proof of Theorem~\ref{thm2.1}.}
Bearing in mind the {\it interior Morrey regularity\/}
(\cite[Theorem~3.3]{DR2}), the
statement of Theorem~\ref{thm2.1} and the
bound \eqref{2.4} follow from Lemma~\ref{lem4.1} through a suitable partition
of unity. \hfill $\diamondsuit$ \smallskip


\paragraph{Proof of Theorem~\ref{thm2.2}.}
The functions $f\equiv 0$ and $\varphi\equiv 0$ lie in $L^{p,\lambda}(\Omega)$
and $W^{(p,\lambda)}(\partial \Omega)$, respectively,
for each $p>1$ and each $\lambda\in(0,n)$. In particular, this holds true for
$p>n$. Thus, bearing in mind
the Aleksandrov--Bakelman--Pucci maximum principle
(\cite[Theorem~2.6.2]{TLN}) it follows that $u(x)=0$ is the
{\it unique solution\/}
of the homogeneous oblique derivative problem
 \eqref{2.1} ($f\equiv0$, $\varphi\equiv0$).
This proves uniqueness of the solution to \eqref{2.1}.

Concerning the strong solvability in the space
$W^{2,p,\lambda}(\Omega)$ of the problem \eqref{2.1},
we note that $L^{p,\lambda}(\Omega)\subset L^{p}(\Omega)$. Therefore,
in view of \cite[Theorem~1.2]{DP1}, there exists a unique solution $u \in
W^{2,p}(\Omega)$ of \eqref{2.1}. Further,
Theorem~\ref{thm2.1} asserts $u\in W^{2,p,\lambda} (\Omega)$.

To derive the estimate \eqref{2.5} we have for the linear
operator
$$
({\mathcal L},{\mathcal B})\colon\ W^{2,p,\lambda} (\Omega)
\to L^{p,\lambda}(\Omega)\times W^{(p,\lambda)}(\partial \Omega)
$$
that
\begin{align*}
\| ({\mathcal L},{\mathcal B})u \|_{L^{p,\lambda}(\Omega)
  \times W^{(p,\lambda)}(\partial \Omega)}  =&\
\| {\mathcal L}u \|_{L^{p,\lambda}(\Omega)}+
\| {\mathcal B}u\|_{W^{(p,\lambda)}(\partial \Omega)}  \\
\leq &\ C \left(
\| u \|_{L^{p,\lambda}(\Omega)}  +
\| Du \|_{L^{p,\lambda}(\Omega)}  +
\| D^2u \|_{L^{p,\lambda}(\Omega)} \right) \\
\leq &\ C \| u \|_{W^{2,p,\lambda} (\Omega)}.
\end{align*}
This shows continuity of $({\mathcal L},{\mathcal B})$. Further,
$({\mathcal L},{\mathcal B})$
is injective and surjective mapping
as it was shown before. Thus, the Banach theorem on
inverse mappings implies continuity of the  operator $({\mathcal L},{\mathcal
B})^{-1}$,
i.e., the bound \eqref{2.5}. \hfill $\diamondsuit$

\subsection*{Acknowledgements.}
The results presented here were obtained during the NATO--CNR-Fellowship of
the third author at the Department of Mathematics, University of Catania,
Italy within the 1998 NATO--CNR Guest Fellowships Programme.

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\end{document}