\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 218, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/218\hfil Solvability of some Neumann-type problems]
{Solvability of some Neumann-type boundary value problems for biharmonic equations}

\author[V. Karachik, B. Turmetov \hfil EJDE-2017/218\hfilneg]
{Valery Karachik, Batirkhan Turmetov}

\address{Valery Karachik \newline
Department of Mathematical Analysis,
South Ural State University, 454080, \newline
Chelyabinsk, Russia}
\email{valerykarachik@mail.ru}

\address{Batirkhan Turmetov (corresponding author) \newline
Department of Mathematics,
Akhmet Yassawi University,
161200, Turkestan, \newline
Kazakhstan}
\email{turmetovbh@mail.ru}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted July 28, 2017. Published September 13, 2017.}
\subjclass[2010]{35J40, 31B30}
\keywords{Neumann-type boundary value problem; biharmonic equation; 
\hfill\break\indent periodic boundary conditions}

\begin{abstract}
 We study some boundary-value problems for inhomogeneous biharmonic equation 
 with periodic boundary conditions. These problems are generalization to periodic
 data of the Neumann-type boundary-value problems considered before by the authors.
 We obtain existence and uniqueness of solutions for the problems under 
 consideration.
\end{abstract}

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


\section{Introduction}

Many stationary processes occurring in physics and mechanics are described
 by equations of elliptic type. One of the important particular cases of
fourth-order elliptic equations is the biharmonic equation.
Solution of the plane deformation problems in elasticity theory in many
cases can be reduced to the integration of biharmonic equations under
corresponding boundary conditions. In addition, many problems of continuous
media mechanics can be reduced to the solution of harmonic and biharmonic equations.
However, convenient analytic expressions for the solutions of these problems
are obtained only for domains of particular forms.

Applications of biharmonic problems in mechanics and physics are described
in numerous investigations (see, for example, \cite{Andersson,Lai,Selvadurai}).
Applications of boundary value problems for biharmonic equations in mechanics
and physics stimulate the study of various boundary value problems for
biharmonic equations. One of the well known boundary value problems for biharmonic
equations is the Dirichlet problem
\cite{Begehr,Gomilko,Kal'menov,Karachik1,Karachik2,Karachik3,Yaginuma}.
Recently other types of boundary value problems for biharmonic equation such
as Riquier problem \cite{Damlakh,Karachik4,Rasulov},
Neumann problem \cite{Bitsadze,Dang,Dezin,Karachik5,Karachik6,Karachik7,
Karachik8,Malakhova,Turmetov1,Turmetov2}, spectral Steklov problem \cite{Gazzola1},
Robin problem \cite{Gomez-Polanco}, generalized Robin problem \cite{Karachik9},
as well as fractional analogous of Neumann problem
\cite{Berdyshev,Turmetov3,Turmetov4,Turmetov5} are begun to investigate intensively.

The theory of polyharmonic (biharmonic) equations and various boundary value
problems for them was described in great detail in \cite{Gazzola2}.
Conditions for the solvability of boundary value problems for elliptic equations
and systems of equations contain the so-called complementing conditions.
It was established that all problems of the given type are Fredholm-type problems.
Therefore, the solvability of these problems for homogeneous boundary conditions
is guaranteed by the orthogonality of the right-hand sides of the equation to
all solutions of the corresponding homogeneous conjugate equation.
In the considered below particular case (biharmonic equation) of the common
problem more detailed results can be obtained. In the present paper a new
class of boundary value problems for inhomogeneous biharmonic equation
${\Delta ^2}u(x) = f(x)$ in the unit ball with periodic boundary conditions
is studied.

This article is organized as follows.
In Section 2 the statement of the main problem \eqref{e2-1}-\eqref{e2-4}
is given. Some preliminary results are cited in Section 3.
The necessary and sufficient conditions for solvability of the
Neumann-type boundary value problems \eqref{e3-1}-\eqref{e3-3}
are given in Theorems \ref{thm3.1}--\ref{thm3.3}.
Auxiliary integral equalities are derived in Lemmas \ref{lem4.1}--\ref{lem4.4} 
In Section 5 uniqueness conditions for the main
problem \eqref{e2-1}-\eqref{e2-4} are given: in case of $k = 1$ in
 Theorem \ref{thm5.1} and in case of $k = 2$ in Theorem \ref{thm5.2}. 
The necessary and
 sufficient existence conditions for the problem \eqref{e2-1}-\eqref{e2-4}
 are obtained in Section 6: in case of $k = 1$ in Theorem \ref{thm6.1}
and in case of $k = 2$ in Theorem \ref{thm6.2}.

\section{Statement of the problem}

Let $\Omega  = \{ x \in \mathbb{R}^n:|x| < 1\} $ be the unit ball,
$n \ge 2$, and $\partial \Omega  = \{ {x \in {\mathbb{R}^n}:|x| = 1} \}$
be the unit sphere. Denote
$\partial {\Omega _ + } = \{ x \in \partial \Omega :{x_n} \ge 0\} $,
$\partial {\Omega _ - } = \{ x \in \partial \Omega :{x_n} \le 0\} $ and
$I =  \cap \{ x \in \partial \Omega :{x_n} = 0\} $. To each point
$x \in \Omega $ we associate the``opposite" point $x^* = \bar \alpha x$,
where $\bar \alpha  = ({\alpha _1},{\alpha _2},\dots,{\alpha _n})$ with
${\alpha _n} =  - 1$ and the other ${\alpha _j}$, $j = 1,2,\dots, n-1$
take one of the values $ \pm 1$. Obviously, if $x \in \partial {\Omega _ + }$
then $x^* \in \partial {\Omega _ - }$. Further, let $\nu $ be the unit normal
to $\partial \Omega $ and $D_\nu ^m = \frac{{{\partial ^m}}}{{\partial {\nu ^m}}}$
($m \ge 1$) be the normal derivative of order $m$.

In the domain $\Omega $ for $k = 1,2$ consider the following boundary value
problems:
\begin{gather}\label{e2-1}
{\Delta ^2}u(x) = f(x),\;x \in \Omega,\\
\label{e2-2}
D_\nu ^mu(x) = g(x),\;x \in \partial \Omega, \\
\label{e2-3}
D_\nu ^{{\ell _1}}u(x) - {( - 1)^k}D_\nu ^{{\ell _1}}u(x^*)
= {g_1}(x),\;x \in \partial {\Omega _ + }, \\
\label{e2-4}
D_\nu ^{{\ell _2}}u(x) + {( - 1)^k}D_\nu ^{{\ell _2}}u(x^*)
= {g_2}(x),\;x \in \partial {\Omega _ + },
\end{gather}
where $1 \le m \le 3$, $1 \le {\ell _1} < {\ell _2} \le 3$, $\ell _1 \ne m$,
$\ell _2 \ne m$.

By a solution of the problem \eqref{e2-1}-\eqref{e2-4} we mean a function
$u(x) \in {C^4}(\Omega ) \cap {C^3}(\bar \Omega )$, which satisfies the
conditions \eqref{e2-1}-\eqref{e2-4} in the classical sense.

Let ${\partial ^\beta }
 = \frac{{{\partial ^{|\beta |}}}}{{\partial x_1^{{\beta _1}}\dots
\partial x_n^{{\beta _n}}}}$, where
$\beta  = \left( {{\beta _1},\dots,{\beta _n}} \right)$
is the multi-index, $|\beta|  =  {{\beta _1+}\dots{+\beta _n}} $ and
${\partial ^0} = I$ is the unit operator.

It is obvious that the necessary condition for existence of the solution
to the problem \eqref{e2-1}-\eqref{e2-4} belonging to the class
${C^3}(\bar \Omega )$ are the following compatibility conditions:
\begin{gather}\label{e2-5}
{\partial ^\beta }{g_1}(\tilde x,0) + {( - 1)^k}{\partial ^\beta }{g_1}
(\tilde \alpha \tilde x,0) = 0,\quad | \beta | \le p, \\
\label{e2-6}
{\partial ^\beta }{g_2}(\tilde x,0) - {( - 1)^k}{\partial ^\beta }{g_2}
 (\tilde \alpha \tilde x,0) = 0,\quad | \beta  | \le q,
\end{gather}
where $\tilde x = ({x_1},\dots,{x_{n - 1}})$,
$\tilde \alpha  = ({\alpha _1},\dots,{\alpha _{n - 1}})$, $p$ and
$q$ take the values $0, 1, 2, 3$ depending on the order of the boundary
operators $D_\nu ^{{\ell_1}}$ and $D_\nu ^{{\ell_2}}$.
Note that analogous problems for the Poisson equation were investigated
in \cite{Sadybekov1,Sadybekov2,Sadybekov3,Turmetov5}.

\section{Preliminary results}

In this section we consider the following Neumann-type problems:
\begin{equation}\label{e3-1}
{\Delta ^2}u(x) = f(x),\;x \in \Omega,
\end{equation}
\begin{equation}\label{e3-2}
D_\nu ^{{m_1}}u(x) = {g_1}(x),\;x \in \partial \Omega,
\end{equation}
\begin{equation}\label{e3-3}
D_\nu ^{{m_2}}u(x) = {g_2}(x),\;x \in \partial \Omega,
\end{equation}
where $1 \le {m_1} < {m_2} \le 3$.

Problems \eqref{e3-1}-\eqref{e3-3} for different values of ${m_1}$ and ${m_2}$
are studied in
\cite{Bitsadze,Dang,Dezin,Karachik5,Karachik6,Karachik7,Karachik8,Malakhova,
Turmetov1,Turmetov2}. The case $f(x) = 0,\;{m_1} = 1,\;{m_2} = 2$ was considered
by Bitsadze in \cite{Bitsadze}. It was established that the necessary and
sufficient condition for solvability of the problem \eqref{e3-1}-\eqref{e3-3}
have the form
\[
\int_{\partial \Omega } {[{g_2}(x) - {g_1}(x)]d{S_x}}  = 0.
\]
Further in \cite{Karachik5} the following statement is established.

\begin{theorem} \label{thm3.1}
Let ${m_1} = 1$, ${m_2} = 2$, $f(x) \in C(\bar \Omega )$,
${g_1}(x) \in {C^1}(\partial \Omega )$, ${g_2}(x) \in C(\partial \Omega )$.
Then for solvability of the problem \eqref{e3-1}-\eqref{e3-3} it is necessary
and sufficient that the  following condition be fulfilled
\begin{equation}\label{e3-4}
\int_{\partial \Omega } {\left[ {{f_2}(x) - {f_1}(x)} \right]d{S_x}}
= \frac{1}{2}\int_{    \Omega } {\left( {1 - |x{|^2}} \right)f(x)dx} .
\end{equation}
If a solution of the problem exists then it is unique up to a constant term.
\end{theorem}

The case ${m_1} = 2$, ${m_2} = 3$ is investigated in \cite{Turmetov1}.
The following statement is proved.

\begin{theorem} \label{thm3.2}
Let ${m_1} = 2$, ${m_2} = 3$, $f(x) \in {C^{\lambda  + 1}}(\bar \Omega )$,
 ${g_1}(x) \in {C^{\lambda  + 2}}(\partial \Omega )$,
${g_2}(x) \in {C^{\lambda  + 1  }}(\partial \Omega )$.
Then for solvability of  problem \eqref{e3-1}-\eqref{e3-3} it is necessary
and sufficient that the following conditions be fulfilled
\begin{gather}\label{e3-5}
\int_{\partial \Omega } {{g_2}(x)\,d{S_x}}
= \frac{{n - 1}}{2}\int_\Omega  {|x{|^2}f(x)\,dx}
- \frac{{n - 3}}{2} \int_\Omega  f(x)\,dx, \\
\label{e3-6}
\int_{\partial \Omega } {{x_j}[{g_2}(x) - {g_1}(x)]\,d{S_x}}
= \frac{{n - 1}}{2}\int_\Omega {x_j}|x|^2f(x)\,dx
- \frac{{n - 3}}{2}\int_\Omega  {x_j}f(x)\,dx
\end{gather}
for $j=1,\dots,n$. If solution of the problem exists,
then it is unique up to the first order polynomials.
\end{theorem}

In \cite{Karachik7} the following statement is obtained.

\begin{theorem} \label{thm3.3}
Let  $m_1 = 1$, $m_2 = 3$, $f(x) = C(\bar \Omega )$,
${g_1}(x) \in C(\partial \Omega )$, ${g_2}(x) \in C(\partial \Omega )$.
Then for solvability of problem \eqref{e3-1}-\eqref{e3-3} it is necessary
 and sufficient that the following condition be fulfilled
\begin{equation}\label{e3-7}
\int_{\partial \Omega } {{g_2}(x)d{S_x}} = \frac{{n - 1}}{2}\int_\Omega  {|x{|^2}f(x)\,dx} - \frac{{n - 3}}{2}\int_\Omega  f(x)\,dx.
\end{equation}
If a solution of the problem exists, then it is unique up to a constant term.
\end{theorem}

\section{Auxiliary integral equalities}

In what follows we need some integral equalities.
Let $f(x) \in C(\bar \Omega )$, ${g_1}(x) \in C(\partial \Omega )$,
${g_2}(x) \in C(\partial \Omega )$. Denote
\begin{equation}\label{e4-0}
\begin{gathered}
{f^ \pm }(x) = \frac{{f(x) \pm f(x^*)}}{2},\;{g^ \pm }(x)
= \frac{{g(x) \pm g(x^*)}}{2},{\rm{ }}\\
{\tilde g^\pm }(x) =\frac{1}{2} \begin{cases}
g(x), &x \in \partial {\Omega_+ },\\
\pm g(x^*), &x \in \partial {\Omega_-}\,.
\end{cases}
\end{gathered}
\end{equation}

\begin{lemma}\label{lem4.1}
Let $f(x) = C(\bar \Omega )$ and $g(x) \in C(\partial \Omega )$. Then
\begin{gather}\label{e4-1}
\int_\Omega  {f(x^*)\,dx = } \int_\Omega  {f(x)\,dx},\\
\label{e4-2}
\int_{\partial \Omega } {g(x^*)dx = } \int_{\partial \Omega } {g(x)dx} .
\end{gather}
\end{lemma}

\begin{proof}
Let $\bar \alpha =({\alpha _1},{\alpha _2},\dots,{\alpha _n})$, where
 ${\alpha _n} =  - 1$ and the other ${\alpha _j}$, $j = \overline {1,n - 1}$
take one of the values $ \pm 1$. Consider the matrix
\[
P = \begin{pmatrix}
{{\alpha _1}}& \ldots &0\\
 \vdots & \ddots & \vdots \\
0& \cdots &{{\alpha _n}}
\end{pmatrix} .
\]

It is obvious that ${P^T} = P$ and $P \cdot {P^T} = E$.
Consequently, $P$ is an orthogonal matrix. It is known (see e.g. \cite{Baker}),
that if $P$ is an orthogonal matrix, then
$$
\int_\Omega  {f(Px)dx = } \int_\Omega  {f(x)dx} ,\quad
\int_{\partial \Omega } {g(Px)d{S_x} = } \int_{\partial \Omega } {g(x)d{S_x}} .
$$

Since $x^* = Px$ then we obtain \eqref{e4-1} and \eqref{e4-2}.
\end{proof}

\begin{corollary} \label{coro4.1}
Let  $f(x) \in C(\bar \Omega )$ and $g(x) \in C(\partial \Omega )$.
Then the following equalities hold:
\begin{gather}\label{e4-3}
\int_\Omega  {{f^ + }(x)dx = } \int_\Omega  {f(x)dx} ,\quad
\int_\Omega  {|x{|^2}{f^ + }(x)dx = } \int_\Omega  {|x{|^2}f(x)dx} , \\
\label{e4-4}
\int_\Omega  {{f^ - }(x)dx = } 0,\int_\Omega  {|x{|^2}{f^ - }(x)dx = 0} , \\
\label{e4-5}
\int_{\partial \Omega } {{g^ + }(x)d{S_x} = } \int_{\partial \Omega } {g(x)d{S_x}} , \\
\label{e4-6}
\int_{\partial \Omega } {{g^ - }(x)d{S_x} = } 0.
\end{gather}
\end{corollary}

\begin{proof}
By the definition of the functions ${f^ \pm }(x)$ we  obtain
$$
\int_\Omega  {{f^ \pm }(x)\,dx = } \frac{1}{2}\int_\Omega  {f(x)\,dx}
 \pm \frac{1}{2}\int_\Omega  f(x^*)\,dx
=  \frac{1}{2}\int_\Omega  {f(x)dx \pm \frac{1}{2}} \int_\Omega  {f(x)\,dx} .
$$
Further, since $|x| = |x^*|$ then \eqref{e4-1} implies equalities \eqref{e4-3}
 and \eqref{e4-4}. Equalities \eqref{e4-5} and \eqref{e4-6} can be proved similarly.
\end{proof}

\begin{lemma}\label{lem4.2}
Let $g(x) \in C(\partial \Omega )$. Then the following equality holds
\begin{equation}\label{e4-7}
\int_{\partial {\Omega _ - }} g(x^*)\,d{S_x}
=  \int_{\partial {\Omega _ + }} {g(x)\,d{S_x}}.
\end{equation}
\end{lemma}

\begin{proof}
To prove \eqref{e4-7} we pass to the spherical coordinate system:
\begin{gather*}
{x_1} = \cos {\theta _1},\quad
{x_2} = \sin {\theta _1}\cos {\theta _2},\dots,\;{x_{n - 1}}
= \sin {\theta _1}\dots\sin{\theta _2}\cos {\theta _{n - 1}},\\
{x_n} = \sin {\theta _1}\sin{\theta _2}\dots\sin {\theta _{n - 2}}
\sin{\theta _{n - 1}},
\end{gather*}
where
\[
0 \le {\theta _j} \le \pi ,\quad
j = 1,2,\dots,\;
n - 2,\; 0 \le {\theta _{n - 1}} \le 2\pi .
\]
The Jacobian of this mapping has the form
\[
J(\theta ) = \sin^{n - 2}{\theta _1}{\sin ^{n - 3}}{\theta _2}
\dots\sin {\theta _{n - 1}}.
\]
Furthermore, we use the following elementary equalities:
\[
\cos(\pi  \pm \theta ) =  - \cos\theta,\quad
\sin(\pi  \pm \theta ) =  \pm \sin \theta .
\]
Since $\partial {\Omega _ - }
= \partial \Omega  \cap \{ x \in \mathbb{R}^n:{x_n} \le 0\}$ if and only if
$\pi \le {\theta _{n - 1}} \le 2\pi$, $0 \le {\theta _j} \le \pi$,
$j = 1,2,\dots,n - 2$, then
\begin{align*}
\int_{\partial {\Omega _ - }} {g(x^*)d{S_x}}
&= \int_0^\pi  {d{\theta _1}} \dots\int_0^\pi  {d{\theta _{n - 2}}}
\int_\pi ^{2\pi } g({\alpha _1}\cos{\theta _1},{\alpha _2}\sin{\theta _1}
\cos{\theta _2},\dots,\\
&\quad  - \sin{\theta _1}\dots\sin{\theta _{n - 1}})J(\theta )d{\theta _{n - 1}}.
\end{align*}

Let us make the change of variables in the last integral,
\begin{gather*}
{\theta _j} = \begin{cases}
\pi - {\xi _j}, &{\alpha _j} =  - 1\\
{\xi _j}, &{\alpha _j} = 1,\,j = 1,2,\dots,n - 2,
\end{cases}, \\
{\theta _{n - 1}} = \pi  + {\xi _{n - 1}}.
\end{gather*}

Note that under this change of variables we obtain the equality
(if ${\alpha _j} =  - 1$, $j = 1,2,\dots,n -2$)
\begin{align*}
J(\xi ) &= \sin^{n - 2}(\pi  - {\xi _1})\sin^{n - 3}(\pi  - {\xi _2})\dots
 \sin(\pi  - {\xi _{n - 2}}) \\
&= \sin^{n - 2}{\xi _1}\sin^{n - 3}{\xi _2}\dots\sin{\xi _{n - 2}},
\end{align*}
i.e. the Jacobian's sign is not changed. Further, since
\begin{gather*}
 - \cos{\theta _1} =  - \cos (\pi  - {\xi _1}) = \cos{\xi _1},\\
 - \sin {\theta _j} =  - \sin (\pi  - {\xi _j})
 =  - \cos \pi \sin {\xi _j} = \sin{\xi _j},
\end{gather*}
then after the change of variables we have
\begin{align*}
\int_{\partial {\Omega _ - }} {g(x^*)\,d{S_x} }
&= \int_0^\pi  {d{\xi _1}} \dots\int_0^\pi  {d{\xi _{n - 2}}} \int_0^\pi
 g(\cos{\xi _1},\sin{\xi _1}\cos{\xi _2},\dots,\\
&\quad \sin{\xi _1}\dots\sin{\xi _{n - 1}})J(\xi )\,d{\xi _{n - 1}} \\
&=\int_{\partial {\Omega _ + }} {g(x)\,d{S_x}}.
\end{align*}
\end{proof}

\begin{corollary} \label{coro4.2}
 Let $g(x) \in C(\partial \Omega )$. Then
\begin{gather}\label{e4-8}
\int_{\partial \Omega } {{{\tilde g}^ + }(x)d{S_x}
= } \int_{\partial {\Omega _ + }} {g(x)d{S_x}} , \\
\label{e4-9}
\int_{\partial \Omega } {{{\tilde g}^ - }(x)d{S_x} = } 0.
\end{gather}
\end{corollary}

\begin{proof}
Using definition of the function ${\tilde g^ + }(x)$ and the equality
\eqref{e4-7}, we have
\begin{align*}
\int_{\partial \Omega } {\tilde g}^ + (x) d{S_x}
&=  \frac{1}{2}\int_{\partial {\Omega _ + }} g(x)d{S_x}
 + \frac{1}{2} \int_{\partial {\Omega _ - }} g(x^*)d{S_x} \\
&= \frac{1}{2} \int_{\partial {\Omega _ + }} g(x)d{S_x}
 + \frac{1}{2} \int_{\partial {\Omega _ + }} g(x)d{S_x}\\
& = \int_{\partial {\Omega _ + }} g(x)d{S_x} .
\end{align*}

Similarly, we obtain
\begin{align*}
\int_{\partial \Omega } {\tilde g}^ - (x)d{S_x}
& = \frac{1}{2}\int_{\partial {\Omega _ + }} g(x)d{S_x}
- \frac{1}{2}  \int_{\partial {\Omega _ - }} g(x^*)d{S_x} \\
&= \frac{1}{2} \int_{\partial {\Omega _ + }} g(x)d{S_x}
 - \frac{1}{2} \int_{\partial {\Omega _ + }} g(x)d{S_x} = 0 .
\end{align*}
\end{proof}

\begin{lemma}\label{lem4.3}
Let $f(x) \in C(\partial \bar \Omega )$. Then
\begin{equation}\label{e4-10}
\int_\Omega  {x_j}f(x^*)dx = {\alpha _j} \int_\Omega  {x_j}f(x)dx,\quad
j = 1,2,\dots,n
\end{equation}
\end{lemma}

\begin{proof}
Since
$x^* = ({\alpha _1}{x_1},{\alpha _2}{x_2},\dots,
{\alpha _{n - 1}}{x_{n - 1}}, - {x_n})$, it follows that
\begin{align*}
\int_\Omega  {{x_j}f(x^*)dx } 
&= \int_{ - 1}^1 \int_{ - \sqrt {1 - x_1^2} }^{\sqrt {1 - x_1^2} }
\dots\int_{ - \sqrt {1 - x_1^2
- \dots - x_{n - 1}^2} }^{\sqrt {1 - x_1^2 - \dots
- x_{n - 1}^2} } {x_j}f({\alpha _1}{x_1},{\alpha _2}{x_2},
\dots,\\
&\quad {\alpha _{n - 1}}{x_{n - 1}}, - {x_n})d{x_n}   \dots d{x_1}.
\end{align*}
In the above integral we make the change of variables
${y_k} = {\alpha _k}{x_k},\;k = 1,2,\dots,n$, where ${\alpha _n} =  - 1$. Then
\begin{align*}
&\int_\Omega  {x_j}f(x^*)dx \\
&=  \int_{ - {\alpha _1}}^{{\alpha _1}} {\int_{ - {\alpha _2}
 \sqrt {1 - y_1^2} }^{{\alpha _2}\sqrt {1 - y_1^2} }
 {\dots\int_{ - {\alpha _n}\sqrt {1 - y_1^2 - \dots
- y_{n - 1}^2} }^{{\alpha _n}\sqrt {1 - y_1^2 - \dots
- y_{n - 1}^2} } {\frac{{{y_j}}}{{{\alpha _j}}}} } }
f({y_1},\dots ,{y_{n - 1}},{y_n})\frac{{d{y_n}}}{{{\alpha _n}}}
\dots\frac{{d{y_1}}}{{{\alpha _1}}} \\
&= {\alpha _j}\int_{ - 1}^1 {\int_{ - \sqrt {1 - y_1^2} }
 ^{\sqrt {1 - y_1^2} } {\dots\int_{ - \sqrt {1 - y_1^2 - \dots
 - y_{n - 1}^2} }^{\sqrt {1 - y_1^2 - \dots - y_{n - 1}^2} } {{y_j}} } }
 f({y_1},\dots,{y_{n - 1}},{y_n})d{y_n}\dots d{y_1}\\
& = {\alpha _j} \int_\Omega  {{x_j}f(x)dx}.
\end{align*}
\end{proof}

\begin{corollary} \label{coro4.3}
If $f(x) \in C(\bar \Omega )$, then for all $j = 1,2,\dots,n$ we have
\begin{gather}\label{e4-11}
\int_\Omega  {{x_j}{f^ + }(x)dx
= \frac{{1 + {\alpha _j}}}{2}} \int_\Omega  {{x_j}f(x)dx,} \\
\label{e4-12}
\int_\Omega  {{x_j}{f^ - }(x)dx
= \frac{{1 - {\alpha _j}}}{2}} \int_\Omega  {{x_j}f(x)dx.}
\end{gather}
\end{corollary}

\begin{proof}
Using \eqref{e4-9} and according to definition \eqref{e4-0} of function
${f^ + }(x)$ we obtain
\begin{align*}
&\int_\Omega  {{x_j}} {f^ + }(x)dx \\
&= \frac{1}{2}\int_\Omega  {{x_j}f(x)\,dx
 + \frac{1}{2}\int_\Omega  {{x_j}f(x^*)\,dx} }  \\
&= \frac{1}{2}\int_\Omega  {{x_j}f(x)\,dx}
 +\frac{{{\alpha _j}}}{2}\int_\Omega  {{x_j}f(x)\,dx} \\
& = \frac{{1 + {\alpha _j}}}{2}\int_\Omega  {{x_j}f(x)\,dx} .
\end{align*}
Similarly we can obtain
\[
\int_\Omega  {{x_j}} {f^ - }(x)\,dx = \frac{1}{2}\int_\Omega  {{x_j}f(x)\,dx
 - \frac{1}{2}\int_\Omega  {{x_j}f(x^*)\,dx} }
= \frac{{1 - {\alpha _j}}}{2}\int_\Omega  {{x_j}f(x)\,dx} .
\]
\end{proof}

\begin{lemma}\label{lem4.4}
Let $g(x) \in C(\partial \Omega )$. Then for $j = 1,2,\dots,n$,
\begin{gather}\label{e4-13}
\int_{\partial \Omega } {{x_j}g(x^*)\,d{S_x}
= } {\alpha _j}\int_{\partial \Omega } {{x_j}g(x)\,d{S_x}}, \\
\label{e4-14}
\int_{\partial {\Omega _ - }} {{x_j}g(x^*)\,d{S_x}
= } {\alpha _j}\int_{\partial {\Omega _ + }} {{x_j}g(x)\,d{S_x}}.
\end{gather}
\end{lemma}

\begin{proof}
To prove this statement we pass to the spherical coordinate system
(see Lemma \ref{lem4.2}). Then we obtain
\begin{align*}
\int_{\partial \Omega } {{x_j}g(x^*)\,d{S_x}} 
&= \int_0^\pi  d{\theta _1} \dots\int_0^\pi  d{\theta _{n - 2}}
 \int_0^\pi  \sin{\theta _1}\dots \sin{\theta _{j - 1}}\cos{\theta _j} \\
&\quad \times g({\alpha _1}\cos{\theta _1},\dots,
 - \sin{\theta _1}\dots \sin{\theta _{n - 1}})J(\theta ) d{\theta _{n - 1}}\\
&\quad + \int_0^\pi  {d{\theta _1}} \dots\int_0^\pi {d{\theta _{n - 2}}}
 \int_\pi ^{2\pi } \sin{\theta _1}\dots \sin{\theta _{j - 1}}\cos{\theta _j} \\
&\quad \times g({\alpha _1}\cos{\theta _1},\dots,
 - \sin{\theta _1}\dots\sin{\theta _{n - 1}})J(\theta ) d{\theta _{n - 1}}.
\end{align*}
For the first integral we make the change of variables
\begin{gather*}
{\theta _j}
=  \begin{cases}
\pi  - {\xi _j}, & {\alpha _j} =  - 1\\
{\xi _j},{\alpha _j} = 1, & j = 1,2,\dots,n - 2,
\end{cases} \\
{\theta _{n - 1}} = {\xi _{n - 1}} - \pi ,
\end{gather*}
and use the equality ${\theta _{n - 1}} = {\xi _{n - 1}} + \pi$
 for the second integral. Note that under these changes of variables we have
\begin{gather*}
\sin {\theta _k} = \sin (\pi  - {\xi _k})
= \sin\pi \cos{\xi _k} - \cos \pi \sin {\xi _k} = \sin {\xi _k},\;k \le n - 2,\\
\sin {\theta _{n - 1}} = \sin ({\xi _{n - 1}} - \pi )
= \sin{\xi _{n - 1}}\cos\pi  - \cos {\xi _{n - 1}}\sin \pi
=  - \sin {\xi _{n - 1}},
\end{gather*}
or\begin{gather*}
\sin {\theta _{n - 1}} = \sin ({\xi _{n - 1}} + \pi )
 = \sin{\xi _{n - 1}}\cos\pi  + \cos{\xi _{n - 1}}\sin \pi
 =  - \sin {\xi _{n - 1}},\\
\cos {\theta _k} = \cos (\pi  - {\xi _k})
= \cos \pi \cos{\xi _k} - \sin \pi \sin {\xi _k} =  - \cos {\xi _k}.
\end{gather*}
Consequently for the monomial ${x_j}$ we obtain:
\begin{itemize}
\item[(a)] if ${\alpha _j} =  - 1$, then
\[
{x_j} \to \sin {\theta _1}\dots
 \sin{\theta _{j - 1}}\cos {\theta _j} \to \sin {\xi _1}
\dots\sin {\xi _{j - 1}}( - \cos{\xi _j}) \to {\alpha _j}{x_j};
\]

\item[(b)] if ${\alpha _j} =  + 1$, then
\[
{x_j} \to {\alpha _j}\sin {\xi _1}\dots\sin{\xi _{j - 1}}\cos {\xi _j}
\to {\alpha _j}{x_j}.
\]
Thus we have the equality
\begin{align*}
&\int_{\partial \Omega } {{x_j}} g(x^*)\,d{S_x} \\
&= \int_0^\pi  {d{\theta _1}} \dots\int_0^\pi  {d{\theta _{n - 2}}}
\int_0^\pi \sin{\theta _1}\dots\sin{\theta _{j - 1}}\cos{\theta _j} \\
&\quad \times g({\alpha _1}\cos{\theta _1},\dots,
- \sin{\theta _1}\dots\sin{\theta _{n - 1}})J(\theta ) d{\theta _{n - 1}} \\
&\quad + \int_0^\pi  {d{\theta _1}} \dots\int_0^\pi  {d{\theta _{n - 2}}}
 \int_\pi ^{2\pi } \sin{\theta _1} \dots\sin{\theta _{j - 1}}\cos{\theta _j} \\
&\quad \times g({\alpha _1}\cos{\theta _1},\dots, - \sin{\theta _1}\dots
\sin{\theta _{n - 1}})J(\theta ) d{\theta _{n - 1}} \\
&= \int_0^\pi  {d{\xi _1}} \dots\int_0^\pi  {d{\xi _{n - 2}}}
\int_\pi ^{2\pi } \sin{\xi _1}\dots\sin{\xi _{j - 1}} \cdot \cos{\xi _j} \\
&\quad \times g(\cos{\xi _1},\dots,\sin{\xi _1}\dots\sin{\xi _{n - 1}})J(\xi )
d{\xi _{n - 1}} \\
&\quad + {\alpha _j}\int_0^\pi  {d{\xi _1}} \dots\int_0^\pi  {d{\xi _{n - 2}}}
 \int_0^\pi  \sin{\xi _1}\dots\sin{\xi _{j - 1}} \cdot \cos{\xi _j}
g(\cos{\xi _1},\dots, \\
&\quad \sin{\xi _1}\dots\sin{\xi _{n - 1}})J(\xi ) d{\xi _{n - 1}}\\
&= {\alpha _j}\int_{\partial \Omega } {{x_j}} g(x)\,d{S_x}.
\end{align*}
\end{itemize}
Thus the equality \eqref{e4-13} is proved.
Consider the equality \eqref{e4-14}. In this case we have
\begin{align*}
&\int_{\partial {\Omega _ - }} {{x_j}} g(x^*)\,d{S_x} \\
&= \int_0^\pi  {d{\theta _1}} \dots\int_0^\pi  {d{\theta _{n - 2}}}
 \int_\pi ^{2\pi } \sin{\theta _1} \dots\sin{\theta _{j - 1}}\cos{\theta _j} \\
&\quad \times g({\alpha _1}\cos{\theta _1},\dots,
 - \sin{\theta _1}\dots\sin{\theta _{n - 1}})J(\theta ) d{\theta _{n - 1}} \\
&= {\alpha _j}\int_0^\pi  {d{\xi _1}} \dots\int_0^\pi  {d{\xi _{n - 2}}}
 \int_\pi ^{2\pi } \sin{\xi _1} \dots\sin{\xi _{j - 1}}\cos{\xi _j} \\
&\quad \times g({\alpha _1}\cos{\xi _1},\dots,
 - \sin{\xi _1}\dots\sin{\xi _{n - 1}})J(\xi ) d{\xi _{n - 1}}=
{\alpha _j}\int_{\partial {\Omega _ + }} {{x_j}} g(x)\,d{S_x}.
\end{align*}
\end{proof}

\begin{corollary} \label{coro4.4}
Let $g(x) \in C(\partial \Omega )$. Then for $j = 1,2,\dots,n$
the following equalities hold:
\begin{gather}\label{e4-15}
\int_{\partial \Omega } {{x_j}{g^ \pm }(x)d{S_x}
= \frac{{1 \pm {\alpha _j}}}{2}} \int_{\partial \Omega } {{x_j}g(x)\,d{S_x},}\\
\label{e4-16}
\int_{\partial \Omega } {{x_j}{{\tilde g}^ \pm }(x)d{S_x}
= \frac{{1 \pm {\alpha _j}}}{2}} \int_{\partial {\Omega _ + }} {{x_j}g(x)\,d{S_x}} .
\end{gather}
\end{corollary}

\begin{proof}
According to Lemma \ref{lem4.4} we have
$$
\int_{\partial \Omega } {{x_j}g(x^*)}\, d{S_x}
= {\alpha _j}\int_{\partial \Omega } {{x_j}g(x)}\, d{S_x}.
$$
Therefore, using \eqref{e4-0} we obtain
\begin{align*}
\int_{\partial \Omega } {{x_j}{g^ \pm }(x)}\, d{S_x}
&= \frac{1}{2}\Big[ {\int_{\partial \Omega } {{x_j}g(x)}\, d{S_x}
\pm \int_{\partial {\Omega}} {{x_j}g(x^*)}\, d{S_x}} \Big]\\
& = \frac{1}{2}\Big[ {\int_{\partial \Omega } {{x_j}g(x)\,d{S_x}}
 \pm {\alpha _j}\int_{\partial {\Omega}} {{x_j}g(x)} \, d{S_x}} \big] \\
& = \frac{{1 \pm {\alpha _j}}}{2}\int_{\partial \Omega } {{x_j}g(x)}\, d{S_x}.
\end{align*}
Similarly we can get
\begin{align*}
\int_{\partial \Omega }{{x_j}{\tilde g^ \pm }(x)}\, d{S_x} 
&= \frac{1}{2}\Big[ {\int_{\partial \Omega_+} {{x_j}g(x)}\, d{S_x}
 \pm \int_{\partial {\Omega_-}} {{x_j}g(x^*)}\, d{S_x}} \Big] \\
&= \frac{1}{2}\Big[ {\int_{\partial \Omega_+} {{x_j}g(x)\,d{S_x}}
  \pm {\alpha _j}\int_{\partial {\Omega_+}} {{x_j}g(x)} \, d{S_x}} \Big] \\
&= \frac{{1 \pm {\alpha _j}}}{2}\int_{\partial \Omega_+} {{x_j}g(x)}\, d{S_x}.
\end{align*}
\end{proof}

\begin{remark} \label{rmk4.5} \rm
Since ${\alpha_n}= -1$, ti follows that \eqref{e4-15} and \eqref{e4-16} imply
\begin{gather*}
\int_{\partial \Omega } {{x_n}{g^ + }(x)} d{S_x} = 0,\quad
\int_{\partial \Omega } {{x_n}{g^ - }(x)} d{S_x}
 = \int_{\partial \Omega } {{x_n}g(x)} d{S_x},\\
\int_{\partial \Omega } {{x_n}{{\tilde g}^ + }(x)} d{S_x} = 0,\quad
\int_{\partial \Omega } {{x_n}{{\tilde g}^ - }(x)} d{S_x}
= \int_{\partial {\Omega _ + }} {{x_n}g(x)} d{S_x}.
\end{gather*}
\end{remark}

\section{Uniqueness conditions}

In this section we study uniqueness of solutions of the problems
\eqref{e2-1}-\eqref{e2-4}.

\begin{theorem} \label{thm5.1}
Let $k = 1$ and solution of problem \eqref{e2-1}-\eqref{e2-4} exist. Then

(1) in the case $m = 1$, ${\ell_1} = 2$, $\ell_2 = 3$,
the solution of  problem \eqref{e2-1}-\eqref{e2-4} is unique up to constant term;

(2) if $m = 2$, $\ell_1 = 1$, $\ell_2 = 3$, or $m = 3$, $\ell_1 = 1$,
$\ell_2 = 2$, then the following cases are possible:
\begin{itemize}
\item[(a)] if for all $1 \le j \le 1,\;{\alpha _j} =  - 1$,
then the solution of homogeneous problem \eqref{e2-1}-\eqref{e2-4}
is a function of the form
$$
u(x) = {c_0} + \sum_{j = 1}^n {{c_j}} {x_j};
$$

\item[(b)] if for some ${j_0} \in \{ 1,2,\dots,n - 1\}$ the equality
${\alpha _{{j_0}}} = 1$ holds, then solution of the homogeneous
problem \eqref{e2-1}-\eqref{e2-4} is a function of the form
$$
u(x) = {c_0} + \sum_{j = 1,j \ne {j_0}}^n {{c_j}} {x_j}.
$$
\end{itemize}
In particular, if ${\alpha _j} = 1$, $1 \le j \le n - 1$ then
$u(x) = {c_0} + {c_n}{x_n}$.
\end{theorem}

\begin{proof}
Let $k = 1$ and function $u(x)$ is a solution of the homogeneous
 problem \eqref{e2-1}-\eqref{e2-4}. Then $u(x)$ is a biharmonic function
that satisfies boundary conditions:
\begin{gather}\label{e5-1}
D_\nu ^mu(x) = 0,\,x \in \partial \Omega, \\
\label{e5-2}
D_\nu ^{{\ell _1}}u(x) =  - D_\nu ^{{\ell _1}}u(x^*),\,x \in \partial {\Omega _ + },\\
\label{e5-3}
D_\nu ^{{\ell _2}}u(x) = D_\nu ^{{\ell _2}}u(x^*),\,x \in \partial {\Omega _ + }.
\end{gather}

If $x \in \partial {\Omega _ - }$ then $x^* \in \partial {\Omega _ + }$
and therefore from the conditions \eqref{e5-2} and \eqref{e5-3} it follows that
$$
D_\nu ^{{\ell _1}}u(x^*) =  - D_\nu ^{{\ell _1}}u(x),\;x \in \partial {\Omega _ - },
\quad
 D_\nu ^{{\ell _2}}u(x^*) = D_\nu ^{{\ell _2}}u(x),\;x \in \partial {\Omega _ - }.
$$
Then for all $x \in \partial \Omega $ the following equalities hold
\begin{gather}\label{e5-4}
D_\nu ^{{\ell _1}}u(x) =  - D_\nu ^{{\ell _1}}u(x^*),\quad x \in \partial \Omega ,\\
\label{e5-5}
D_\nu ^{{\ell _2}}u(x) = D_\nu ^{{\ell _2}}u(x^*),\;x \in \partial \Omega.
\end{gather}

On the other hand differentiating \eqref{e5-4} along the normal $\nu $ give us
\begin{equation}\label{e5-6}
D_\nu ^{{\ell _2}}u(x) =  - D_\nu ^{{\ell _2}}u(x^*),\quad x \in \partial \Omega.
\end{equation}
Then from the equalities \eqref{e5-5} and \eqref{e5-6} it follows that
\[
D_\nu ^{{\ell _2}}u(x) = 0,x \in \partial \Omega .
\]
Thereby the function $u(x)$ is a solution of the problem
\begin{gather}\label{e5-7}
{\Delta ^2}u(x) = 0,\quad x \in \Omega , \\
\label{e5-8}
 {D_\nu ^mu(x)} \big|_{\partial \Omega } = 0,\quad
{\;D_\nu ^{{\ell _2}}u(x)} \big|_{\partial \Omega } = 0.
\end{gather}

Furthermore when we use the results of Section 3.
The following cases are possible:

(1) if $m = 1$, $\ell_2 = 3$, then by Theorem \ref{thm3.3},
the function $u(x) = {c_0} \equiv const$ is a unique solution of the
problem \eqref{e5-7}-\eqref{e5-8}. Obviously, this function satisfies
all conditions of the homogeneous problem \eqref{e2-1}-\eqref{e2-4}
 for $k = 1$. Consequently, if $m = 1, \ell_1 = 2, \ell_2 = 3$
then solution of the homogeneous problem \eqref{e2-1}-\eqref{e2-4} is a
function $u(x) = {c_0}$.

(2) if $m = 2,\ell_2 = 3$ then according to Theorem \ref{thm3.2}, the unique solution of
the homogeneous problem \eqref{e5-7}-\eqref{e5-8} is a function of the form:
$$
u(x) = {c_0} + \sum_{j = 1}^n {{c_j}} {x_j},
$$
where ${c_j}$ are constants, $j = 0,1,\dots,n$.
In this case ${\ell_1} = 1$ and for all $x \in \partial \Omega $,
\[
{ {D_\nu ^1u(x)} \big|_{\partial \Omega }}
= { {r\frac{{\partial u(x)}}{{\partial r}}} \big|_{\partial \Omega }}
= \sum_{i = 1}^n {{x_i}} \frac{{\partial u}}{{\partial {x_i}}}(x).
\]
Since $u(x^*) = {C_0} + \sum_{j = 1}^n {{c_j}} {\alpha _j}{x_j}$, it follows that
\[
0 = D_\nu ^1u(x) +  {D_\nu ^1u(x^*)} |_{\partial \Omega }
= \sum_{j = 1}^n {{c_j}} {x_j} + \sum_{j = 1}^n {{c_j}{\alpha _j}} {x_j}
= \sum_{j = 1}^n {(1 + {\alpha _j}){c_j}} {x_j}.
\]
Further, if for all $1 \le j \le n$: ${\alpha _j} =  - 1$, then ${c_j}$ are
arbitrary numbers and if for some ${j_0} \in \{ 1,2,\dots,n - 1\}$,
${\alpha _{{j_0}}} = 1$ then for the equality
$$
D_\nu ^1u(x) + D_\nu ^1u(x^*) = 0
$$
to be correct it is necessary that ${c_{{j_0}}} = 0$.
Hence, if ${\alpha _j} =  - 1,\;1 \le j \le n$ then the function
$$
u(x) = {c_0} + \sum_{j = 1}^n {{c_j}} {x_j}
$$
is a solution of the homogeneous problem \eqref{e2-1}-\eqref{e2-4}.
If for some ${j_0} \in \{ 1,2,\dots,n - 1\}$, ${\alpha _{{j_0}}} = 1$,
 then the solution of the homogeneous problem \eqref{e2-1}-\eqref{e2-4}
is a function of the form
$$
u(x) = {c_0} + \sum_{j = 1,j \ne {j_0}}^n {{c_j}} {x_j}.
$$
In particular, if ${\alpha _j} = 1$ for all $j = 1,2, \ldots ,n - 1$, then
$$
u(x) = {c_0} + {c_n}{x_n}.
$$

(3) Let $m = 3$, $\ell_1 = 1$, $\ell_2 = 2$. Then as in the case (2)
the function of the form
$$
u(x) = {c_0} + \sum_{j = 1}^n {{c_j}} {x_j}
$$
is a solution of the homogeneous problem \eqref{e5-7}-\eqref{e5-8}.
 Making the same arguments as in the case $m = 2$, $\ell_1 = 1$, $\ell_2 = 3$
we obtain:
If ${\alpha _j} =  - 1,\;1 \le j \le n$ then solution of the homogeneous
 problem \eqref{e2-1}-\eqref{e2-4} is the function
\[
u(x) = {c_0} + \sum_{j = 1}^n {{c_j}} {x_j}.
\]

If for some ${j_0} \in \{ 1,2,\dots,n - 1\}$, ${\alpha _{{j_0}}} = 1$,
then solution of the homogeneous problem \eqref{e2-1}-\eqref{e2-4}
is a function of the form
\[
u(x) = {c_0} + \sum_{j = 1,j \ne {j_0}}^n {{c_j}} {x_j}.
\]
In particular, if ${\alpha _j} = 1$ for all $1 \le j \le n - 1$ then the
solution has the form $u(x) = {c_0} + {c_n}{x_n}$.
\end{proof}

The following statement can be proved similarly.

\begin{theorem} \label{thm5.2}
Let $k = 2$ and a solution of  problem \eqref{e2-1}-\eqref{e2-4} exist. Then

(1) in the case $m = 1, \ell_1 = 2, \ell_2 = 3$ solution
of the problem \eqref{e2-1}-\eqref{e2-4} is unique up to constant term;

(2) if $m = 2, \ell_1 = 1, \ell_2 = 3$ or $m = 3$, $\ell_1 = 1$,
$\ell_2 = 2$, then the following cases are possible:
\begin{itemize}
\item[(a)] if for all $1 \le j \le n - 1$, ${\alpha _j} = 1$
then the solution of the homogeneous problem  \eqref{e2-1}-\eqref{e2-4}
is function of the form:
\[
u(x) = {c_0} + \sum_{j = 1}^{n - 1} {{c_j}} {x_j};
\]

\item[(b)] if for some ${j_0} \in \{ 1,2,\dots,n\}$, $\alpha_{j_0} =  - 1$
 then solution of the homogeneous problem \eqref{e2-1}-\eqref{e2-4} is a
function of the form
$$
u(x) = {c_0} + \sum_{j = 1,j \ne {j_0}}^n {{c_j}{x_j}} .
$$
\end{itemize}
In particular, if ${\alpha _j} =  - 1,\;1 \le j \le n$, then  $u(x) = {c_0}$.
\end{theorem}

\section{Existence conditions}

In this section we present results on existence of a solution of
 problem \eqref{e2-1}-\eqref{e2-4}.

\begin{theorem}\label{thm6.1}
Let $k = 1$ and functions $f(x),\;g(x),\;{g_j}(x)$ $j = 1,2$ be smooth enough
and compatibility conditions \eqref{e2-5}, \eqref{e2-6} be fulfilled.
Then the necessary and sufficient conditions for solvability of
 problem \eqref{e2-1}-\eqref{e2-4} have the following form:

(1) if  $m = 1$, $\ell_1 = 2$, $\ell_2 = 3$, then
\begin{equation}\label{e6-1}
\frac{1}{2}\int_\Omega  {(1 - |x{|^2})f(x)dx
= } \int_{\partial {\Omega _ + }} {{g_1}(x)d{S_x}}
 - \int_{\partial \Omega } {g(x)d{S_x}} ;
\end{equation}

(2) if  $m = 2$, $\ell_1 = 1$, $\ell_2 = 3$, then
\begin{gather}\label{e6-2}
\begin{aligned}
&\frac{1}{2}\int_\Omega  (1 - |x{|^2})f(x)dx \\
&=  \int_{\partial \Omega } {g(x)d{S_x}}  - \int_{\partial {\Omega _ + }}
{{g_1}(x)d{S_x}},, 
\end{aligned}\\
\label{e6-3}
\begin{aligned}
&\frac{{n - 1}}{2}\int_\Omega  {x_j}|x{|^2}f(x)\,dx
- \frac{{n - 3}}{2}\int_{\Omega } {x_j}f(x)\,dx\\
&= \int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)\,d{S_x} - }
 \int_{\partial \Omega } {{x_j}g(x)\,d{S_x}}
\end{aligned}
\end{gather}
for all $j$ such that ${\alpha _j}=- 1$;

(3) if  $m = 3$, $\ell_1 = 1$, $\ell_2 = 2$, then
\begin{gather}\label{e6-4}
\frac{{n - 1}}{2}\int_\Omega |x|^2f(x)\,dx
-\frac{{n - 3}}{2}\int_\Omega  {f(x)\,dx}
= \int_{\partial \Omega } {g(x)\,d{S_x},} \\
\label{e6-5}
\begin{aligned}
&\frac{{n - 1}}{2}\int_\Omega  {x_j}|x|^2f(x)\,dx
- \frac{{n - 3}}{2}\int_{\Omega } {{x_j}f(x)\,dx} \\
& = \int_{\partial \Omega } {x_j}g(x)\,d{S_x}
 - \int_{\partial {\Omega_+ }} {{x_j}{g_2}(x)\,d{S_x},}
\end{aligned}
\end{gather}
for all $j \in \{ 1,2,\dots,n\}$ such that ${\alpha _j}= - 1$.
\end{theorem}

\begin{proof}
We introduce two auxiliary functions
\[
v(x) = \frac{1}{2}[u(x) + u(x^*)],\quad
w(x) = \frac{1}{2}[u(x) - u(x^*)].
\]
It is obvious that $u(x) = v(x) + w(x)$. It is easy to see that the functions
$v(x)$ and $w(x)$ are solutions of the following Neumann-type problems:
\begin{gather}\label{e6-6}
{\Delta ^2}v(x) = {f^ + }(x),\;x \in \Omega ,\;
{ {\;D_\nu ^mv(x)} \big|_{\partial \Omega }}
= {g^ + }(x),\; { {D_\nu ^{{\ell_1}}v(x)} \big|_{\partial \Omega }}
= {\tilde g_1}^ + (x), \\
\label{e6-7}
{\Delta ^2}w(x) = {f^ - }(x),\;x \in \Omega ,\;{ {D_\nu ^mw(x)}
\big|_{\partial \Omega }} = {g^ - }(x),\;{ {D_\nu ^{{\ell_2}}w(x)}
\big|_{\partial \Omega }} = {\tilde g_2}^ - (x).
\end{gather}

Indeed applying the biharmonic operator ${\Delta ^2}$ to the function $v(x)$,
we obtain
\[
{\Delta ^2}v(x) = \frac{1}{2}[{\Delta ^2}u(x) + {\Delta ^2}u(x^*)]
= \frac{1}{2}[f(x) + f(x^*)] = {f^ + }(x),\;x \in \Omega .
\]

Further, taking the boundary conditions \eqref{e2-2},  \eqref{e2-3} into account,
we have
\begin{gather*}
D_\nu ^mv(x) = \frac{1}{2}[D_\nu ^mu(x) + D_\nu ^mu(x^*)]
= \frac{1}{2}[g(x) + g(x^*)] = {g^ + }(x),\quad x \in \partial \Omega ,\\
D_\nu ^{{\ell_1}}v(x) = \frac{1}{2}[D_\nu ^{{l_1}}u(x) + D_\nu ^{{\ell_1}}u(x^*)]
= \frac{1}{2}g(x),\quad x \in \partial {\Omega _ + },\\
\begin{aligned}
D_\nu ^{{\ell_1}}v(x)
&= \frac{1}{2}[D_\nu ^{{\ell_1}}u(x) + D_\nu ^{{\ell_1}}u(x^*)] \\
&= \frac{1}{2}[D_\nu ^{{\ell_1}}u(x^*) + D_\nu ^{{\ell_1}}u(x)]
 = \frac{1}{2}g(x^*),\;x \in \partial {\Omega_- }.
\end{aligned}
\end{gather*}
Similarly, for function $w(x)$ we obtain
\begin{gather*}
{\Delta ^2}w(x) = \frac{1}{2}[{\Delta ^2}u(x) - {\Delta ^2}u(x^*)]
= \frac{1}{2}[f(x) - f(x^*)] = {f^ - }(x),\quad x \in \Omega ,\\
D_\nu ^mw(x) = \frac{1}{2}[D_\nu ^mu(x) - D_\nu ^mu(x^*)]
 = \frac{1}{2}[g(x) - g(x^*)] = {g^ - }(x),\;x \in \partial \Omega ,\\
D_\nu ^{{\ell_2}}w(x) = \frac{1}{2}[D_\nu ^{{\ell_2}}u(x)
 - D_\nu ^{{\ell_2}}u(x^*)] = \frac{1}{2}{g_2}(x),\;x \in \partial {\Omega _ + },\\
\begin{aligned}
D_\nu ^{{\ell_2}}w(x)
&= \frac{1}{2}[D_\nu ^{{\ell_2}}u(x) - D_\nu ^{{\ell_2}}u(x^*)]\\
& =  - \frac{1}{2}[D_\nu ^{{\ell_2}}u(x^*) - D_\nu ^{{\ell_2}}u(x)]
=  - \frac{1}{2}\tilde g_2^ - (x^*),\;x \in \partial {\Omega _ - }.
\end{aligned}
\end{gather*}
Note that if the function $f(x)$ is a smooth enough function defined on the
 domain $\bar \Omega$, and function $g(x)$ is defined on the sphere
$\partial \Omega$, then it is obvious that the functions ${f^ \pm }(x)$ and
${g^ \pm }(x)$ have the same properties. Moreover, if functions ${g_1}(x)$
and ${g_2}(x)$ are smooth on $\partial {\Omega _ + }$, then because of
compatibility conditions \eqref{e2-5}, \eqref{e2-6} the functions
${\tilde g_1}^ \pm (x)$ and ${\tilde g_2}^ \pm (x)$  have the same properties.
Further, to study the solvability of the problems \eqref{e6-6} and
\eqref{e6-7} we use the statements of Theorems \ref{thm3.1}, \ref{thm3.3}.

(1) if $m = 1$, $\ell_1 = 2$, $\ell_2 = 3$ then the necessary and sufficient
conditions for solvability of the problems \eqref{e6-6} and \eqref{e6-7},
respectively, are:
\begin{gather}\label{e6-8}
\frac{1}{2}\int_\Omega  {(1 - {{| x |}^2}){f^ + }(x)\,dx
 = } \int_{\partial \Omega } {[{{\tilde g}^ + }_1(x) - {g^ + }(x)]\,d{S_x}},\\
\label{e6-9}
\frac{1}{2}\int_\Omega  {[(n - 1){{| x |}^2} - (n - 3)]{f^ - }(x)\,dx
= } \int_{\partial \Omega } {{{\tilde g}^ - }_2(x)\,d{S_x}}.
\end{gather}

From equalities \eqref{e4-3}, \eqref{e4-5} and \eqref{e4-8} we obtain
\[
\frac{1}{2}\int_\Omega  {(1 - {{| x |}^2}){f^ + }(x)\,dx
= } \frac{1}{2}\int_\Omega  {(1 - {{| x |}^2})f(x)\,dx} .
\]
Similarly, from  \eqref{e4-3}, \eqref{e4-5} and \eqref{e4-8}, it follows that
\begin{gather*}
\frac{1}{2}\int_\Omega  {(1 - {{| x |}^2}){f^ + }(x)\,dx
= } \frac{1}{2}\int_\Omega  {(1 - {{| x |}^2})f(x)\,dx} ,\\
\int_{\partial \Omega } {{{\tilde g}^ + }_1(x)\,d{S_x}
 - \int_{\partial \Omega } {{g^ + }(x)}\,d{S_x}}
= \int_{\partial {\Omega _ + }} {{g_1}(x)}\, d{S_x}
- \int_{\partial \Omega } {g(x)}\, d{S_x}.
\end{gather*}
Consequently, condition \eqref{e6-9} always holds and condition \eqref{e6-8}
can be rewritten in the form \eqref{e6-1}.

(2) if $m = 2$, $\ell_1 = 1$, $\ell_2 = 3$, then the necessary and sufficient
condition for solvability of the problem \eqref{e6-6} has the form
\[
\frac{1}{2}\int_\Omega  {(1 - {{| x |}^2}){f^ + }(x)\,dx
= } \int_{\partial \Omega } {[{g^ + }(x) - {{\tilde g}_1}^ + (x)]\,d{S_x}} ,
\]
which can be rewritten in the form
\[
\frac{1}{2}\int_\Omega  {(1 - {{| x |}^2})f(x)\,dx
= } \int_{\partial \Omega } {g(x)d{S_x}}
- \int_{\partial {\Omega _ + }} {{g_1}(x)\,d{S_x}.}
\]
For  problem \eqref{e6-7} we obtain the conditions:
\begin{gather}\label{e6-10}
\frac{1}{2}\int_\Omega  {[(n - 1){{| x |}^2} - (n - 3)]{f^ - }(x)\,dx
= } \int_{\partial \Omega } {{{\tilde g}^ - }_2(x)\,d{S_x}} , \\
\label{e6-11}
\begin{aligned}
&\frac{1}{2}\int_\Omega  {{x_j}[(n - 1){{| x |}^2} - (n - 3)]{f^ - }(x)dx }
\\
&= \int_{\partial \Omega } {{x_j}[{{\tilde g}^ - }_2(x) - {g^ - }(x)]d{S_x}} ,\quad
j = 1,2,\dots,n.
\end{aligned}
\end{gather}
From equalities \eqref{e4-3} and \eqref{e4-8}, the condition \eqref{e6-10} always
holds. Further, using \eqref{e4-11}, \eqref{e4-14} and \eqref{e4-15} we have
\begin{gather*}
\frac{1}{2}\int_\Omega  {{x_j}[(n - 1){{| x |}^2} - (n - 3)]{f^ - }(x)dx
= } \frac{{1 - {\alpha _j}}}{2}\int_\Omega  {{x_j}[(n - 1){{| x |}^2}
- (n - 3)]f(x)dx,} \\
\int_{\partial \Omega } {{x_j}{{\tilde g}^ - }_2(x)} d{S_x}
 = \frac{{1 - {\alpha _j}}}{2}\int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)d{S_x}} ,\\
\int_{\partial \Omega } {{x_j}{g^ - }(x)} d{S_x}
= \frac{{1 - {\alpha _j}}}{2}\int_{\partial \Omega } {{x_j}g(x)d{S_x}} .
\end{gather*}

Then equality \eqref{e6-11} can be rewritten in the form
\begin{equation}\label{e6-12}
\begin{aligned}
&\frac{{1 - {\alpha _j}}}{4}\int_\Omega  {{x_j}[(n - 1){{| x |}^2}
 - (n - 3)]f(x)dx} \\
&= \frac{{1 - {\alpha _j}}}{2}[\int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)d{S_x}}
- \int_{\partial \Omega } {{x_j}g(x)d{S_x}} ],\;j = 1,2,\dots,n.
\end{aligned}
\end{equation}
If for all $1 \le j \le n - 1$, ${\alpha _j} = 1$ then
 condition \eqref{e6-12} always holds for these indexes and in this case
 condition \eqref{e6-11} for $j = n$ can be rewritten in the form
\[
\frac{{n - 1}}{2}\int_\Omega  {{x_n}{{| x |}^2}f(x)dx - }
\frac{{n - 3}}{2}\int_{\partial \Omega } {{x_n}f(x)dx}
= \int_{\partial {\Omega _ + }} {{x_n}{g_2}(x)d{S_x} - }
 \int_{\partial \Omega } {{x_n}g(x)d{S_x}} .
\]

If for some ${j_0} \in \{ 1,2,\dots,n\}$, ${\alpha _{{j_0}}} =  - 1$,
then for this ${j_0}$, condition \eqref{e6-11} can be rewritten in the form
\begin{align*}
&\frac{{n - 1}}{2}\int_\Omega  {{x_{{j_0}}}{{| x |}^2}f(x)dx - }
\frac{{n - 3}}{2}\int_{\partial \Omega } {{x_{{j_0}}}f(x)dx} \\
&= \int_{\partial {\Omega _ + }} {{x_{{j_0}}}{g_2}(x)d{S_x} - }
\int_{\partial \Omega } {{x_{{j_0}}}g(x)d{S_x}} .
\end{align*}

(3) if $m = 3$, $\ell_1 = 1$, $\ell_2 = 2$, then by Theorem \ref{thm3.3} the
problem's solvability condition has the form
\[
\frac{1}{2}\int_\Omega  {[(n - 1)|x{|^2} - (n - 3)]{f^ + }(x)dx
= } \int_{\partial \Omega } {{g^ + }(x)d{S_x}} .
\]
According to \eqref{e4-2} and \eqref{e4-4} the last condition can be rewritten
in the form
\[
\frac{1}{2}\int_\Omega  {[(n - 1){{| x |}^2} - (n - 3)]f(x)dx
= } \int_{\partial \Omega } {g(x)d{S_x}} .
\]

Further, using Theorem \ref{thm3.2} the solvability condition of problem \eqref{e6-7}
can be rewritten in the  form
\begin{gather}\label{e6-13}
\frac{1}{2}\int_\Omega  {[(n - 1){{| x |}^2} - (n - 3)]{f^ - }(x)dx
= } \int_{\partial \Omega } {{g^ - }(x)d{S_x}}, \\
\label{e6-14}
\begin{aligned}
&\frac{1}{2}\int_\Omega  {{x_j}[(n - 1)|x{|^2} - (n - 3)]{f^ - }(x)dx }\\
&= \int_{\partial \Omega } {{x_j}[{g^ - }(x) - {{\tilde g}^ - }_2(x)]d{S_x}},\quad
 j = 1,2,\dots,n.
\end{aligned}
\end{gather}
From \eqref{e4-3} and \eqref{e4-5} it follows that  condition \eqref{e6-13}
 always holds. From \eqref{e4-11}, \eqref{e4-14} and \eqref{e6-14} we obtain
\begin{gather*}
\frac{1}{2}\int_\Omega  {{x_j}{{| x |}^2}{f^ - }(x)dx
= } \frac{{1 - {\alpha _j}}}{2}\int_\Omega  {{x_j}{{| x |}^2}f(x)dx,} \\
\int_{\partial \Omega } {{x_j}{g^ - }(x)} d{S_x} 
= \frac{{1 - {\alpha _j}}}{2}\int_{\partial \Omega } {{x_j}g(x)d{S_x}} ,\\
\int_{\partial \Omega } {{x_j}{{\tilde g}_2}^ - (x)} d{S_x} 
= \frac{{1 - {\alpha _j}}}{2}\int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)d{S_x}} .
\end{gather*}
Then \eqref{e6-14} can be rewritten as follows
\begin{equation}\label{e6-15}
\begin{aligned}
&\frac{{1 - {\alpha _j}}}{4}\int_\Omega  {{x_j}[(n - 1)|x{|^2} - (n - 3)]f(x)dx} \\
& = \frac{{1 - {\alpha _j}}}{2}\Big[ {\int_{\partial \Omega } {{x_j}g(x)d{S_x}} 
 - \int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)d{S_x}} } \Big],\quad
j = 1,2,\dots,n.
\end{aligned}
\end{equation}

If ${\alpha _j} = 1$ then \eqref{e6-15}  holds, and if ${\alpha _j} =  - 1$ 
then this condition can be rewritten in the form
\begin{align*}
&\frac{{n - 1}}{2}\int_\Omega  {{x_j}|x{|^2}f(x)dx 
- } \frac{{n - 3}}{2}\int_\Omega  {{x_j}f(x)dx} \\
&= \int_{\partial \Omega } {{x_j}g(x)d{S_x} 
- } \int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)d{S_x}} ,\quad j = 1,2,\dots,n.
\end{align*}
Thus  equality \eqref{e6-5} and, consequently, the theorem are proved.
\end{proof}

The following statement can be proved similarly to Theorem \ref{thm6.1}.

\begin{theorem} \label{thm6.2}
Let $k = 2$ and the functions $f(x)$, $g(x)$, ${g_j}(x)$, $j = 1,2$ be
 smooth enough on the domains $\bar \Omega$, $\partial \Omega $ and 
$\partial {\Omega _ + }$, respectively, and the compatibility 
conditions \eqref{e2-5} and \eqref{e2-6} hold. 
Then the necessary and sufficient conditions for solvability of
 problem \eqref{e2-1}-\eqref{e2-4} have the form:

(1) if $m = 1$, $\ell_1 = 2$, $\ell_2 = 3$, then
$$
\frac{{n - 1}}{2}\int_\Omega  {{{| x |}^2}f(x)dx
- } \frac{{n - 3}}{2}\int_\Omega  {f(x)dx}  
= \int_{\partial {\Omega _ + }} {{g_2}(x)d{S_x}} ;
$$

(2) if $m = 2$, $l_1 = 1$, $l_2 = 3$,
then
$$
\frac{{n - 1}}{2}\int_\Omega  {{{| x |}^2}f(x)dx
- } \frac{{n - 3}}{2}\int_\Omega  {f(x)dx}  
= \int_{\partial {\Omega _ + }} {{g_2}(x)d{S_x}} ,
$$
and
$$
\frac{{n - 1}}{2}\int_\Omega  {{x_j}{{| x |}^2}f(x)dx
 - } \frac{{n - 3}}{2}\int_\Omega  {{x_j}f(x)dx}  
= \int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)d{S_x}}  
- \int_{\partial \Omega } {{x_j}g(x)d{S_x}},
$$
for all $j \in \{ 1,2,\dots,n - 1\} $ for which ${\alpha _j} = 1$;

(3) if $m = 3$, $l_1 = 1$, $l_2 = 3$, then
$$
\frac{{n - 1}}{2}\int_\Omega  {{{| x |}^2}f(x)dx 
- } \frac{{n - 3}}{2}\int_\Omega  {f(x)dx}  
= \int_{\partial \Omega } {g(x)d{S_x}} ,
$$
and
$$
\frac{{n - 1}}{2}\int_\Omega  {{x_j}{{| x |}^2}f(x)dx 
- } \frac{{n - 3}}{2}\int_\Omega  {{x_j}f(x)dx}  
= \int_{\partial \Omega } {{x_j}g(x)d{S_x}}  
- \int_{\partial {\Omega _ + }} {{x_j}{g_2}(x)d{S_x}},
$$
for all  $j \in \{ 1,2,\dots,n - 1\} $ such that ${\alpha _j} = 1$.
\end{theorem}

\subsection*{Acknowledgements}
This research was supported by the target program 0085 / PTSF-14 
from the Ministry of Science and Education of the Republic of 
Kazakhstan and by Act 211 Government of the Russian Federation, 
contract no. 02.A03.21.0011.


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\end{document}