\documentstyle{SibMatJ}

\topmatter

\HeHasNoConflicts

  \Author       Nazarov
    \Initial    S.
    \Initial    A.
    \Email      srgnazarov\@yahoo.co.uk
    \Email      srgnazarov108\@gmail.com
    \ORCID      0000-0002-8552-1264
    \AffilRef   1
  \endAuthor

   \Affil 1
    \Organization Institute of Problems of Mechanical Engineering
    \City         St. Petersburg
    \Country      Russia
  \endAffil

\author    S.~A.~Nazarov\endauthor
\title
Deformation of a~Thin Elastic Plate with a Fixed Edge and Attached Rods. Part~1:~The~Static~Problem
\endtitle

\thanks
This work is supported by the Ministry of Science and Higher Education of the Russian Federation (Project 124041500009--8).
\endthanks

\translator  A.~P.~Ulyanov\endtranslator
\iauthor     Nazarov~S.A.\endiauthor
\UDclass     517.956.8+517.956+539.3(3)\endUDclass
\opages      481--505\endopages

\pforename   S.A.\endpforename
\psurname    Nazarov\endpsurname
\pemail      srgnazarov\@yahoo.co.uk\endpemail

\datesubmitted  November 14, 2024\enddatesubmitted
\daterevised    November 14, 2024\enddaterevised
\dateaccepted   April 25, 2025\enddateaccepted

\affil       S.~A.~Nazarov\\
             Institute of Problems of Mechanical Engineering,
             St. Petersburg, Russia
\endaffil

\email        srgnazarov\@yahoo.co.uk;\quad  srgnazarov108\@gmail.com\endemail

    \orcid   0000-0002-8552-1264\endorcid

\keywords
junction of plate and rods,
dimension reduction,
exponential and power-law boundary layers,
asymptotics
\endkeywords

\abstract
We construct asymptotics for
the stress-strain state of a~thin horizontal plate
clamped along its edge,
with vertical rods attached to it.
Made of an~isotropic and homogeneous elastic material,
this structure is loaded by gravity.
Using the dimension reduction procedure and analysis of the boundary layer,
which exhibits exponential behavior near the plate edges and power-law behavior near the junction zones,
we find the main and correction terms in the asymptotics for
the deflection of the plate and rigid displacement of the rods,
and their longitudinal deformation.
We derive a weighted and anisotropic Korn inequality that is asymptotically sharp
and provides justification for the asymptotic formulas.
\endabstract

\endtopmatter

\specialhead\Label{Section 1}
1. Statement of the problem
\endspecialhead

Consider several domains
$\omega_0$
and
$\omega_j$
of the plane
bounded by simple closed smooth
(of class $C^\infty$)
contours
$\partial\omega_0$
and
$\partial\omega_j$,
for
$j=1,\dots,J$,
as well as distinct points
$P^1,\dots, P^J$
inside
$\omega_0$.
Thin isotropic and homogeneous
(with Lam\'{e} constants
$\lambda\geq0$,
$\mu>0$,
and density
$\rho>0$)
cylindrical plate and rods
with variable cross-sections
$\omega^h_j(z)$
given by positive profile functions
$H_j\in C^\infty[-\ell_j,0]$
are defined as
\iftex
$$
\align
\Omega^h_0
&=\{ x=(x_1,x_2,x_3):\,
y:=(x_1,x_2)\in\omega_0,\ z:=x_3\in(0,h)\},
\tag1
\\
\Omega^h_{j}
&=\bigl\{ x: \, h^{-1}H(z)^{-1}y^j\in\omega_j,\ z\in(-
\ell_j,0]\bigr\},\quad j=1,\dots,J.
\tag2
\endalign
$$
\else
$$
\Omega^h_0=\{ x=(x_1,x_2,x_3):
y:=(x_1,x_2)\in\omega_0,z:=x_3\in(0,h)\},
\tag1
$$
$$
\Omega^h_{j}=\big\{ x: \, h^{-1}H(z)^{-1}y^j\in\omega_j, z\in(-
\ell_j,0]\},\quad j=1,\dots,J.
\tag2
$$
\fi
By scaling,
we reduce the characteristic size of the longitudinal section~$\omega_0$
of the plate~\Tag(1) to~1,
thereby making dimensionless
the Cartesian coordinate systems
$x\in{\Bbb R}^3$
and
$y^j\in{\Bbb R}^2$
centered at
$\Cal  O$
and
$P^j$
respectively,
as well as all geometric parameters,
in~particular,
small
$h>0$
and fixed positive
$\ell_1,\dots,\ell_J$.
To simplify exposition (see \Par*{Section~9}),
assume that
the domain
$\omega_j$
includes a disk
${\Bbb B}^2_{{\bold r}_j}=\{ y^j: \,|y^j|<{\bold r}_j\}$
of radius
${\bold r}_j>0$
and impose the relations
$$
\aligned
&\int\limits_{\omega_j}y_1^j\,dy^j=\int\limits_{\omega_j}y_2^j\,dy^j=0,
\quad
I_{12}(\omega_j):=\int\limits_{\omega_j}y_1^jy^j_2\,dy^j=0,
\\
&H_j(z)=1\ \ \text{for}\  z\in(-\delta_H,0),\ \delta_H>0.
\endaligned
\tag3
$$
The first couple of them can be achieved with small adjustments of the points~$P^j$,
while the third,
by a~rotation of the system
$y^j=\bigl(y^j_1,y^j_2\bigr)$.

Deformations of the elastic junction
$\Pi^h=\Omega^h_0\cup\Omega^h_1\cup\dots\cup\Omega^h_J$
under the action of gravity
are described by the Lam\'{e} system of
three partial differential equations
$$
-\mu\Delta_x u^h(x)-(\lambda+\mu)\nabla_x\nabla_x^\top u^h(x)=f^h(x):=
\rho ge_{(3)}, \quad x\in\Pi^h.
\eqno{(4)}
$$
Here
$\nabla_x=\operatorname{grad}$
is the gradient operator,
$\nabla_x^\top=\operatorname{div}$
is the divergence operator,
$\Delta_x=\nabla_x^\top\nabla_x$
is the Laplace operator,
$g>0$
is acceleration due to gravity,
and~$e_{(q)}$
is the unit vector of the axis~$x_q$.
We~interpret the displacement vector
$u^h=\bigl(u^h_1,u^h_2,u^h_3\bigr)^\top$
as a~column in a~fixed coordinate system~$x$,
with
$\top$
indicating the transpose,
while the Cartesian components of the strain and stress tensors
are
$$
\varepsilon_{pq}(u^h)=\frac{1}{2}\, \bigg( \frac{\partial u^h_p}{\partial
x_q}+\frac{\partial u^h_q}{\partial x_p} \bigg),\quad
\sigma_{pq}(u^h)=2\mu\varepsilon_{pq}(u^h)+\lambda\delta_{p,q}
\sum\limits_{i=1}^3\varepsilon_{ii}(u^h),
\quad p,q=1,2,3,
$$
where
$\delta_{p,q}$
is the Kronecker symbol.
The plate edges
$\Gamma^h_0=\partial \omega_0\times(0,h)$
are rigidly clamped,
while the remaining part of the surface
$\partial\Pi^h$
is free of external forces,
and so the boundary conditions
\iftex
$$
\align
&u^h(x)=0\in{\Bbb R}^3 ,  \quad x\in\Gamma^h_0,
\tag5
\\
&\sigma^{(n)}_{q}(u^h;x):=\sum\limits_{p=1}^3n_p(x)\sigma_{pq}(u^h;x)=0,
\quad x\in\partial \Pi^h\setminus\overline{\Gamma^h_0},\ q=1,2,3,
\tag6
\endalign
$$
\else
$$
u^h(x)=0\in{\Bbb R}^3 ,  \quad x\in\Gamma^h_0,
\tag5
$$
$$
\sigma^{(n)}_{q}(u^h;x):=\sum\limits_{p=1}^3n_p(x)\sigma_{pq}(u^h;x)=0,
\quad x\in\partial \Pi^h\setminus\overline{\Gamma^h_0},\ q=1,2,3,
\tag6
$$
\fi
are satisfied.
Here
$n=(n_1, n_2, n_3)^\top$
is the unit outer normal vector,
which is defined almost everywhere on the piecewise smooth surface
$\partial\Pi^h$;
i.e.,
$$
\sigma^{(n)}(u^h)=\big( \sigma^{(n)}_1(u^h),
\sigma_2^{(n)}(u^h),\sigma_3^{(n)}(u^h)\big)^\top
$$
is the normal stress vector.
Denote by
$L(\nabla_x)$
and
$B(x,\nabla_x)$
the $3\times3$ matrices of differential operators
on the left-hand sides of~\Tag(4) and~\Tag(6).

The variational formulation of problem \Tag(4)--\Tag(6)
appeals to the integral identity [1,\,2]
$$
E(u^h,\psi^h;\Pi^h)=(f^h,\psi^h)_{\Pi^h}
\quad \text{for all}\ \psi^h\in H^1_0\bigl(\Pi^h;\Gamma^h_0\bigr)^3,
\eqno{(7)}
$$
where
$(\cdot,\cdot)_{\Pi^h}$
denotes the natural inner product in the Lebesgue space
$L^2(\Pi^h)$,
which can be either scalar or vector,
$H^1_0\bigl(\Pi^h;\Gamma^h_0\bigr)$
is the Sobolev space of functions vanishing on~$\Gamma^h_0$.
The last upper index~$3$ in~\Tag(7)
indicates the number of components of a~test vector function
$\psi^h=\bigl(\psi^h_1,\psi^h_2,\psi^h_3\bigr)^\top$,
but it is absent from the notation for the inner product and the norms.
Moreover,
$E(u^h,u^h;\Pi^h)$
is twice the elastic energy stored by the body
$\Pi^h$,
$$
E(u^h,\psi^h;\Pi^h)=\sum\limits_{p,q=1}^3(\sigma_{pq}(u^h),
\varepsilon_{pq}(\psi^h))_{\Pi^h}.
$$

The main goal of this article is
to construct asymptotics for the displacement field,
in~particular,
the main terms for the deflection of the plate
$\Omega^h_0$
and rigid displacements of the rods.%
\footnote"${}^{1)}$"{For the first time the author saw the structure described here
in an~exposition of the Inquisition's torture tools
in the Religion and Atheism Museum,
which occupied the Kazan Cathedral in Leningrad before the perestroika,
but the same body
$\Pi^h$
depicts,
for instance,
a~leaky barn roof with stalactite icicles.}

Many works address
elastic junctions of bodies with various limiting dimensions;
the best understood are the junctions of massive bodies with thin rods;
see [3--8] and~[9] on static and spectral problems of elasticity.
There are publications about junctions of plates and rods [10--12] and [13--15]
(as in a~popular joke [16, p.~293],
they treat the cases
$J=1$
and
$J\rightarrow\infty$
respectively),
in which the main asymptotic terms for elastic fields
are found by passing to various limits.
In this article we focus mainly on analyzing
the boundary layer phenomenon
near the rods-to-plate junction zones,
which enables us to construct higher-order asymptotic terms,
and in~particular,
to describe the translations and rotations of the rods
as a~consequence of the plate getting deformed.

The asymptotic analysis of junctions of bodies
with distinct limiting dimensions
(for plate and rods,
respectively~2 and~1)
is substantially complicated by the circumstance that
the flattened and elongated elastic bodies
respond quite differently to longitudinal and transverse loads,
which are usually in discord in architecture, engineering constructions, and furniture.
An~absolutely new point is precisely to construct
boundary layers near the rods-to-plate junction zones.
These layers are described by solutions to the elasticity problem
in the union
$\Xi^j$
of the unit layer and a~semi-infinite cylinder
(\Par*{Section~5}).
Mentioning the articles [17--19] and [20],
which studied scalar boundary problems on the junction
$\Pi^h$
and similar ones,
we should emphasize that
for them solutions to the Neumann problem on the unbounded domain
$\Xi^j$
are quite simple to study,
and so the procedure for constructing the asymptotics itself
differs substantially from the vector problem presented below.

\specialhead\Label{Section 2}
2. The two-dimensional model of a~plate
\endspecialhead

The classical Kirchhoff--Love model (see [21--26] among others)
for a~bent of a~thin plate clamped along its edge
consists of Sophie Germain's biharmonic equation [27, Section~30]
in the domain
$\omega_0\subset{\Bbb R}^2$
equipped with Dirichlet's double condition on its boundary,
\iftex
$$
\alignat2
&A_0\Delta^2_{y} w_3(y)=g\rho,&\quad& y\in\omega_0,
\tag8
\\
&w_3(y)=0,\ \ \partial_{n^0} w_3(y)=0,&\quad& y\in\partial\omega_0.
\tag9
\endalignat
$$
\else
$$
A_0\Delta^2_{y} w_3(y)=g\rho,\quad y\in\omega_0,
\tag8
$$
$$
w_3(y)=0,\quad \partial_{n^0} w_3(y)=0,\quad  y\in\partial\omega_0.
\tag9
$$
\fi
Here
$\partial_{n^p}$
is the derivative along the outer normal
$n^p=\big(n^p_1,n^p_2\big)^\top$
on the boundary of the domain
$\omega_p\subset{\Bbb R}^2$.
Moreover,
the reduced cylindrical rigidity of the plate,
as well as
the modified Lam\'{e} constant, the Poisson ratio
$\nu\in[0,1/2)$,
and the fundamental solution of the biharmonic operator on the plane,
that are used below,
are as follows:
$$
A_0=\frac{\mu}{3}\frac{\lambda+\mu}{\lambda+2\mu},\quad
\lambda_0=\frac{2\lambda\mu}{\lambda+2\mu}\leq\lambda,\quad
\nu=\frac{\lambda}{2(\lambda+\mu)},\quad
\Phi_3(y)=\frac{1}{8\pi}|y|^2\ln\frac{1}{|y|}.
\eqno{(10)}
$$
Problem \Tag(8),~\Tag(9)
admits a~unique solution
$w_3\in C^\infty(\overline{\omega_0})$.

Denote by
$G^j\in H^2(\omega_0)$
the Green's function of problem \Tag(8),~\Tag(9)
with a~singularity at the point~$P^j$:
$$
G^j(y)=A_0^{-1}\Phi_3(y^j)+G^j_0(y),\quad G^j_0\in
C^\infty(\overline{\omega_0}).
\eqno{(11)}
$$
The $J\times J$ matrix
${\bold G}=\big(G^j_0(P^k)\big)^J_{j,k=1}$
is symmetric and positive definite
because
$$
A_0\big(\Delta_y G^j,\Delta_y G^k\big)_{\omega_0}
=
A_0\lim\limits_{R\rightarrow+0}\int\limits_{{\Bbb S}^1_R(P^j)}
\fortex{\mskip-12mu}\bigg(\frac{\partial G^k}{\partial r_j}(y)\Delta_y G^j(y)-G^k(y)
\frac{\partial}{\partial r_j} \Delta_y G^j(y) \bigg)\,ds_y=G^k(P^j).
$$

The well-known procedure of dimension reduction in a~thin plate (see [28--32],
and this list is by no means exhaustive)
often uses the asymptotic ansatz
$$
u^h_{(0)}(y,z)={\bold W}^h_{(0)}(\zeta,\nabla_{y})w(y)+\cdots
\,:=\,\sum\limits_{k=0}^4h^{k-2} \Cal {W}^k_{(0)}
(\zeta,\nabla_{y})w(y)+\cdots,
\eqno{(12)}
$$
where the dots replace higher-order asymptotic terms,
$\zeta=h^{-1}(z-h/2)\in(-1/2,1/2)$
is the dilated transverse coordinate, the column
$w=(w_1,w_2,w_3)^\top$
contains the longitudinal displacement and deflection of the plate
averaged over its thickness,
while the $3\times3$ matrices
$\Cal {W}^k_{(0)}$
of differential operators
are prescribed by the equalities
\iftex
$$
\align
&\mskip-10mu\Cal {W}^0_{(0)}w(y)=e_{(3)}w_3(y),\\
&\mskip-10mu\Cal {W}^1_{(0)}(\zeta,\nabla_{y})w(y)=
\sum\limits_{p=1}^2e_{(p)}\bigg(w_p(y)-\zeta\frac{\partial w_3}{\partial
y_p}(y)\bigg),
\\
&\mskip-10mu\Cal {W}^2_{(0)}(\zeta,\nabla_{y})w(y)=-
e_{(3)}\frac{\lambda}{\lambda+2\mu}
\Biggl(\zeta\sum\limits_{p=1}^2\frac{\partial w_p}{\partial y_p}(y)-
\left(\frac{\zeta^2}{2}-\frac{1}{24}\right)\Delta_{y}w_3(y)\Biggr),
\tag13
\\
&\mskip-10mu\Cal {W}^1_{(0)}(\zeta,\nabla_{y})w(y)
=\Cal {X}^0(\zeta)
\hbox{\small$\scriptscriptstyle\pmatrix \partial/\partial y_1&0&2^{-
1/2}\partial/\partial y_2&0& 0&0\\0&\partial/\partial y_2&2^{-
1/2}\partial/\partial y_1&0&0&0\\
0&0&0&\partial^2/\partial y_1^2&2\partial^2/\partial y_1\partial
y_2&\partial/\partial y^2_2\endpmatrix$}^{\mskip-5mu\top}\mskip-3mu w(y).
 \tag14
\endalign
$$
\else
$$
\Cal {W}^0_{(0)}w(y)=e_{(3)}w_3(y),\quad
\Cal {W}^1_{(0)}(\zeta,\nabla_{y})w(y)=
\sum\limits_{p=1}^2e_{(p)}\bigg(w_p(y)-\zeta\frac{\partial w_3}{\partial
y_p}(y)\bigg),
$$
$$
\Cal {W}^2_{(0)}(\zeta,\nabla_{y})w(y)=-
e_{(3)}\frac{\lambda}{\lambda+2\mu}
\Biggl(\zeta\sum\limits_{p=1}^2\frac{\partial w_p}{\partial y_p}(y)-
\left(\frac{\zeta^2}{2}-\frac{1}{24}\right)\Delta_{y}w_3(y)\Biggr),
\tag13
$$
$$
\Cal {W}^1_{(0)}(\zeta,\nabla_{y})w(y)
=\Cal {X}^0(\zeta)
\pmatrix \partial/\partial y_1&0&2^{-
1/2}\partial/\partial y_2&0& 0&0\\0&\partial/\partial y_2&2^{-
1/2}\partial/\partial y_1&0&0&0\\
0&0&0&\partial^2/\partial y_1^2&2\partial^2/\partial y_1\partial
y_2&\partial/\partial y^2_2\endpmatrix^\top w(y).
\tag14
$$
\fi
We will not need the last term of the sum in \Tag(12)
and the full expression for the $3\times6$ matrix
$\Cal {X}^0(\zeta)$,
but they can be found in [31, Chapter~4, Section~2] for instance
or,
like all other formulas of this section and the next one,
in any other books with technical and mathematical theories
of thin elastic bodies.
Only two components of that matrix are used in~\Par*{Section~5},
namely
$$
\Cal {W}^1_{(0)i}(\zeta,\nabla_{y})e_{(3)}w_3(y)=\frac{1}{6}
\biggl(\frac{3\lambda+4\mu}{\lambda+2\mu}\zeta^3-
\frac{11\lambda+12\mu}{\lambda+2\mu}
\frac{\zeta}{4}\biggr)\frac{\partial}{\partial y_i}\Delta_y w_3,\quad i=1,2.
\eqno{(15)}
$$

The vector
$w^\prime=(w_1,w_2)^\top$
of mean longitudinal displacements of the plate
is determined by the two-dimensional Lam\'{e} system
with the Dirichlet boundary conditions
\iftex
$$
\alignat2
&L^\prime(\nabla_y)w^\prime(y):=-\mu\Delta_{y}w^\prime(y)-
(\mu+\lambda_0)\nabla_y\nabla_y^\top w^\prime(y)=f^\prime(y) ,
&\quad& y\in\omega_0,
\tag16
\\
&w^\prime(y)=g^\prime (y),&\quad& y\in\partial\omega_0.
\tag17
\endalignat
$$
\else
$$
L^\prime(\nabla_y)w^\prime(y):=-\mu\Delta_{y}w^\prime(y)-
(\mu+\lambda_0)\nabla_y\nabla_y^\top w^\prime(y)=f^\prime(y) ,
\quad y\in\omega_0,
\tag16
$$
$$
w^\prime(y)=g^\prime (y),\quad y\in\partial\omega_0.
\tag17
$$
\fi
Owing to the assumed way of loading,
below we only need the singular solution to
the planar elasticity problem \Tag(16), \Tag(17) for
$f^\prime=0$:
we describe the influence of the rods
by using the fundamental matrix for the two-dimensional Lam\'{e} system
(the Somigliana tensor)
given by
$$
\Phi^\prime(y)=
(\Phi^{\prime1}(y),\Phi^{\prime2}(y))=\Phi^0\ln r
+\Phi^1(\varphi),
\eqno{(18)}
$$
where
$(r,\varphi)\in{\Bbb R}_+\times{\Bbb S}^1_1$
is a~system of polar coordinates,
while the symmetric $2\times2$ matrices,
the nondegenerate numerical
$\Phi^0$
and the smooth
$\Phi^1$
on the unit circle
${\Bbb S}^1_1$,
will not be needed.

\specialhead\Label{Section 3}
3. The one-dimensional model of the rods
\endspecialhead

The asymptotic analysis (see [31--35] among others)
of the elasticity problem for a~thin rod,
which incorporates the classical dimension reduction procedure,
uses the following asymptotic ansatz for the displacement vector:
$$
u^h_{(j)}(y^j,z_j)={\bold W}^h_{(j)}
(\eta^j,z,\partial_z)\,w^j(z)+\cdots\,:=\,\sum\limits_{k=0}^4h^{k-2}
\Cal {W}^k_{(j)}
(\eta^j,z,\partial_z)w^j(z)+\cdots.
\eqno{(19)}
$$
Here the $3\times4$ matrices
$\Cal {W}^k_{(j)}$
of differential operators
are defined as
$$
\aligned
&\Cal {W}^0_{(j)}w^j(z)=\sum\limits_{p=1}^2e^j_{(p)}w_p^j(z),
\\\fortex{\noalign{\vskip-4mm}}
&\Cal {W}^1_{(j)}(\eta^j,z,\partial_z)\,w^j(z)=
d^6(\eta^j)w_4^j(z)+e_{(3)}
\bigg(w^j_3(z) -\sum\limits_{p=1}^2\eta^j_p
\frac{\partial w_p^j}{\partial z}(z)\bigg),
\\\fortex{\noalign{\vskip-2mm}}
&\Cal {W}^3_{(j)}(\eta^j,z,\partial_{z})w^j(z)=
\Cal {X}^j(\eta^j, z)\Cal {D}(\partial_{z}) w^j,
\\
&\Cal {W}^4_{(j)}(\eta^j, z,\partial_{z})w^j(z)=
\Cal {X}^{j\prime}(\eta^j, z)\Cal {D}(\partial_{z})\partial_{z}w^j(z).
\endaligned
\tag20
$$
This involves a~rotation around the rod axis;
namely,
the last column of the $3\times6$ matrix
$$
d(x)=(d^\prime(x),d^6(x))
=(d^\dag(x),d^\equiv(x))
=\pmatrix
0&-x_3&0&1& 0&-2^{-1/2}x_2\\0&0&-x_3&0&1&2^{-1/2}x_1\\
1&x_1&x_2&0&0&0\endpmatrix,
\eqno{(21)}
$$
which generates three translations and three rotations;
as well as the diagonal matrix
$\Cal {D}(\partial_{z})
=\operatorname{diag}\bigl\{\partial^2_{z},
\partial^2_{z}, \allowmathbreak
\partial_{z},\partial_{z}\bigr\}$
and the $3\times4$ matrix
$\Cal {X}(\eta^j, z)$,
which reads:
$$
\frac{1}{2}\pmatrix
\nu((\eta^j_1)^2-
(\eta^j_2)^2)H_j(z)^4&2\nu\eta^j_1\eta^j_2H_j(z)^4&-
2\nu\eta^j_1H_j(z)^2&0\\
2\nu\eta^j_2\eta^j_1H_j(z)^4&\nu((\eta^j_2)^2-
(\eta^j_1)^2)H_j(z)^4&-2\nu\eta^j_2H_j(z)^2&0\\
0&0&0&\sqrt{2}H_j(z)^4\Psi^j(\eta^j)\endpmatrix.
\eqno{(22)}
$$
It includes Poisson's ratio~$\nu$,
the profile
$H_j(z)$,
and the Saint-Venant function~$\Psi^j$,
meaning the solution to the Neumann problem
\iftex
$$
\alignedat2
&-\Delta_{\eta^j}\Psi^j(\eta^j)=0,&\quad&\eta^j\in\omega_j,
\\
&\partial_{n^j}\Psi^j(\eta^j)=\eta^j_2n^j_1(\eta^j)-
\eta^j_1n^j_2(\eta^j),&\quad&\eta^j\in\partial\omega_j,
\endalignedat
\tag23
$$
\else
$$
\gathered
-\Delta_{\eta^j}\Psi^j(\eta^j)=0,\  \eta^j\in\omega_j,
\\
\partial_{n^j}\Psi^j(\eta^j)=\eta^j_2n^j_1(\eta^j)-
\eta^j_1n^j_2(\eta^j),\  \eta^j\in\partial\omega_j
\endgathered
\tag23
$$
\fi
with zero mean-value.
The fourth expression on the list in~\Tag(20)
and the $3\times4$ matrix
$\Cal {X}^{j\prime}$
are not of further use.

The functions
$w_1^1$,
$w_2^1$,
and
$w_3^1$
are the averaged transverse and longitudinal displacements in the rod,
while
$w_4^1$
is its twisting.
They satisfy the Kirchhoff--Clebsch system (see [21,\,36--41] etc.)
of four ordinary differential equations,
which decouples by restrictions~\Tag(3):
\iftex
$$
\align
\partial^2_{z}A^j_k(z)\partial^2_{z} w^j_k(z)
&=f^j_k(z),\quad
z\in\Upsilon_j:=(-\ell_j,0),\ k=1,2,
\tag24
\\
-\partial_{z}A^j_l(z)\partial_{z} w^j_l(z)
&=f^j_3(z),\quad
z\in\Upsilon_j,\ l=3,4.
\tag25
\endalign
$$
\else
$$
\partial^2_{z}A^j_k(z)\partial^2_{z} w^j_k(z)=f^j_k(z),\quad
z\in\Upsilon_j:=(-\ell_j,0),\ k=1,2,
\tag24
$$
$$
-\partial_{z}A^j_l(z)\partial_{z} w^j_l(z)=f^j_3(z),\quad
z\in\Upsilon_j,\ l=3,4.
\tag25
$$
\fi

The coefficients are of the form
$$
\aligned
&A^j_k(z)=\mu\frac{3\lambda+2\mu}{\lambda+\mu}H_j(z)^4
\int\limits_{\omega_j}\big(\eta^j_k\big)^2\,d\eta^j,
\\
&A^j_3(z)=\mu\frac{3\lambda+2\mu}{\lambda+\mu}H_j(z)^2|\omega_j|,
\quad  A^j_4(z)=\frac{\mu}{2}H_j(z)^4G(\omega_j);
\endaligned
\tag26
$$
see [31, Chapter~5, Section~2] for instance.
This involves the area
$|\omega_j|$
and the moments of inertia
$I_{kk}(\omega_j)$
of the figure
$\omega_j$,
see the penultimate formula in~\Tag(3),
as well as its torsional stiffness~[42]
$$
G(\omega_j)=\int\limits_{\omega_j}
\bigg(\biggl|\frac{\partial \Psi^j}{\partial
\eta^j_1}(\eta^j)-\eta^j_2
\biggr|^2+\biggl|\frac{\partial \Psi^j}{\partial
\eta^j_2}(\eta^j)+\eta^j_1\biggr|^2\bigg)d\eta^j.
\eqno{(27)}
$$

Boundary conditions~(6) imply the following conditions
at the lower endpoints of the segment~$\Upsilon_j$:
\iftex
$$
\align
&A_k^j(z_j)\partial^2_{z_j}w^j_k(z_j)\big|_{z_j=-\ell_j}=0,\quad
\partial_{z_j}A^j_k(z_j)\partial^2_{z_j}w^j_k(z_j)\big|_{z_j=-
\ell_j}=0,\quad k=1,2,
\tag28
\\
&A_l^j(z_j)\partial_{z_j}w^j_l(z_j)\big|_{z_j=-\ell_j}=0,\quad    l=3,4.
\tag29
\endalign
$$
\else
$$
A_k^j(z_j)\partial^2_{z_j}w^j_k(z_j)\big|_{z_j=-\ell_j}=0,\quad
\partial_{z_j}A^j_k(z_j)\partial^2_{z_j}w^j_k(z_j)\big|_{z_j=-
\ell_j}=0,\quad k=1,2,
\tag28
$$
$$
A_l^j(z_j)\partial_{z_j}w^j_l(z_j)\big|_{z_j=-\ell_j}=0,\quad    l=3,4.
\tag29
$$
\fi
Conditions at the points
$z_j=0$
will be found as a~result of asymptotic analysis.

\specialhead\Label{Section 4}
4. The classical boundary layer near the plate edges
\endspecialhead

In~\Par*{Section~6},
as the main terms of asymptotics on the plate
for the solution to problem \Tag(4)--\Tag(6)
we will take the sum
$$
u^h_{(0)}(x)={\bold W}^h_{(0)}(\zeta,\nabla_y)e_{(3)}(w_3(y)
+hw^\circ_3(y))+\cdots,
\eqno{(30)}
$$
where the differential operator
${\bold W}^h_{(0)}(\zeta,\nabla_y)$
is defined in \Tag(12), \Tag(13)
and~$w_3$
is a~solution to problem~\Tag(8),~\Tag(9),
while the function
$w^\circ_3$
is to be found.
Ansatz~\Tag(30) is suitable only outside some neighborhood of the edge~$\Gamma^h_0$
and the ends
$\Gamma^h_j=\omega^h_j(0)$
of rod~\Tag(2) attached to the plate.
Near these sets
we have to construct boundary layers,
one two-dimensional and one three-dimensional.
Let us discuss the first of~them.

Considering the plate edges,
we reproduce an~approach and results of~[43].
In~a~neighborhood~$\Cal {U}$
of the arc
$\partial\omega_0$
introduce local curvilinear coordinates
$(n,s)$,
where~$n$
is the oriented distance to
$\partial\omega_0$,
$n<0$
in
$\omega_0\cap\Cal {U}$,
while~$s$
is the arc length measured along the contour counterclockwise.
Accordingly,
$e_{(n)}$,
$e_{(s)}$,
and
$e_{(z)}=e_{(3)}$
are the unit vectors of the axes~$n$, $s$, and~$z$.
Keep the scale for the coordinate~$s$
and introduce the dilated coordinates
${\bold y}=({\bold y}_1,{\bold y}_2)^{\top}=h^{-1} (n, z-h/2)$,
which we interpret henceforth as Cartesian.
Replacing
$x\mapsto({\bold y},s)$
and passing formally to
$h=0$
transform the domain~\Tag(1) into the direct product
${\bold P}_-\times\partial\omega_0$,
while problem \Tag(4)--\Tag(6) restricted to the plate~$\Omega^h_0$,
into the following two-dimensional
(plane and out-of plane; see~[38])
mixed boundary value problems of elasticity
on the half-strip
${\bold P}_-=\{{\bold y}:\, {\bold y}_1<0,\ |{\bold y}_2|<1/2\}$
with the parameter
$s\in\partial\omega_0$:
\iftex
$$
\align
 &\alignedat2
&{-}\mu\Delta_{{\bold y}}{\bold v}^\#({\bold y})-
(\lambda+\mu)\nabla_{{\bold y}}\nabla_{{\bold y}}\cdot
{\bold v}^\#({\bold y})=0,&\quad& {\bold y}\in{\bold P}_-,
\\
&{\bold v}^\#(0,{\bold y}_2)={\bold g}^\#({\bold y}_2),&\quad&
|{\bold y}_2|<\frac{1}{2},
\\
&\mu\frac{\partial{\bold v}_1}{\partial{\bold y}_2}({\bold y})
+\mu\frac{\partial{\bold v}_2}{\partial{\bold y}_2}({\bold y})=0,\ \
(\lambda+2\mu)\frac{\partial{\bold v}_2}{\partial{\bold y}_2}({\bold y})
+\lambda
\frac{\partial{\bold v}_1}{\partial{\bold y}_1}({\bold y})=0,&\quad&
{\bold y}_1<0,\  {\bold y}_2=\pm\frac{1}{2};
\endalignedat
\tag31
\\\noalign{\vskip3mm}
&\alignedat2
&{-}\mu\Delta_{{\bold y}}{\bold v}_s({\bold y})=0,&\quad&
{\bold y}\in{\bold P}_-,
\\
&\mu \frac{\partial{\bold v}_s}{\partial{\bold y}_2}({\bold y})=0,&\quad&
{\bold y}_1<0,\  {\bold y}_2=\pm\frac{1}{2},
\\
&{\bold v}_s(0,{\bold y}_2)={\bold g}_s({\bold y}_2),
&\quad&
|{\bold y}_2|<\frac{1}{2}.
\endalignedat
\tag32
\endalign
$$
\else
$$
\aligned
&{-}\mu\Delta_{{\bold y}}{\bold v}^\#({\bold y})-
(\lambda+\mu)\nabla_{{\bold y}}\nabla_{{\bold y}}\cdot
{\bold v}^\#({\bold y})=0,\quad {\bold y}\in{\bold P}_-,
\\
&{\bold v}^\#(0,{\bold y}_2)={\bold g}^\#({\bold y}_2),\quad
|{\bold y}_2|<\frac{1}{2},
\\
&\mu\frac{\partial{\bold v}_1}{\partial{\bold y}_2}({\bold y})
+\mu\frac{\partial{\bold v}_2}{\partial{\bold y}_2}({\bold y})=0,\ \
(\lambda+2\mu)\frac{\partial{\bold v}_2}{\partial{\bold y}_2}({\bold y})
+\lambda
\frac{\partial{\bold v}_1}{\partial{\bold y}_1}({\bold y})=0,\quad
{\bold y}_1<0,\  {\bold y}_2=\pm\frac{1}{2};
\endaligned
\tag31
$$
$$
\aligned
&{-}\mu\Delta_{{\bold y}}{\bold v}_s({\bold y})=0,\quad
{\bold y}\in{\bold P}_-,
\\
&\mu \frac{\partial{\bold v}_s}{\partial{\bold y}_2}({\bold y})=0,\quad
{\bold y}_1<0,\  {\bold y}_2=\pm\frac{1}{2},
\\
&{\bold v}_s(0,{\bold y}_2)={\bold g}_s({\bold y}_2),
\quad
|{\bold y}_2|<\frac{1}{2}.
\endaligned
\tag32
$$
\fi
Here
${\bold v}^\#=({\bold v}_n,{\bold v}_z)^\top$
and
${\bold v}_s$
are respectively the vector of longitudinal displacements
and the transverse displacement (deflection)
in the planes orthogonal to the lateral surface
$\Gamma^h_0$
of the plate
$\Omega^h_0$.

According to \Tag(13) and \Tag(9),
as well as to the anticipated equality
$$
w^\circ_3(y)=0,\quad y\in\partial\omega_0,
\eqno{(33)}
$$
the terms of orders~$h^{-2}$
and
$h^{-1}$
in ansatz~\Tag(13)
satisfy boundary conditions~\Tag(5) on~$\Gamma^h_0$,
but the term of order
$h^0=1$
leaves a~discrepancy,
which must be compensated with the term
$h^0{\bold v}({\bold y},s)$
found from problems~\Tag(31) and~\Tag(32) with the right-hand sides
$$
{\bold g}_z({\bold y}_2)=-
\frac{\lambda}{\lambda+2\mu}
\bigg(\frac{{\bold y}_2^2}{2}-\frac{1}{24}\bigg)
\Delta_yw_3(0,s),
\quad
{\bold g}_n({\bold y}_2)=
-{\bold y}_2\frac{\partial w^\circ_3}{\partial n}(0,s),
\quad
{\bold g}_s({\bold y}_2)=0.
$$
Here the functions
$w_3$
and
$w^\circ_3$
are expressed in the local coordinates
$(n,s)$,
while their traces
on~$\partial\omega_0$
correspond to the value
$n=0$.
It is clear that
${\bold v}_s=0$.
Moreover,
the Kondrat'ev theory~[44] (see~[43] for details,
as well as [45, Chapter~5, Section~7] and [46, Example~1.12 and Theorem~3.4])
shows that problem \Tag(31) admits a~solution in the form
$$
\aligned
{\bold v}^\#({\bold y})
= {\bold K}_n(\nu)e_{(n)}+{\bold K}_z(\nu)
e_{(z)}
+({\bold K}(\nu)\Delta_{{\bold y}} w_3(0,s)-
\partial_nw_3^\circ(0,s))
({\bold y}_2e_{(n)}-{\bold y}_1e_{(z)})+\widetilde{\bold v}^\#({\bold y}).
\endaligned
\tag34
$$
As
${\bold y}_1\rightarrow-\infty$,
the remainder
$\widetilde{\bold   v}^\#({\bold y})$
decays exponentially (see the weighted H\"{o}lder estimates in~[47]);
${\bold K}_n(\nu)$,
${\bold K}_z(\nu)$,
and
${\bold K}(\nu)$
are some coefficients;
furthermore,
${\bold K}(0)=0$
and
${\bold K}(\nu)>0$
for
$\nu\in(0,1/2)$;
see~[43] for details.

In order to satisfy the natural property of the two-dimensional boundary layer,
namely its exponential decay away from the plate edges,
we first need to remove the linear term in~\Tag(34).
The constant terms are accounted for
when we construct the next-order corrections in ansatz~\Tag(30); see \Par*{Section~7}.
That
first term,
imitating the rotation of the half-strip
${\bold P}_-$
at infinity,
is cancelled by imposing the boundary condition
$$
\partial_n w^\circ_3(y)={\bold K}(\nu)\Delta_y w_3(y),\quad
y\in\partial\omega_0.
\eqno{(35)}
$$

Near the lower endpoints of the rods
the boundary layer phenomenon also arises (see [34,\,48,\,49] among others)
and it is described by means of solutions to
the Neumann problem for the Lam\'{e} system in the half-cylinder
${\bold P}^j_+=\omega_j(-\ell_j)\times{\Bbb R}_+$.
We obtain the half-cylinder
${\bold P}^j_+$
itself
as a~result of the coordinate change
$x\mapsto \xi^j_{(+)}=h^{-1}(y^j,z+\ell_j)$.
In~the case
$\partial^m_{z}H_j(-\ell_j)\not=0$
for some
$m\in{\Bbb N}$
the surface turns out ``slightly curved,''
but flattens out in the limit as
$h\rightarrow+0$.
Thus,
the profile
$H_j$
does not affect the main asymptotic terms,
but significantly complicates the structure of the higher-order terms
due to the necessity of accounting for
the additional discrepancies in the boundary conditions on
$\partial {\bold P}^j_+ \setminus \omega_j(-\ell_j)$.
As~[50] shows,
we can ensure that
the boundary layer decays exponentially
by adjusting the matrices
$\Cal {X}^j$
and~$\Cal {X}^{j\prime}$
and choosing boundary conditions \Tag(28) and~\Tag(29).

Explicit formulas for higher-order terms are not needed in this article,
but we introduce the last requirement in~\Tag(3)
in order to simplify the structure of the power-law boundary layer
studied in \Par*{Section~5}.

\specialhead\Label{Section 5}
5. The boundary layer near the rods-to-plate junction zone
\endspecialhead

Dilate the coordinates as
$$
x\mapsto \xi^j=(\eta^j,\zeta)=h^{-1}(y-P^j,z-h/2)
$$
and pass formally to
$h=0$.
As a~result,
we move a~part of the junction boundary $\Pi^h$
to infinity and transform it into the union
$\Xi^j$
of the unit layer
$\Xi_0=\{\xi^j :\,  \eta^j\in{\Bbb R}^2,\ |\zeta|<1/2\}$
and the half-cylinder
$\Xi^j_-=\{\xi^j :\,  \zeta\leq-1/2,\ \eta^j\in\omega_j\}$.
Simultaneously convert
system~\Tag(4) and boundary conditions~\Tag(6)
into the problem
\iftex
$$
\alignedat2
&L(\nabla_{\xi^j})v^j(\xi^j)=f^j(\xi^j),&\quad&  \xi^j\in\Xi^j,
\\
&B(\xi^j,\nabla_{\xi^j})v^j(\xi^j)=g^j (\xi^j),&\quad&   \xi^j\in\partial\Xi^j,
\endalignedat
\tag36
$$
\else
$$
\gathered
L(\nabla_{\xi^j})v^j(\xi^j)=f^j(\xi^j),\quad  \xi^j\in\Xi^j,
\\
B(\xi^j,\nabla_{\xi^j})v^j(\xi^j)=g^j (\xi^j),\quad   \xi^j\in\partial\Xi^j,
\endgathered
\tag36
$$
\fi
whose variational formulation is the integral identity [51--53]
$$
E(v^j,\psi^j;\Xi_j)=(f^j,\psi^j)_{\Xi_j}+(g^j,\psi^j)_{\partial\Xi_j}\quad
\text{for all}\   \psi^j\in{\goth E}_j.
\eqno{(37)}
$$
The space
${\goth E}_j$
is obtained by completing the linear space
$C^\infty\bigl(\,\overline{\Xi^j}\,\bigr)^3$
of infinitely differentiable compactly supported vector-valued functions 
in the ``energy'' norm
$\|v^j;\goth{E}_j\|=\bigl(E(v^j,v^j;\Xi_j)+\|v^j;L^2(\goth{K}_j)\|^2\bigr)^{1/2}$,
where~$\goth{K}_j$
is some nonempty compact set in
$\overline{\Xi^j}$.

Let us mention the weighted Korn inequality [52--56]
$$
\triplenorm{v^j;\Xi_0}^2+\triplenorm{v^j;\Xi^j_-}^2\leq K_j\left(E(v^j,v^j;\Xi^j)+
\|v^j; L^2({\goth K}_j)\|^2\right)\quad \text{for all}\   v^j\in
C^\infty\bigl(\,\overline{\Xi_j}\,\bigr)^3,
$$
in which the norms on the layer and the half-cylinder read as follows: 
$$
\align
\triplenorm{v^j;\Xi_0}^2&=\int\limits_{\Xi_0}
\Biggl(\,\sum\limits_{p=1}^2\biggg(\biggl|
\frac{\partial v^j_p}{\partial \eta^j_p}\biggr|^2+
\biggl|\frac{\partial v^j_p}{\partial \eta^j_{3-p}}\biggr|^2
+\frac{1}{{\goth R}_0(\xi^j)^2}
\biggl(\biggl|\frac{\partial v^j_3}{\partial \eta^j_p}\biggr|^2+
\biggl|\frac{\partial v^j_p}{\partial \zeta}\biggr|^2\biggr)
+\frac{1}{\goth{R}_0(\xi^j)^2}\bigl|v^j_p\bigr|^2\bigggr)
\\\fortex{\noalign{\vskip-2ex}}
&\qquad\qquad+\bigl|\partial_{\zeta} v^j_3\bigr|^2+\goth{R}_9(\xi^j)^{-4}
\bigl|v^j_3\bigr|^2\Biggr)d\xi^j\quad
\text{for}\ \goth{R}_0(\xi^j)=(1+\rho_j)
(1+|{\ln \rho_j}|),\  \rho_j=|\eta^j|,
\\
\triplenorm{v^j;\Xi^j_-}^2&=\int\limits_{\Xi^j_-}
\Biggl(\,\sum\limits_{p=1}^2
\biggg(\biggl|\frac{\partial v^j_p}{\partial
\eta^j_p}\biggr|^2+\frac{1}{{\goth R}_j(\xi^j)^2}
\biggl(\biggl|\frac{\partial v^j_3}{\partial \eta^j_p}\biggr|^2+
\biggl|\frac{\partial v^j_p}{\partial \zeta}
\biggr|^2+\biggl|\frac{\partial v^j_p}{\partial
\eta^j_{3-p}}\Big|^2\biggr)\biggg)
\\\fortex{\noalign{\vskip-2.5ex}}
&\qquad\qquad+\biggl|\frac{\partial v^j_3}{\partial
\zeta}\biggr|^2+\frac{1}{{\goth R}_j(\xi^j)^4}
\sum\limits_{p=1}^2|v^j_p|^2+\frac{1}{{\goth R}_j(\xi^j)^2}|v^j_3|^2
\Biggr)d\xi^j\quad
\text{for}\ {\goth R}_j(\xi^j)=1+|\zeta|.
\endalign
$$
Observe that
rotations around the axis~$\zeta$,
namely, the last column
$d^6(\xi^j)$
of the matrix in \Tag(21),
lie outside the introduced space
because for them the integral over the layer diverges.
It is not difficult to verify (see~[52,\,56] for instance)
that
the remaining rigid motions,
i.e.,
all five columns of the $3\times5$ block
$d^\prime(\xi^j)$,
for which all integrals converge,
belong to the space~${\goth E}_j$
because they can be approximated by
compactly supported vector-valued functions in the energy norm.
Since the bilinear form in~\Tag(37)
degenerates only on the rigid displacements,
the observation we made
together with simple trace inequalities
and the Riesz Representation Theorem for linear functionals in a~Hilbert space
lead to the following statement:

\proclaim{Lemma~1}\Label{L1}
Suppose that
the right-hand sides of problem {\rm\Tag(36)}
are subject to%
\footnote"${}^{2)}$"{%
Requirements \Tag(38) on the right-hand sides can be relaxed.}
\iftex
$$
\alignedat2
&{\goth R}_0f^j_p,{\goth R}^2_0f^j_3\in L^2(\Xi_0),&\quad&
{\goth R}_0g^j_p,{\goth R}^2_0g^j_3\in
L^2(\partial\Xi_0\cap\partial\Xi^j),
\\
&{\goth R}^2_jf^j_p,{\goth R}_jf^j_3\in L^2(\Xi^j_-),&\quad&
{\goth R}^2_jg^j_p,{\goth R}_jg^j_3\in L^2 (\partial\Xi^j_-
\cap\partial\Xi^j)
\endalignedat
\tag38
$$
\else
$$
\gathered
{\goth R}_0f^j_p,{\goth R}^2_0f^j_3\in L^2(\Xi_0),\quad
{\goth R}_0g^j_p,{\goth R}^2_0g^j_3\in
L^2(\partial\Xi_0\cap\partial\Xi^j),
\\
{\goth R}^2_jf^j_p,{\goth R}_jf^j_3\in L^2(\Xi^j_-),\quad
{\goth R}^2_jg^j_p,{\goth R}_jg^j_3\in L^2 (\partial\Xi^j_-
\cap\partial\Xi^j)
\endgathered
\tag38
$$
\fi
for
$p=1,2$
and to the following five orthogonality conditions:
$$
\int\limits_{\Xi^j}d^\prime(\xi^j)^\top f^j(\xi^j)d\xi^j+
\int\limits_{\partial\Xi^j}\!d^\prime(\xi^j)^\top
g^j(\xi^j)\,ds_{\xi^j}=0\in{\Bbb R}^5.
\eqno{(39)}
$$
\goodbreak\noindent
Then the variational problem {\rm\Tag(37)}
has a~solution
$v^j\in{\goth E}_j$
defined up to the term
$d^\prime(\xi^j)b^{j\prime}$
with any column
$b^{j\prime}\in{\Bbb R}^5$;
however,
when the orthogonality conditions
$(v^j,d^\prime)_{{\goth K}_j}=0\in{\Bbb R}^5$
are met,
it becomes unique and satisfies the estimate
$\|v^j; {\goth E}_j\|\leq c^j_{{\goth K}}{\goth N}_j$,
where
${\goth K}_j\subset \overline{\Xi^j}$
is a~nonempty compact set,
while
${\goth N}_j$
is the sum of the norms of functions {\rm\Tag(38)}
in the mentioned spaces.
\endproclaim

If the right-hand sides~$f^j$
and~$g^j$
are smooth,
then their differential properties
pass to the solutions everywhere except on the edge
$\partial\omega_j\times\{-1/2\}$
on the boundary
$\partial\Xi^j$;
see [51,\,57--60].

It is known (see [46, Example~1.12 and Proposition~3.2] for instance)
that by \Par{L1}{Lemma~1} the homogeneous version of problem~\Tag(36)
(for
$f^j=0$
and
$g^j=0$)
admits~12 solutions with polynomial growth as
$\zeta\rightarrow-\infty$
and a~certain well-defined behavior in the layer;
see below.

The number 12 is exactly the number of the monomials
\iftex
$$
\align
&{\bold e}_{(3)},\ -z{\bold e}_{(1)},\ -z{\bold e}_{(2)}\
\text{and}\
{\bold e}_{(1)},\ {\bold e}_{(2)},\ \frac{1}{\sqrt{2}}{\bold e}_{(4)},
\tag40
\\
 &\aligned
&{\bold Z}^1(z)=
\frac{z}{A^j_3(0)}{\bold e}_{(3)},\quad
{\bold Z}^2(z)=\frac{1}{A^j_1(0)}\frac{z^3}{6}{\bold e}_{(1)},\quad
{\bold Z}^3(z)=\frac{1}{A^j_2(0)}\frac{z^3}{6}{\bold e}_{(2)},
\\
&{\bold Z}^4(z)=-
\frac{1}{A^j_1(0)}\frac{z^2}{2}{\bold e}_{(1)},\quad {\bold Z}^5(z)=
-\frac{1}{A^j_2(0)}\frac{z^2}{2}
{\bold e}_{(2)},\quad {\bold Z}^6(z)=2^{-1/2}
\frac{z}{A^j_4(0)}{\bold e}_{(4)},
\endaligned
\tag41
\endalign
$$
\else
$$
{\bold e}_{(3)},\ -z{\bold e}_{(1)},\ -z{\bold e}_{(2)}\
\text{and}\
{\bold e}_{(1)},\ {\bold e}_{(2)},\ \frac{1}{\sqrt{2}}{\bold e}_{(4)},
\tag40
$$
$$
\aligned
&{\bold Z}^1(z)=
\frac{z}{A^j_3(0)}{\bold e}_{(3)},\quad
{\bold Z}^2(z)=\frac{1}{A^j_1(0)}\frac{z^3}{6}{\bold e}_{(1)},\quad
{\bold Z}^3(z)=\frac{1}{A^j_2(0)}\frac{z^3}{6}{\bold e}_{(2)},
\\
&{\bold Z}^4(z)=-
\frac{1}{A^j_1(0)}\frac{z^2}{2}{\bold e}_{(1)},\quad {\bold Z}^5(z)=
-\frac{1}{A^j_2(0)}\frac{z^2}{2}
{\bold e}_{(2)},\quad {\bold Z}^6(z)=2^{-1/2}
\frac{z}{A^j_4(0)}{\bold e}_{(4)},
\endaligned
\tag41
$$
\fi
which are the solutions to the following system of
homogeneous ordinary differential equations \Tag(24) and~\Tag(25)
with the coefficients frozen at $z=0$:
$$
\Cal {D}(\partial_{z})A^j(0)\Cal {D}(\partial_{z}){\bold Z}^q(z)=0,
\quad
z\in{\Bbb R},\ q=1,\dots,6.
$$
In \Tag(40) and \Tag(41) we see the unit vectors
${\bold e}_{(m)}=
(\delta_{1,m},\delta_{2,m},\delta_{3,m},\delta_{4,m})^\top$
of the space
${\Bbb R}^4$
and the diagonal matrix
$A^j=\operatorname{diag}\bigl\{A^j_1,A^j_2,A^j_31,A^j_4\bigr\}$.
By using the $3\times4$ matrix
${\bold W}^1_{(j)}$
of differential operators
prescribed by \Tag(19), \Tag(20), and \Tag(22) with the parameter
$h=1$,
define by means of the group of monomials \Tag(41)
the following vector-valued functions
with polynomial behavior with respect to the variable~$\zeta$:
$$
\Cal {Z}^q_{(j)}(\xi^j)={\bold W}^1_{(j)}(\eta^j,\partial_{\zeta})
{\bold Z}^q(\zeta), \quad q=1,\dots,6.
\eqno{(42)}
$$
It is not difficult to see that,
first,
\Tag(42) with the first group \Tag(40)
gives rise to rigid motions,
i.e.,
the columns of matrix \Tag(21),
and, second,
vector-valued functions \Tag(42) themselves are annihilated by the operator~$L(\nabla_{\xi^j})$
in the entire cylinder
$\omega_j\times{\Bbb R}$
and by the operator
$B(\eta^j,\nabla_{\xi^j})$
on its surface.

Our immediate objective is to show that
displacement fields \Tag(42) generate respectively
the longitudinal and transverse forces
applied at infinity in the half-cylinder
$\Xi^j_-$,
as well as bending moments and torque, and to establish existence of the indicated solutions
in the form
$$
\Cal {V}^q_{(j)}(\xi^j)=\chi_j(\xi^j)\Cal {Z}^q_{(j)}(\xi^j)
+\chi_0(\xi^j)\Cal {Y}^q(\xi^j)+\widehat{\Cal  V}^q_{(j)}(\xi^j),\quad
q=1,\dots,6,
\eqno{(43)}
$$
\spacing{.9}{by finding suitable vector-valued functions
$\Cal {Y}^1,\dots,\Cal {Y}^6$
in the layer.
The cut-off functions
$\chi_j(\xi^j)=\chi(-\zeta)$
and
$\chi_0(\xi^j)=\chi(\rho_j/R_j)$
are determined from the standard cut-off function
$\chi\in C^\infty({\Bbb R})$
obeying the equalities
$\chi(t)=0$
for
$t<1$
and
$\chi(t)=1$
for
$t>2$,
while the radius
$R_j$
is chosen so that
the disk
${\Bbb B}^2_{R_j}=\{\eta^j:\,\rho_j<R_j\}$
includes
$\overline{\omega_j}$.}

Let us calculate the integrals
${\goth I}^q_{jp}(R)$
from the formula
$$
\left({\goth I}^{q}_{11}(R),\dots,{\goth I}^{q}_{j6}(R)\right)^\top
=\int\limits_{\omega_j}
d(\eta^j,R)^\top\sigma^{(\zeta)}
\bigl(\Cal {Z}^q_{(j)};\eta^j,-R\bigr)\,d\eta^j,\quad
R>\frac{1}{2},
\eqno{(44)}
$$
where
$\Sigma^j_R(\Cal {Z}):=\sigma^{(\zeta)}(\Cal {Z})
=(\sigma_{13}({\Cal Z}), \sigma_{23}(\Cal {Z}),\sigma_{33}(\Cal {Z}))^\top$
is the normal stress vector on the cross-section
$\omega_j(-R)$
of the cylinder.
First, consider the most important integrals in \Tag(44)
for
$p=1$.
By~\Tag(20), \Tag(22), and  \Tag(3) we have
$$
\Sigma^j_R
\bigl(\Cal {Z}^1_{(j)};\eta^j\bigr)
=\frac{1}{A^j_3(0)}e_{(3)}
\bigg(\lambda+2\mu+\lambda
\biggl(\frac{\partial\Cal {X}_{13}}{\partial\eta^j_1}(\eta^j)+
\frac{\partial\Cal {X}_{13}}{\partial\eta^j_1}(\eta^j)\biggr)\bigg)
\frac{\partial \zeta}{\partial \zeta}
=\frac{\mu}{A^j_3(0)}\frac{3\lambda+2\mu}{\lambda+\mu};
$$
therefore,
$$
{\goth I}^{1}_{j1}(R)=1\ \text{ and }\ {\goth I}^1_{jm}(R)=0
\,\text{ for } m=2,\dots,6.
\tag45
$$
\goodbreak

We need slightly more complicated calculations using~\Tag(27) and~\Tag(26)
to derive the equalities
$$
{\goth I}^{6}_{j6}(R)=1\ \text{ and }\ {\goth I}^{6}_{jm}(R)=0
\,\text{ for } m=1,\dots,5,
\eqno{(46)}
$$
because
$\Sigma^j_R\bigl(\Cal {Z}^6_{(j)};\eta^j\bigr)
=\mu\bigl(d^6(\eta^j,0)+2^{1/2}
(\nabla_{\eta^j}^\top,0)^\top\Psi^j(\eta^j)\bigr)$
and
$$
\aligned
A^j_4(0){\goth I}^{6}_{j6}(R)
&=\frac{\mu}{2}
\int\limits_{\omega_j}\bigg(\big|\eta^j_2\big|^2+\big|\eta^j_1\big|^2
-\eta^j_2\frac{\partial\Psi^j}{\partial\eta^j_1}(\eta^j)+
\eta^j_1\frac{\partial\Psi^j}{\partial\eta^j_2}(\eta^j)\bigg)\,d\eta^j
\\
&=\frac{\mu}{2}G(\omega_j)-
\frac{\mu}{2}\int\limits_{\omega_j}
\bigg(|\nabla_{\eta^j}\Psi^j(\eta^j)|^2
-\eta^j_2\frac{\partial\Psi^j}{\partial\eta^j_1}(\eta^j)+
\eta^j_1\frac{\partial\Psi^j}{\partial\eta^j_2}(\eta^j)\bigg)
\,d\eta^j=A^j_4(0);
\endaligned
$$
moreover,
the last integral vanishes by the Green's formula and~\Tag(23).

Furthermore,
for
$p=1,2$
and
$m=1,\dots,6$
we find that
the integral
${\goth I}^{3+p}_{jm}(R)$
is equal to
$$
\int\limits_{\omega_j}\!
d^m(\eta^j,-R)^\top
\Sigma^j_R\bigl(\Cal {Z}^{3+p}_{(j)};\eta^j\bigr)
d\eta^j
=\delta_{3+p,m}
\int\limits_{\omega_j}\!\bigl(0,0,(\lambda+2\mu-2\lambda\nu)\eta^j_p\bigr)
\bigl(Re_{(p)}+\eta^j_pe_{(3)}\bigr)d\eta^j=\delta_{3+p,m}.
\tag47
$$

Finally,
the last group of the necessary equalities,
$$
{\goth I}^{1+p}_{jm}(R)=\delta_{1+p,m}, \quad p=1,2,\  m=1,\dots,6,
$$
requires much longer calculations
because for the monomials~${\bold Z}^{1+p}$
of degree~3 in the list~\Tag(41)
the normal stress vector
$\Sigma^j_R\bigl(\Cal {Z}^{1+p}_{(j)};\eta^j\bigr)$
depends on the third row of the matrix
$\Cal {X}^\prime$,
for which we omit the bulky explicit expression.
Therefore,
as in [31, Chapter~5, Sections~1 and~2] for instance,
we have to account for the fact that
the required elements of the mentioned matrix
are found from the Neumann problem
for the Laplace operator in the domain
$\omega_j$,
and after multiple integration by parts
to reduce the required quantity to the integral of
$\big(\eta^j_p\big)^2$
with the same multiplier as in~\Tag(47).
For several reasons,
we do not reproduce the appropriate calculations.
First, below we use essentially only~\Tag(45).
Second,
a~quite elementary method for calculating integrals
is developed in [31, Chapter~5, Section~3]
even for anisotropic rods.
Moreover,
for general boundary value problems
in the case of formally self-adjoint elliptic second-order systems in cylinders,
[61] established simple relations
between the Green's formulas for the original problem
and its one-dimensional model.
We present the calculations in \Tag(44)--\Tag(47)
in order to make further arguments more accessible.

As we mentioned,
the behavior of solutions to problem \Tag(36) in the cylindrical outlet to infinity
is studied using the Kondrat'ev theory~[44] (also see [45, Chapters~3 and~6] and [46, Section~3]),
which covers angular and conical domains,
but not an outlet to infinity as a~layer.
An~approach to constructing expansions of
concrete mathematical physics problems in a~sector of a~layer,
and in~particular in the whole layer,
was developed in [62--65] and~[53].
An~important observation is that,
for an~elastic layer, the algorithm for constructing expansions at infinity in~$\Xi_0$
coincides with the dimension reduction procedure for the plate
$\Omega^h_0$
of \Par*{Section~2}.

By analogy with~\Tag(41),
introduce a~set of solutions to the two-dimensional model \Tag(8),~\Tag(16):
\iftex
$$
\alignedat3
&{\bold Y}^1(y)=-\Phi(y)e_{(3)},&\quad& {\bold Y}^2(y)=\Phi(y)e_{(1)},&\quad&
{\bold Y}^3(y)=\Phi(y)e_{(2)},
\\
&{\bold Y}^4(y)=\frac{\partial \Phi}{\partial
y_1}(y)e_{(3)},&\quad&{\bold Y}^5(y)=
\frac{\partial \Phi}{\partial y_1}(y)e_{(3)},&\quad&{\bold Y}^6(y)
=\big(d^6(-\nabla^\top_y,0)^\top\Phi(y)^\top\big)^\top.
\endalignedat
\tag48
$$
\else
$$
\aligned
&{\bold Y}^1(y)=-\Phi(y)e_{(3)},\quad {\bold Y}^2(y)=\Phi(y)e_{(1)},\quad
{\bold Y}^3(y)=\Phi(y)e_{(2)},
\\
&{\bold Y}^4(y)=\frac{\partial \Phi}{\partial
y_1}(y)e_{(3)},\quad{\bold Y}^5(y)=
\frac{\partial \Phi}{\partial y_1}(y)e_{(3)},\quad{\bold Y}^6(y)
=\big(d^6(-\nabla^\top_y,0)^\top\Phi(y)^\top\big)^\top.
\endaligned
\tag48
$$
\fi
Here~$\Phi$
is the~ $3\times3$ matrix
$\operatorname{diag}
\big\{\Phi^\prime,A^{-1}_0\Phi_3\big\}$,
which involves the Somigliana tensor and the fundamental solution of the operator~$A_0\Delta_y^2$
from formulas~\Tag(18) and~\Tag(10),
while several transpose signs arose due to the convention about disposition of differentiation and matrix products.
Put
$$
\Cal {Y}^q(\xi^j)={\bold W}^1_{(0)}(\zeta,\nabla_{\eta^j})
\bold  {Y}^q(\eta^j),
\quad q=1,\dots,6,
\eqno{(49)}
$$
where the matrix operator
${\bold W}^1_{(0)}$
is given in \Tag(12) and \Tag(13) for
$h=1$.
Let us present the calculations,
which in accordance with~\Tag(48)
give to fields~\Tag(49) the meaning of
respectively transverse and longitudinal forces,
as well as bending moments and torque
applied at infinity in the layer~$\Xi_0$.
Precisely these fields appear in~\Tag(43)
for the required solutions to homogeneous problem \Tag(36)
for the unbounded body~$\Xi^j$.
\goodbreak

Calculate the integral
${\goth I}^q_{0m}(R)$
from the formula
$$
\bigl({\goth I}^q_{01}(R),\dots,{\goth I}^q_{06}(R)\bigr)^\top
=\int\limits_{\circledcirc_R}
d(\eta^j,z)^\top\sigma^{(\rho)}(\Cal {Y}^q;\xi^j)ds_{\xi^j},\quad R>R_j,
$$
where
$(\rho,\varphi,\zeta)$
are cylindrical coordinates,
while
$\circledcirc_R={\Bbb S}^1_R\times(-1/2,1/2)$
is a~finite cylindrical surface
on which the normal stress vector
$\Sigma^0_R(\Cal {Y})=\sigma^{(\rho)}(\Cal{Y})$
takes the form
$$
\sigma^{(\rho)}(\Cal{Y})=\sigma^{(1)}(\Cal{Y})\cos\varphi+
\sigma^{(2)}(\Cal{Y})\sin\varphi;
$$
furthermore,
$\sigma^{(k)}(\Cal{Y})=(\sigma_{k1}(\Cal{Y}),\sigma_{k2}(\Cal{Y}),
\sigma_{k3}(\Cal {Y}))^\top$.

We start with the most important integrals for
$q=1$.
According to \Tag(13) and \Tag(15), we have
$$
\aligned
\sigma_{ki}(\Cal {Y}^1;\xi^j)&=2\zeta\frac{\mu}{A_0}
\biggl(\frac{\partial^2 \Phi_3}{\partial\eta^j_i
\partial\eta^j_k}(\eta^j)+\delta_{k,i}\frac{\lambda}{\lambda+2\mu}
\Delta_{\eta^j}\Phi_3(\eta^j)\biggr)+O\biggl(\frac{1}{\rho_j}\biggr),
\\
\sigma_{k3}(\Cal {Y}^1;\xi^j)&=\frac{\mu}{A_0}\,\frac{\lambda+\mu}
{\lambda+2\mu}\biggl(\frac{1}{2}-2\zeta^2\biggr)
\frac{\partial}{\partial\eta^j_k}\Delta_{\eta^j}\Phi_3(\eta^j),
\quad k,i=1,2.
\endaligned
\tag50
$$
Integrating over
$\zeta\in(-1/2,1/2)$
and using the relation
$\partial_{\rho_j}=\cos\varphi\partial_{\eta_1^j}\allowmathbreak
+\sin\varphi\partial_{\eta_2^j}$,
we obtain
$$
{\goth I}^1_{01}(R)=\frac{\mu}{A_0}\frac{\lambda+\mu}{\lambda+2\mu}
\int\limits_{-1/2}^{1/2}
\biggl(\frac{1}{2}-2\zeta^2\biggr)\,d\zeta
\int\limits_{{\Bbb S}^1_R}\frac{\partial}{\partial\rho_j}\Delta_{\eta^j}
\Phi_3(\eta^j)ds_{\eta^j} +O
\biggl(\frac{1}{R}\biggr)=1+O\biggl(\frac{1}{R}\biggr),
$$
where the integral over the circle
is calculated according to the third formula in~\Tag(10).
The relations
${\goth I}^1_{0m}(R)=O(R^{-1})$
for
$m=2,\dots,5$
are immediate from~\Tag(50),
but in the case
$m=6$,
we must additionally account for the oddness
in one of the variables~$\eta^j_i$.

Calculate the integral
${\goth I}^{3+p}_{0m}(R)$
for
$p=1,2$.
We obtain expressions for the stress
$\sigma_{kq}(\Cal {Y}^{p+3};\xi^j)$
by differentiating the right-hand side of \Tag(50) with respect to~$-\eta^j_p$.
Observe that
$$
\aligned
&\cos\varphi\frac{\partial^3\Phi_3}{\partial(\eta^j_1)^3}+
\sin\varphi\frac{\partial^3\Phi_3}{\partial\eta^j_2\partial(\eta^j_1)^2}=\cos
\varphi
\Delta_{\eta^j}\frac{\partial\Phi_3}{\partial\eta^j_1}-
\biggl(\undersetbrace{}\to{\cos\varphi\frac{1}{\partial\eta^j_2}-
\sin\varphi\frac{1}{\partial\eta^j_1}}
\biggr)\frac{\partial^2\Phi_3}{\partial\eta^j_1\partial\eta^j_2},
\\
&\cos\varphi\frac{\partial^3\Phi_3}{\partial\eta^j_1\partial(\eta^j_2)^2}+
\sin\varphi\frac{\partial^3\Phi_3}{\partial(\eta^j_2)^3}=\sin\varphi
\Delta_{\eta^j}\frac{\partial\Phi_3}{\partial\eta^j_2}+
\biggl(\undersetbrace{}\to{\cos\varphi\frac{1}{\partial\eta^j_2}-
\sin\varphi\frac{1}{\partial\eta^j_1}}
\biggr)\frac{\partial^2\Phi_3}{\partial\eta^j_1\partial\eta^j_2},
\endaligned
$$
where the underbraces highlight the differential expression
$\rho_j^{-1}\partial_\varphi$,
which vanishes upon integration over the circle.
Eventually~\Tag(10) for~$\Phi_3$ shows that
$$
{\goth I}^{3+p}_{03+p}(R)=\frac{4 \mu}{A_0}\frac{\lambda+\mu}{\lambda+2\mu}
\int\limits_{{\Bbb S}^1_R}\frac{\partial\eta^j_p}{\partial\rho_j}
\Delta_{\eta^j}\frac{\partial\Phi_3}{\partial\eta^j_p}(\eta^j)\,ds_{\eta^j}
-
\int\limits_{{\Bbb S}^1_R}\eta^j_p\frac{\partial}{\partial\rho_j}
\Delta_{\eta^j}
\frac{\partial\Phi_3}{\partial\eta^j_p}(\eta^j)\,ds_{\eta^j}
+O\left(\frac{1}{R}\right)=1+O\left(\frac{1}{R}\right).
$$
Moreover,
taking into account the oddness of the integrand in one of the variables~$\eta^j_k$
and~$\zeta$,
we discover that
${\goth I}^{3+p}_{0m}(R)=O(R^{-1})$
for
$m=1,\dots,6$
with
$m\not=3+p$.

We find the integrals
${\goth I}^{1+p}_{0m}(R)$
for
$p=1,2$
by using the relations
$$
\sigma_{ki}(\Cal {Y}^{1+p};\xi^j)=\sigma^\prime_{ki}(\Phi^\prime
e_{(p)};\eta^j)+O\bigl(\rho_j^{-2}\bigr),
\quad \sigma_{k3}(\Cal {Y}^1;\xi^j)
=O\bigl(\rho_j^{-2}\bigr),\quad k,i=1,2,
$$
and a~calculation
where the integrals are interpreted in the sense of distributions
(see [66] for instance),
which gives meaning to the equality
$L^\prime(\nabla_y)\Phi^\prime(y)={\Bbb I}_2\delta(y)$
with the Dirac $\delta$-function
$$
\align
{\goth I}^{1+p}_{0m}(R)
&=\int\limits_{-1/2}^{1/2}d\zeta
\int\limits_{{\Bbb S}^1_R}
e^{\prime\top}_{(m)}\sigma^{\prime(\rho_j)}(\Phi^\prime
e_{(p)};\eta^j)\,ds_{\eta^j}
+O\left(\frac{1}{R}\right)
=\int\limits_{{\Bbb B}^2_R}
e^{\prime\top}_{(m)}L^\prime(\nabla_y)\Phi^\prime(\eta^j)
e_{(p)}\,d\eta^j+O\left(\frac{1}{R}\right)
\\
&=\delta_{1+p,m}+O(R^{-1}),\quad   m=2,3,
\\
{\goth I}^{1+p}_{0m}(R)&=O(R^{-1}),\quad   m=1,4,5,6.
\endalign
$$
Here
$e^{\prime}_{(m)}=\big(\delta_{1,m},\delta_{2,m}\big)^\top$
and
$\sigma^{\prime(\rho_j)}(\Phi^{\prime p})$
is the normal stress vector on the circle
found from the two-dimensional stress tensor
$\big(\sigma^\prime_{ki}(\Phi^{\prime p})\big)_{k,i=1}^2$
with Lam\'{e} constants
$\lambda_0$
and~$\mu$.

Finally,
the last required equalities
${\goth I}^{6}_{0m}(R)=\delta_{6,m}+O(R^{-1})$
for
$m=1,\dots,6$
are derived while using the definition of the Dirac $\delta$-function
and the ensuing relation
$$
\aligned
\int\limits_{{\Bbb B}^2_R}d^{6\prime}(\eta^j)^\top
L^\prime(\nabla_{\eta^j})(
d^{6\prime}(-\nabla_{\eta^j})^\top\Phi^\prime({\eta^j})^\top)^\top
d{\eta^j}&=
\int\limits_{{\Bbb B}^2_R}d^{6\prime}(\eta^j)^\top
d^{6\prime}(-\nabla_{\eta^j})\delta(\eta^j)\,d{\eta^j}
\\
&=\int\limits_{{\Bbb B}^2_R}\delta(\eta^j)\,d^{6\prime}(\nabla_{\eta^j})^\top
d^{6\prime}(\eta^j)
\,d{\eta^j}=\int\limits_{{\Bbb B}^2_R}\delta(\eta^j)\,d{\eta^j}=1.
\endaligned
$$
Furthermore,
$d^{6\prime}(y)=2^{-1/2}(-y_2,y_1)^\top$
is the truncated last column of matrix \Tag(21),
while the equality
$d^{6\prime}(\nabla_y)^\top d^{6\prime}(y)=1$
is due to the factors~$2^{-1/2}$
in~\Tag(21).

Let us now state the central proposition of this section.

\proclaim{Theorem~1}
The homogeneous problem {\rm\Tag(36)}
has six solutions {\rm\Tag(30)}
with the asymptotic terms {\rm\Tag(42),~\Tag(49)}
and the ``energy'' terms
$\widehat{\Cal  V}^q_{(j)}\in{\goth E}_j$,
which admit the representations
$$
\widehat{\Cal
V}^q_{(j)}(\xi^j)=\chi_j(\xi^j)\,d(\xi^j)t^q_{(j)}+\chi_0(\xi^j)
{\bold W}^1_{(0)}(\zeta,\nabla_{\eta^j})\Cal
{T}^q_{(j)}(\nabla_{\eta^j})\Phi(\eta^j)
+\widetilde{\Cal  V}^q_{(j)}(\xi^j).
$$
The column
$t^q_{(j)}\in{\Bbb R}^6$
and the coefficients of the differential operators
$$
\aligned
\Cal  {T}^q_{(j)}(\nabla_{\eta^j})\Phi
&=\pmatrix
\Cal  {T}^{q1}_{(j)}(\nabla_{\eta^j})\Phi^{\prime i} + \Cal
{T}^{q2}_{(j)}(\nabla_{\eta^j})\Phi^{\prime 2}\\
\big(T^{q3}_{(j)11}\partial_{\eta^j_1}^2+2T^{q3}_{(j)12}\partial_{\eta^j_1}
\partial_{\eta^j_2}+
T^{q3}_{(j)22}\partial_{\eta^j_2}^2\big)\Phi_3\endpmatrix,
\\
\Cal  {T}^{qk}_{(j)}(\nabla_{\eta^j})
&=\pmatrix
{T}^{qk}_{(j)1}\partial_{\eta^j_1}+{T}^{qk}_{(j)0}\partial_{\eta^j_2}\\
{T}^{qk}_{(j)0}\partial_{\eta^j_1}+{T}^{qk}_{(j)2}\partial_{\eta^j_2}
\endpmatrix
\endaligned
\tag51
$$
depend on the indices
$q=1,\dots,6$
and
$j=1,\dots,J$
{\rm (}as well as $k=1,2${\rm)},
the remainder
$\widetilde{\Cal  V}^q_{(j)}(\xi^j)$
vanishes exponentially at infinity in the cylinder
$\Xi_-^j$
together with its derivatives,
while in the layer
$\Xi_0$
for all
$\alpha^\prime=(\alpha_1,\alpha_2)\in{\Bbb N}_0^2$
and
$\alpha_3\in{\Bbb N}_0=\{0,1,2,\dots\}$
it satisfies the estimates
$$
\big|\partial_\zeta^{\alpha_3}\nabla^{\alpha^\prime}_{\eta^j}
\widetilde{\Cal V}^q_{(j)m}(\xi^j)\big|
\leq C_{l\alpha}\rho_j^{\delta_{m,3}-2-\alpha_1-\alpha_2}\bigl(1+(1-
\delta_{m,3})|{\ln\rho_j}|\bigr),\quad \rho_j>R_j,\  m=1,2,3.
\eqno{(52)}
$$
Moreover,
the components
$\varepsilon_{ik}(\widehat{\Cal  V}^q_{(j)})$
of the strain {\rm(}and stress\/{\rm)} tensor
vanish exponentially at infinity in the half-cylinder,
while in the layer,
they do as
$O(\rho_j^{-2})$.
\endproclaim
\goodbreak

\demo{Proof}
\Par{L1}{Lemma~1} and the calculations already made show that solutions \Tag(30) exist.
Indeed,
the right-hand sides of the problems for the ``energy'' terms
$\widehat{\Cal  V}^q_{(j)}$
satisfy \Tag(38),
while integration by parts in the truncated body
$\Xi^j(R)=\{\xi^j:\,\zeta>-R,\  \rho_j<R \}$
and passage to the limit as
$R\rightarrow+\infty$
establish \Tag(39) for
$q=1,\dots,5$.
The latter claim is guaranteed by the preceding calculations
because the outer normal on the cross-section
$\omega_j(-R)$
and on the cylindrical surface
$\circledcirc_R$
equals
$-e_{(3)}$
and
$\rho_j^{-1}\eta^j$
respectively,
and so the integrals over these parts of the boundary
cancel each other in the limit.
Moreover,
\Tag(46) verifies the compatibility conditions \Tag(39)
in the case
$q=6$.

The asymptotic expression in the cylinder~$\Xi^j_-$
is a~classical result%
%
\footnote"${}^{3)}$"{%
A~different approach,
although not yielding pointwise exponential estimates for the remainders,
is proposed~in~[67].}
%
of the Kondrat'ev theory~[44] (also see [45, Chapter~5] and [46, Section~3; 47]),
while the~representation in the layer is established in~[53],
which in~particular presents procedures
for constructing infinite series expansions
and estimating asymptotic remainders.
Note that
the rigid motion is removed from the expansion in the layer~$\Xi_0$
and put into the cylinder~$\Xi^j_-$,
see the term
$\chi_j(\xi^j)d(\xi^j)t^q_{(j)}$,
and so the expansion coefficients are uniquely determined.
The expression
$\chi_0 {\bold W}^1_{(0)}\Cal  {T}^q_{(j)}\Phi$
includes all power-logarithmic solutions $\rho_j^{-1}U^\prime(\varphi)$
to the two-dimensional Lam\'{e} system
and
$U^1_3(\varphi)\ln \rho_j+U^0_3(\varphi)$
to the biharmonic equations
with the exception of the field generated by
the rotational moment around the applicate axis,
present only for the solution~$\Cal {Y}^6_{(j)}$,
as~well as the vertical translation taken to $\Xi^j_-$.
Finally,
a~straightforward calculation of strains
while using \Tag(13), \Tag(14) and \Tag(51), \Tag(52)
guarantees the last claim of the theorem.
\qed\enddemo
%\medskip

We now describe another set of solutions to the problem
in the unbounded elastic body
$\Xi^j$.
Consider a~vector polynomial
$\Cal {P}(y)$
defined similarly to~\Tag(51)
and consisting of the linear vector-valued function
$\Cal {P}^\prime(y)=(\Cal {P}_1(y),\Cal {P}_2(y))^\top$
satisfying the Lam\'{e} system
and the quadratic trinomial
$\Cal {P}_3(y)$
annihilated by the biharmonic operator
$\Delta^2_y$.
Following the scheme presented above,
construct a~solution
${\bold V}(\Cal {P};\xi^j)$
to problem \Tag(36) that admits the representation
$$
{\bold V}_{(j)}(\Cal {P};\xi^j)
=\chi_0(\xi^j)
{\bold W}^1_{(0)}(\zeta,\nabla_{\eta^j})\bigl(\Cal {P}(\eta^j)+
\Cal {T}^{\Cal {P}}_{(j)}(\nabla_{\eta^j})\Phi(\eta^j)\bigr)
+\chi_j(\xi^j)d(\xi^j){\bold b}^{\Cal {P}}_{(j)}+
\widetilde{\bold   V}_{(j)}(\Cal {P};\xi^j).
\tag53
$$
Here
${\bold b}^{\Cal {P}}_{(j)}\in{\Bbb R}^6$
is a~column
and
$\Cal {T}^{\Cal {P}}_{(j)}$
is the differential operator defined by~\Tag(51),
which depend on the polynomial~$\Cal {P}$
and the index~$j$,
while the remainder
$\widetilde{\bold   V}_{(j)}(\Cal {P};\xi^j)$,
exponentially  decaying in the semi-infinite cylinder~$\Xi^j_-$,
satisfies
\Tag(52) in the layer~$\Xi_0$.

\specialhead\Label{Section 6}
6. The construction of the main asymptotic terms
\endspecialhead

We complement ansatz \Tag(30)
for the restriction~$u^{h0}$
of the solution to problem \Tag(4)--\Tag(6) on the plate~$\Omega^h_0$
with the asymptotic ans\"atze
$$
\aligned
u^h_{(j)}(x)
&=h^{-2}d^\dag(y^j,z)d_{(3)}^\dag(\nabla_{y^j},0)^\top
w_3(P^j)+h^{-1}
d^\dag(y^j,z)d_{(3)}^\dag(\nabla_{y^j},0)^\top w^\circ_3(P^j)
\\
&\qquad+h{\bold W}^1_{(j)}(\eta^j,\partial_{z}){\bold e}_{(3)}w^j_3(z)+\cdots
\endaligned
\tag54
$$
for the restrictions~$u^{hj}$
to the rods~$\Omega^h_j$.
Here
$d^\dag(x)$
and
$d_{(3)}^\dag(x)$
are the $3\times3$ blocks of matrix \Tag(21)
and its bottom row,
while
$w_3$
is the solution to problem \Tag(8),~\Tag(9).
The first term in \Tag(54),
$$
d^\dag(y^j,z_j)d_{(3)}^\dag(\nabla_{y^j},0)^\top w_3(P^j)
=e_{(3)}w_3(P^j)+
\big(y^j_1e_{(3)}-ze_{(1)}\big)\partial_{y^j_1} w_3(P^j)+
\big(y^j_2e_{(3)}-ze_{(2)}\big)\partial_{y^j_2} w_3(P^j),
$$
is exactly the main term of the rigid displacement of the rod,
while the second term remains to be determined.
The third term describes
the longitudinal extension of the rod under the action of gravity,
and so the function~$w^j_3$
can be found from \Tag(25) for~$l=3$
with the right-hand side
$f^j_3(z_j)=g\rho|\omega_j|H_j(z_j)^2$.
\goodbreak\noindent
Boundary conditions~\Tag(29) for
$l=3$
yield
$$
A^j_3(0)\partial_{z_j}w^j_3(0)=F_j:=-g\rho|\omega_j|
\int\limits_{-\ell_j}^0H_j(z_j)^2\,dz_j.
\eqno{(55)}
$$
The equality
$w^j_3(0)=0$
fully fixes the function
$w^j_3$,
while
$h^2F_j$
is the total weight of the rod
$\Omega^h_j$.
\goodbreak

Apply the method of matched asymptotic expansions
(see [30, Chapter~2; 68,\,69] among others)
and take the sum
$$
\aligned
u^h(x)
&=h^{-2}d^\dag(h\xi^j)d_{(3)}^\dag(\nabla_{y^j},0)^\top w_3(P^j)+h^{-1}
d^\dag(h\xi^j)d_{(3)}^\dag(\nabla_{y^j},0)^\top w^\circ_3(P^j)
\\
&\qquad+hF_j \big(\Cal {V}^1_{(j)}(\xi^j)-(8\pi)^{-1} \ln h
\bold{V}\big(\eta^j\mapsto\rho_j^2;\xi^j\big)\big)+\cdots
\endaligned
\tag56
$$
as the main terms of the inner expansion suitable in the vicinity of the end 
$\omega^h_j(0)$,
at which the rod is attached to the plate.
The first two terms of \Tag(56) coincide with the same terms of outer expansions \Tag(30) and \Tag(54),
while the third term,
which includes the solution
$\Cal {V}^1_{(j)}$
to the homogeneous problem \Tag(36)
constructed in \Par*{Section~5},
accounts for its representation \Tag(43) in the half-cylinder~$\Xi^j_-$
and Taylor's formula
$$
w^j_3(z)=A_3^j(0)^{-1}F^jz+O(z^2)=hA_3^j(0)^{-1}
F^j\zeta+O(h^2\zeta^2)\ \ \text{as}\  z\rightarrow-0,
$$
which follows from~\Tag(55).

\spacing{.85}{Therefore,
expansions \Tag(56) and \Tag(54) have been matched.
The expressions in \Tag(43) and \Tag(53)
for the vector-valued functions~$\Cal {V}^1_{(j)}$
and
${\bold V}_{(j)}(\Cal {P};\cdot)$
in the layer
$\Xi_0$
for
$\Cal {P}(\eta^j)\re=\rho_j^2\re=
\bigl(\eta^j_1\bigr)^2\re+\bigl(\eta^j_2\bigr)^2$,
as well as the~relation}
$$\Phi_3(\eta^j)=h^{-2}(8\pi)^{-1}r_j^2(\ln h-\ln r_j),
$$
which explains the appearance of the factors~$h$
and
$\ln h$
in the last term of ansatz \Tag(56),
show that
matching the inner and outer expansions in the plate body
guarantees for the correction term
$w^1_3$
of ansatz \Tag(30) near the points
$P^1,\dots,P^J$
the expansions
$$
w^1_3(y)=d^\dag(y^j,0)d_{(3)}^\dag(\nabla_{y^j},0)^\top w^1_3(P^j)+
F^jA_0^{-1}\Phi_3(y^j)+O(r_j^2)\ \ \text{as}\  r_j\rightarrow+0.
\eqno{(57)}
$$
By the definition in \Tag(11) of the Green's functions~$G^j$,
the solution to the homogeneous equation~\Tag(8) in the punctured domain
$\omega_0\setminus\{P^1,\dots,P^J\}$ satisfying~\Tag(57)
becomes
$$
w^\circ_3(y)=w^\bullet_3(y)+F_1G^1(x)+\dots+F_JG^J(x),
$$
where
$w^\bullet_3\in C^\infty(\overline{\omega_0})$
is a~biharmonic function satisfying boundary conditions \Tag(33) and \Tag(35).

Finally,
the correction term in the asymptotics of rigid displacements of the rod
looks like
$$
d_{(3)}^\dag(\nabla_{y^j},0)^\top w^\circ_3(P^j)
=d_{(3)}^\dag(\nabla_{y^j},0)^\top\biggl(  w^\bullet_3(y)+
\sum\limits_{k=1}^J F_k\bigl(G^k_0(y)+(1-\delta_{j,k})\Phi_3(y-
P^k)\bigr)\biggr)\biggr|_{y=P^j}.
$$

\specialhead\Label{Section 7}
7. Higher-order asymptotic terms
\endspecialhead

Owing to the geometric symmetry requirements introduced in \Par*{Section~1},
the main asymptotic terms constructed in the previous section
indicate the vertical deformation of the elements of the junction~$\Pi^h$,
namely,
the deflection of the plate and the downward displacements of the rods
with rotations and longitudinal dilations.
Let us verify that
the next asymptotic terms also generate other types of deformations in the plate and the rods.

Recall first of all that
we should also obey the conditions of exponential decay for the boundary layer near the plate edges,
i.e.,
to remove the terms
$K_z(\nu)e_{(z)}$
and
$K_n(\nu)e_{(n)}$
from~\Tag(34) for the vector~${\bold v}^\#$.
While we account for the first term by imposing the boundary condition
$$
w^2_3(y)=-K_z(\nu)\Delta_yw_3(y),\quad y\in\partial\omega_0,
$$
in the problem for the next correction term
$h^2w^2_3(y)$
for~\Tag(30),
the second term requires us to introduce the term
$hw^{1\prime}(y)$,
in which the vector-valued function
$hw^{1\prime}=h\big(w^1_1,w^1_1\big)^\top$
is a~solution to the homogeneous version (with $f^\prime=0$)
of the two-dimensional Lam\'{e} system \Tag(16) with the Dirichlet data
$$
w^1_n(y)=-K_z(\nu)\Delta_yw_3(y),
\quad
w^1_s(y)=0,\quad y\in\partial\omega_0,
$$
and describes the longitudinal deformation of the plate.
We have Taylor's formula
$$
w^{1\#}(y)=w^{1\#}(P^j)+\sum\limits_{p,q=1}^2y^j_p\frac{\partial
w^1_q}{\partial y^j_p}(P^j)+O(r_j^2)
\quad  \text{as}\  r_j\rightarrow+0,
\eqno{(58)}
$$
whose main terms must be matched with
the inner expansion in a~neighborhood of the rod end
$\overline{\omega^h_j(0)}
=\partial\Omega^h_0\cap\overline{\partial\Omega^h_j}$.
Complement~\Tag(30) with the expression
$h{\bold W}^1_{(0)} \bigl(w^1_1,w^1_2,0\bigr)^\top$
and split the components generated by the linear part of \Tag(58)
into the groups
$$
{\bold W}^1_{(0)}(\zeta,\nabla_y)
\Biggl(\,
\sum\limits_{p=1}^2w_p^1(P^j)e_{(p)}+\frac{1}{2}
\biggl(\frac{\partial w^1_2}{\partial y^j_1}(P^j)-
\frac{\partial w^1_1}{\partial y^j_2}(P^j)\biggr)
\pmatrix{c} Y^-(y^j)\\ 0\endpmatrix\Biggr)
\eqno{(59)}
$$
and
$$
{\bold W}^1_{(0)}(\zeta,\nabla_y)
\Biggl(\,\sum\limits_{p=1}^2
\frac{\partial w^1_p}{\partial y^j_p}(P^j)y^j_pe_{(p)}+\frac{1}{2}
\biggl(\frac{\partial w^1_2}{\partial y^j_1}(P^j)+
\frac{\partial w^1_1}{\partial y^j_2}(P^j)\biggr)
\pmatrix Y^+(y^j)\\ 0\endpmatrix\Biggr).
\eqno{(60)}
$$
Here
$Y^\pm(y)=(\pm y_2,y_1)^\top$.
The sum in \Tag(59) is the rigid longitudinal displacement
$d^\equiv(y,0)b^\equiv$
with a~suitable column
$b^\equiv\in{\Bbb R}^3$.
The sum in \Tag(60),
which we denote by
${\bold W}^1_{(0)}(\zeta,\nabla_y)\Cal {P}_{(1j)}(y^j)$,
is a~linear vector-valued function
that creates a~longitudinal deformation of the plate at~$P^j$.
Therefore,
we should complement the inner expansion~\Tag(56)
with terms~\Tag(59)
and the field
$h^2{\bold V}(\Cal {P}_{(1j)};\xi^j)$
constructed at the end of \Par*{Section~5},
which by~\Tag(53) induces the additional rigid displacement
$h^2d(hy^j,hz){\bold b}^{\Cal {P}_{(1j)}}$
of the rod
$\Omega^h_j$,
including its twisting in general.
However,
in the ansatz for
$u^{hj}=u^h\big|_{\Omega^h_j}$
we have already had to include a~similar field
$$
-(8\pi)^{-1}F^jh\ln h\,
d(hy^j,hz){\bold b}^{\eta^j\mapsto\rho_j^2}_{(j)}
\eqno{(61)}
$$
due to the subtrahend in the inner expansion \Tag(56) in the rod-to-plate junction zone.
We emphasize the logarithmic factor in \Tag(61); see \Par*{Section~8}.

The above suggests the following conclusion:
in all terms of the inner expansion near the rod end~$\omega^h_j(0)$
except the last term in \Tag(56)
in the expression on the half-cylinder~$\Xi^j_-$,
only linear vector-valued functions of the form~$d(\xi^j)b^{\dots}$
appear;
as usual,
in asymptotic series with respect to the powers of a~small parameter~$h$
we ignore the terms exponentially  decaying at infinity.
This conclusion could be challenged only for two reasons.
First,
even for cylindrical rods,
meaning that
$H_j=1$
in~\Tag(2),
the asymptotic structure
$$
w^j_3(z_j)e_{(3)}-
\nu\sum\limits_{p=1}^2\eta^j_p\partial_{z_j}w^j_3(z_j)e_{(p)}
+\Cal {X}^{\prime 3}(\eta^j)\partial^2_{z_j}w^j_3(z_j)e_{(p)}
$$
inherited from ansatz \Tag(19)
for
$w^j_3(z_j)=\frac{1}{2}\rho gz_j\big(z_j+2\ell_j)$
leaves discrepancies
in the boundary conditions on the lateral surface of the cylinder.
Second,
discrepancies also arise in the boundary conditions on the lower ends of the rod,
and they are compensated
by constructing the boundary layer in the half-cylinder~${\bold P}^J_+$ (see the end of \Par*{Section~4})
whose exponential decay may require
inhomogeneous boundary conditions~\Tag(28) and~\Tag(29).
In~both cases the variable cross-sections
$\omega^h_j(z_j)$
aggravate these effects.
Even then,
it is reasonable to conjecture that
the loading of the rod induced by this method
is self-balanced and therefore it does not pass to the plate.
However,
the justification of that is absent in this article.

\specialhead\Label{Section 8}
8. Korn's inequality \iftex[70]\else{\rm[70]}\fi\ for junctions
\endspecialhead

An asymptotically sharp,
and therefore anisotropic and weighted, Korn inequality
for constructions made from plates and rods of various shapes
with various types of junctions is justified in [55]; also see [56,~Section~5].
However,
the junction
$\Pi^h$
clamped only along the plate edges
was unfortunately not considered there;
in the articles mentioned in \Par*{Section~1},
both plates and rods are clamped.
Let us fill this gap.

On the plate
$\Omega^h_0$
we have (see [71] as well as [56, Theorem~2.5])
the estimate
$$
\triplenorm{u^h;\Omega^h_0}_0^2\leq K_0E\bigl(u^h,u^h;\Omega^h_0\bigr),
\eqno{(62)}
$$
where the factor
$K_0$
is independent of both the field
$u^h\in
H^1_0\bigl(\Omega^h_0;\Gamma^h_0\bigr)^3$
and the parameter
$h\in(0,h_0]$
for some
$h_0>0$,
while the norm on the left-hand side is given as
$$
\aligned
\triplenorm{u^h;\Omega^h_0}_h^2
&=\int\limits_{\Omega^h_0)}
\Biggl(\,\sum\limits_{k=1}^2\bigg(\biggl|\frac{\partial u^h_k}{\partial y_k}\biggr|^2+
\biggl|\frac{\partial u^h_k}{\partial y_{3-k}}\biggr|^2+
\frac{h^2}{{\bold R}_h^2}
\biggl(\biggl|\frac{\partial u^h_k}{\partial z}\biggr|^2+
\biggl|\frac{\partial u^h_3}{\partial y_k}\biggr|^2\biggr)+
\frac{1}{{\bold R}_h^2}|u^h_k|^2\biggl)
\\\fortex{\noalign{\vskip-3mm}}
&\qquad\qquad +\biggl|\frac{\partial u^h_3}{\partial
z}\biggr|^2+\frac{h^2}{{\bold R}_h^4}|u^h_3|^2\Biggr)\,dx
\quad \text{for}\ \ {\bold R}_h(y)
=h+\operatorname{dist}(y,\partial\omega_0).
\endaligned
\tag63
$$
The weight
${\bold R}_h(y)$,
acquiring order~$h$
in the nearest vicinity of the boundary sections of
$\omega_0$,
incorporates the Dirichlet boundary condition~\Tag(5).
By the simple relation
$$
r_j|{\ln r_j}|\leq c_jh(1+|{\ln h}|)
\ \text{ on the disk}\
{\Bbb B}^2_{h{\bold r}_j}=
\big\{y: \eta^j\in{\Bbb B}^2_{{\bold r}_j}\big\}\subset\omega^h_j,
$$
the anisotropic Korn inequality \Tag(62)
and the one-dimensional Hardy's inequality ``with logarithm,''
$$
\int\limits_0^1\frac{|U(r)|^2}{|{\ln r}|^2}\frac{dr}{r}\leq4
\int\limits_0^1\biggl|\frac{dU}{dr}(r)\biggr|^2rdr\quad\text{for all}\   U\in
C^\infty_c[0,1)
\eqno{(64)}
$$
(see the original article [72]
and the book [73] for instance)
yield the following estimate
on the circular cylinder
$
{\bold Q}^h_j={\Bbb B}^2_{h{\bold r}_j}(P^j) \times(0, h)\subset \Omega^h_0
$
of small height and radius:
$$
h^{-2}(1+|{\ln h}|)^{-2}\bigl(
\bigl\|u^h_1;L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2+
\bigl\|u^h_2;L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2
+h^2\bigl\|u^h_3;L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2\bigr)
\leq
C_jE\bigl(u^h,u^h;\Omega^h_0\bigr).
\tag65
$$

On that cylinder,
represent the field~$u^h$
as
$$
u^h(x)=d(y-P^j, z-h/2){\bold b}^h_{(j)}+u^{h\bot}_{(j)}(x),\quad x\in
{\bold Q}^h_j,
\eqno{(66)}
$$
defining the column
${\bold b}^h_{(j)}\in{\Bbb R}^6$
so as to meet the orthogonality conditions
$$
\int\limits_{{\bold Q}^h_j}d\biggl(y-P^j,z-\frac{h}{2}\biggr)^\top
u^{h\bot}_{(j)}(x)\,dx=0\in{\Bbb R}^6,
\eqno{(67)}
$$
which
according to the results of [52,\,74]
(also see [56, Section~2])
show that
\iftex
$$
\align
&\aligned
h^{-2}&\bigl\|u^{h\bot}_{(j)};
L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2+
\bigl\|\nabla_xu^{h\bot}_{(j)}; L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2
\\
&\leq c_jE
\bigl(u^{h\bot}_{(j)},u^{h\bot}_{(j)}; L^2
\bigl({\bold Q}^h_j\bigr)\bigr)
=c_jE\bigl(u^h,u^h; L^2\bigl({\bold Q}^h_j\bigr)\bigr),
 \endaligned
\tag68
\\
 &\aligned
h^{-2}
&\bigl\|u^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-2h,0)\bigr)\bigr\|^2
+\bigl\|\nabla_xu^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-h,0)\bigr)\bigr\|^2
\\
&\leq
C_j\big(E\bigl(u^{h\bot}_{(j)},u^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-h,h)\bigr)\bigr)
+h^{-2}\bigl\|u^{h\bot}_{(j)} ;
L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2\big).
 \endaligned
\tag69
\endalign
$$
\else
$$
\aligned
h^{-2}&\bigl\|u^{h\bot}_{(j)};
L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2+
\bigl\|\nabla_xu^{h\bot}_{(j)}; L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2
\\
&\leq c_jE
\bigl(u^{h\bot}_{(j)},u^{h\bot}_{(j)}; L^2
\bigl({\bold Q}^h_j\bigr)\bigr)
=c_jE\bigl(u^h,u^h; L^2({\bold Q}^h_j\bigr),
 \endaligned
\tag68
$$
$$
\aligned
h^{-2}
&\bigl\|u^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-2h,0)\bigr)\bigr\|^2
+\bigl\|\nabla_xu^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-h,0)\bigr)\bigr\|^2
\\
&\leq
C_j\big(E\bigl(u^{h\bot}_{(j)},u^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-h,h)\bigr)\bigr)
+h^{-2}\bigl\|u^{h\bot}_{(j)} ;
L^2\bigl({\bold Q}^h_j\bigr)\bigr\|^2\big).
 \endaligned
\tag69
$$
\fi
Here
$$\Omega^{h}_j(-2h,0)=\omega^{h}_j\times(-2h,0)=\{ x \in\Omega^{h}_j:z\in(-2h,0)\},
$$
while the coefficients~$h^{-2}$
appear as a~consequence of dilating the coordinates
before applying the available versions of Korn's inequalities.

Extend \Tag(66) to the rod~$\Omega^h_j$
and introduce the field
${\bold u}^{h}_{(j)}(x)=\chi(1-h^{-1}z) u^{h\bot}_{(j)}(x)$,
equal to zero for
$x\in\omega^h_j(0)\subset \partial\Omega^h_0$
and satisfying by \Tag(69) the relation
$$
\align
E\bigl({\bold u}^{h}_{(j)},{\bold u}^{h}_{(j)};\Omega^h_j\bigr)&\leq 2
\Bigl(E\bigl(u^{h\bot}_{(j)},u^{h\bot}_{(j)};\Omega^h_j\bigr)
+c_j\Bigl(h^{-2}\bigl\|u^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-h,0)\bigr)\bigr\|^2
+\bigl\|\nabla_xu^{h\bot}_{(j)};
L^2\bigl(\Omega^{h}_j(-h,0)\bigr)\bigr\|^2\Bigr)\Bigr)
\\
&\leq C_j \bigl(E\big(u^h,u^h;\Omega^h_j\big)
+E\big(u^h,u^h;\Omega^h_0\big)\bigr).
\endalign
$$

Now apply Korn's inequality (see [54,\,71]
and [56, Section~2, Subsection~6])
for a~thin rod with a~clamped end
and then,
as in [75],
by estimate \Tag(68) for
$u^{h\bot}_{(j)}$,
rearrange it as
$$
\align
\triplenorm{u^{h\bot}_{(j)};\Omega^h_j}_h^2\,:=\,&\int\limits_{\Omega^h_j}
\Biggl(\,\sum\limits_{k=1}^2
\!\bigg(\biggl|\frac{\partial u^{h\bot}_{(j)k}}{\partial
y_k}\biggr|^2\!+\frac{h^2}{h^2+z^2}\biggl(
\biggl|\frac{\partial u^{h\bot}_{(j)k}}{\partial y_{3-k}}\biggr|^2\!+
\biggl|\frac{\partial u^{h\bot}_{(j)k}}{\partial z}\biggr|^2\!+
\biggl|\frac{\partial u^{h\bot}_{(j)3}}{\partial y_k}\biggr|^2\biggr)
+\frac{h^2}{(h^2+z^2)^2}|u^{h\bot}_{(j)k}|^2\bigg)
\\\fortex{\noalign{\vskip-3mm}}
&\qquad+\biggl|\frac{\partial u^{h\bot}_{(j)3}}{\partial
z}\biggr|^2+\frac{1}{h^2+z^2}
\bigl|u^{h\bot}_{(j)3}\bigr|^2\Biggr)\,dx
\,\leq\,  C_j E(u^h,u^h;\Omega^h).
\tag70
\endalign
$$

Inspect the components
${\bold b}^h_{(j)q}$
of the column in~\Tag(66).
Appreciating the structure of~\Tag(21)
and the symmetries of the circular cylinder~${\bold Q}^h_j$,
we find that
$$
\aligned
\bigl|{\bold b}^h_{(j)1}\bigr|
&=\frac{1}{|{\bold Q}^h_j|}
\biggg|\int\limits_{{\bold Q}^h_j} e_{(3)}^\top d
\biggl(y-P^j,z-\frac{h}{2}\biggr)
{\bold b}^h_{(j)}\,dx\biggg|=\frac{1}{|{\bold Q}^h_j|}
\biggg|\int\limits_{{\bold Q}^h_j}\big(u^{h0}_3(x)-u^{hj\bot}_3(x)
\big)dx\biggg|
\\
&\leq
c_jh^{-3/2}\big(\bigl\|u^{h0}_3;
L^2\bigl({\bold Q}^h_j\bigr)\bigr\|
+\bigl\|u^{hj\bot}_3; L^2\bigl({\bold Q}^h_j\bigr)\bigr\|\big)
\leq C_jh^{-1/2}(1+|{\ln h}|){\bold E}^{1/2}_h,
\\
\bigl|{\bold b}^h_{(j)p+1}\bigr|
&=\biggg(\,\int\limits_{{\bold Q}^h_j}z^2\,dx\biggg)^{-1}
\biggg|\int\limits_{{\bold Q}^h_j}
ze_{(p)}^\top d\biggl(y-P^j,z-\frac{h}{2}\biggr)
{\bold b}^h_{(j)}dx\biggg|
\\
&\leq
c_jh^{-5}\bigl|{\bold Q}^h_j\bigr|^{1/2}
\big(\bigl\|zu^{h0}_p;
L^2\bigl({\bold Q}^h_j\bigr)\bigr\|
+\bigl\|zu^{hj\bot}_p; L^2\bigl({\bold Q}^h_j\bigr)\bigr\|\big) \leq
C_jh^{-3/2}(1+|{\ln h}|){\bold E}^{1/2}_h,
\\
\bigl |{\bold b}^h_{(j)p+3}\bigr|
&=\frac{1}{\bigl|{\bold Q}^h_j\bigr|}
\biggg|\int\limits_{{\bold Q}^h_j}\big(u^{h0}_p(x)-u^{hj\bot}_p(x) \big)\,dx\biggg|
\leq  C_jh^{-1/2}(1+|{\ln h}|){\bold E}^{1/2}_h,
\\
|{\bold b}^h_{(j)6}|
&=
\biggg(\frac{1}{2}
\int\limits_{{\bold Q}^h_j}r_j^2dx\biggg)^{-1}
\biggg|\int\limits_{{\bold Q}^h_j}
d^6(y-P^j,0)^\top\big(u^{h0}_p(x)-u^{hj\bot}_p(x) \big)dx\biggg|
\\
&\leq
c_jh^{-7/2}\sum\limits_{p=1}^2\Big(\bigl\|\bigl(y_{3-p}-P^j_{3-p}\bigr)u^{h0}_p;
L^2\bigl({\bold Q}^h_j\bigr)\bigr\|
+\bigl\|\bigl(y_{3-p}-P^j_{3-p}\bigr)
u^{hj\bot}_p; L^2\bigl({\bold Q}^h_j\bigr)\bigr\|\Big)
\\
&\leq
C_jh^{-3/2}(1+|{\ln h}|){\bold E}^{1/2}_h,
\endaligned
\tag71
$$
where
$p=1,2$
and
${\bold E}_h$
is the elastic energy
$\frac{1}{2} E(u^h,u^h;\Omega^h_0)$
stored in the plate.
To process the norms in the space
$L^2({\bold Q}^h_j)$,
we used \Tag(65) and \Tag(68).
\goodbreak

Using \Tag(71),
estimate the Sobolev norms of the rigid displacement
$d({\dots}){\bold b}^h_{(j)}$
with carefully tailored weight factors
which are smaller than those arising in estimate \Tag(70) for the component
$u^{h\bot}_{(j)}$
of~\Tag(66).
Combining the result with Korn's inequality \Tag(62), \Tag(63) on the plate,
by analogy with~[75]
we arrive at the following statement:

\proclaim{Theorem~2}\Label{T2}
On the junction
$\Pi^h$,
the following anisotropic weighted Korn inequality holds:
$$
\align
&\triplenorm{u^h;\Omega^h_0}^2_h
\\
&\quad+
\sum\limits_{j=1}^J\,\int\limits_{\Omega^h_j}
\Biggl(\,\sum\limits_{k=1}^2\!\bigg(
\biggl|\frac{\partial u^h_k}{\partial y_k}\bigg|^2\!\!+\frac{h^2(1+|{\ln h}|)^{-
2}}{h^2+z^2}\biggl(\biggl|\frac{\partial u^h_k}{\partial y_ {3-k}}\biggr|^2\!\!+
\biggl|\frac{\partial u^h_k}{\partial z}\biggr|^2\!\!
+\biggl|\frac{\partial u^h_3}{\partial y_k}\biggr|^2\biggr)
+\frac{h^2(1+|{\ln h}|)^{-2}}{(h^2+z^2)^2}\big|u^h_k\big|^2\bigg)
\\\fortex{\noalign{\vskip-3mm}}
&\qquad\qquad\qquad+\biggl|\frac{\partial u^h_3}{\partial z}\biggr|^2\!\!+
\frac{h(1+|{\ln h}|)^{-2}}{(h^2+ z^2)^{1/2}}
\big|u^h_3\big|^2\Biggr)\,dx
\,\leq\, K_\Pi E(u^h, u^h;\Pi^h),
\tag72
\endalign
$$
where
$\triplenorm{u^h;\Omega^h_0}_h$
is given in~{\rm\Tag(63)},
while the factor~$K_\Pi$
is independent of the field
$u^h\in H^1_0\bigl(\Pi^h;\Gamma^h_0\bigr)^3$
and the parameter
$h\in(0,h_\Pi]$
for some
$h_\Pi>0$.
\endproclaim

\demo{Remark}
The inequality in \Tag(72) disregards the details of loading the junction~$\Pi^h$
by gravity;
however,
in a~certain sense it is asymptotically sharp.
Indeed,
the functional
$E(\cdot, \cdot;\Omega^h_0)$
and the squared norm
$\triplenorm{\,{\cdot}\,;\Omega^h_0}_h$,
calculated for the main term
${\bold W}^h_{(0)}(\zeta,\nabla_{y})e_{(3)}w_3(y)$
of~\Tag(30),
acquire order
$h^{-1}$,
while each integral in the sum over
$j=1,\dots,J$,
found from the term
$h^{-2}d^\dag(y^j,z)d^\dag(\nabla_y,0)^\top w_3(P^j)$
of~\Tag(54),
equals
$O(h^{-1}(1+|{\ln h}|)^{-2})$.
As for junctions of other shapes (see the survey [56])
the logarithmic factor coming from Hardy's inequality~\Tag(64)
cannot be detected by using asymptotic ans\"atze for solutions to problems with smooth right-hand sides,
because the logarithm is attached only to higher-order asymptotic terms; see~\Tag(61).
The tricks of [56, Section~5] enable us to
remove the logarithm from the multipliers of
the derivatives of displacements in the integrand of \Tag(72),
but the author is unaware whether
that is possible for the displacements themselves.
\enddemo

Using Korn's inequality of \Par{T2}{Theorem~2},
the justification of the asymptotic representations of
the solution to problem \Tag(4)--\Tag(6) obtained in \Par*{Section~6}
follows the standard scheme;
see the books [5,\,30],
the articles [4,\,8,\,9], and other publications.
To reduce the volume of this article,%
%
\footnote"${}^{4)}$"{Sharp estimates for asymptotic remainders
will be presented in full
in the following publications of the author,
devoted to the analysis of eigenoscillations of the junction
$\Pi^h$,
i.e.,
to the spectral problem of elasticity.}
%
we restrict ourselves to stating the result
for the most interesting attributes of the displacement field~$u^h$.

\proclaim{Theorem~3}
%
{\rm (1)}
The restriction
$u^{h0}$
to the plate
$\Omega^h_0$
of the solution to problem {\rm\Tag(4)--\Tag(6)}
satisfies the estimate
$$
\triplenorm{
u^{h0}-\big(h^{-2}\Cal {W}^0_{(0)}+
h^{-
1}\Cal {W}^1_{(0)}+\Cal {W}^2_{(0)}\big)e_{(3)}w_3;\Omega^h_0}
^2_h \leq C^0_\Omega,
\eqno{(73)}
$$
which involves the norm {\rm\Tag(63)},
the matrix differential operators {\rm\Tag(13)},
and the solution to problem {\rm\Tag(8),~\Tag(9)},
while the factor~$C^0_\Omega$
is independent of the parameter
$h\in\bigl(0,h^0_\Omega\bigr]$
for some
$h^0_\Omega>0$.

{\rm (2)}
The longitudinal displacements
$u^{hj}_3$
in the rods
$\Omega^h_j$,
for
$j=1,\dots,J$,
satisfy the estimates
$$
\big\|u^{hj}_3
- h^{-2}d^\dag_{(3)}(y^j)d^\dag_{(3)}
\bigl(\nabla_y^\top,0\bigr)w_3(P^j);
L^2\bigl(\Omega^h_j\bigr)\big\|
+
h^{-1/2}\big\|\varepsilon_{33}\big(u^{hj}\big)- h^{-1}\partial_{z_j}w^j_3;
L^2\bigl(\Omega^h_j\bigr)\big\|\leq C^j_\Omega,
\tag74
$$
in which
$d^\dag_{(3)}$
is the bottom row of the block
$d^\dag$
of the matrix {\rm\Tag(21)},
the functions~$w_3$
and~$w^j_3$
are indicated in \Par*{Section~{\rm6}},
while the factor~$C^j_\Omega$
is independent of the parameter
$h\in\bigl(0,h^j_\Omega\bigr]$
for some
$h^j_\Omega>0$.
\endproclaim

The derivation of \Tag(73) generally repeats the material of Chapter~4 of~[31];
the changes due to the adjunction of the rods are negligible
because the displacement
$e_{(3)}w_3(P^j)$
satisfies both the equations and the boundary conditions in the rod,
while the order
$1=h^0$
of the bound is determined by the boundary layer near the edge
$\Gamma^h_0$;
see \Par*{Section~4} and [31, Chapter~4].
We can refine the asymptotics of the deflection itself; i.e.,
$$
\bigl\|u^h_3-h^{-2}w_3-h^{-1}w^\circ_3;
L^2\bigl(\Omega^h_0\bigr)\bigr\|\leq c^0_\Omega.
$$
Together with that,
due to the
$O(|{\ln r_j}|)$
singularities of the second derivatives of
$w^\circ_3$,
similar improvements in the asymptotic expansions for stresses and  strains
are impossible:
in them we need to invoke the boundary layers of \Par*{Section~5}.
These boundary layers determine the order of the bound in~\Tag(74),
which in its simplified version referring to
the longitudinal displacement and deformation of the rod
is derived with the use of the procedure of [31, Chapter~5].

\specialhead\Label{Section 9}
9. Discussion
\endspecialhead

The equalities in~\Tag(4) enabled us to substantially simplify
calculations and formulations of the results.
When the first equalities are violated,
the Kirchhoff--Clebsch system
$$
\Cal {D}(\partial_{z}){A}^j(z)\Cal {D}(\partial_{z})
w^j(z)=f^j(z),\quad z\in(-\ell_j,0),
\eqno{(75)}
$$
fails to decouple into separate scalar equations \Tag(24) and \Tag(25),
but this circumstance has very small effect on
the main terms of asymptotics for the stress-strain state of the junction~$\Pi^h$
described in \Par*{Section~6}.
The interaction of the longitudinal deformation of the rods
$\Omega^h_j$
with their bending and twisting (see [33] and [31,~Chapter~5, Section~3] for instance)
coming from the fullness of the matrix~${A}^j$
in system~\Tag(75),
in~particular changes the problem of seeking
the longitudinal deformation of the plate
$\Omega^h_0$.
Together with that,
the considerable smallness is preserved
of the horizontal displacements in the plate
in comparison with its deflection
and accordingly translations of the rods in the vertical direction.
Precisely the same effects arise
in the case of an~anisotropic and inhomogeneous material,
for which the structure of system~\Tag(75) is preserved;
see [31, Chapter~5, Section~3] among others.

A~plate with variable cross-section
or a~plate made of an~anisotropic inhomogeneous material,
a~composite for instance,
might have a~more significant influence on
the asymptotic structure of the stress-strain state of the junction
$\Pi^h$
because the vector-valued function
$w^0=\bigl(w^0_1,w^0_2,w^0_3\bigr)^\top$
is found from the system of differential equations
$$
\Cal {D}^0(\nabla_y)\Cal {A}^0(y)\Cal {D}^0(\nabla_y)^\top
w^0(y)=f^0(y),\quad y\in\omega_0,
$$
with boundary conditions \Tag(9) and~\Tag(17).
Here
$\Cal {D}^0(\nabla_y)$
is the~$3\times6$ matrix of differential operators mentioned in~\Tag(14);
$\Cal {A}^0$
is the~symmetric positive definite matrix function of size~$3\times6$
calculated from the elastic moduli of the plate material.
In general the matrix
$\Cal {A}^0$
is entirely full,
and therefore the force of gravity
$\rho ge_{(3)}$
creates not only deflection,
but also longitudinal displacements in the plate.
However, according to the results of [31, Chapter~4, Section~2, Subsection~4]
for a~homogeneous anisotropic cylindrical plate,
or even a~composite plate with different layers of constant thickness and elastic properties,
there is a~coefficient~$\alpha$,
not necessarily in the segment
$[0,1]$,
such that
the translation of the origin of the applicate axis to the point
$z^0=h\alpha$
makes the matrix
$\Cal {A}^0$
block diagonal,
while its bottom row becomes
$\big(0,0,\Cal {A}^0_3\big)$;
in other words,
gravity does not induce in the main
a~longitudinal deformation of the plate,
and so the asymptotic formulas obtained in this article are largely preserved
under the translation of the global Cartesian coordinate system~$x$.
If the matrix $\Cal {A}^j(y)$ is full,
for the rod
$\Omega^h_j$
not only is a~longitudinal deformation characteristic,
but also its deflection and twisting are,
see [31, Chapter~5; 33] among others,
and we have to account for this circumstance
while constructing boundary layers near the point~$P^j$.

The interpretations of the junction
$\Pi^h$
mentioned in \Par*{Section~1}
implicitly assume that the endpoints of the rod have conical tips,
i.e.,
$H_j(-\ell_j)=0$
and
$\partial_{z_j}H_j(-\ell_j)>0$.
In~this case, \Tag(24), \Tag(27), or \Tag(75) acquire degenerating coefficients,
and according to the results of [27,\,76]
boundary conditions \Tag(28) and \Tag(29) for them
are not necessary,
while the asymptotic structures change insignificantly.
\goodbreak

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\jour                Russian Acad. Sci. Izv. Math.
\yr                  1994
\vol                 42
\issue               1
\pages               183--217
\endref

\ref\no 64
\by                  Nazarov~S.A.
\paper               On the flow the water under a~still stone
\jour                Sb. Math.
\yr                  1995
\vol                 186
\issue               11
\pages               1621--1658
\endref

\ref\no 65
\by                  Nazarov~S.A. and Pileckas~K.
\paper               The asymptotic properties of the solution to the Stokes problem in domains that are layer-like at infinity
\jour                J.~Math. Fluid Mech.
\yr                  1999
\vol                 1
\issue               2
\pages               131--167
\endref

\ref\no 66
\by         Vladimirov~V.S.
\book       Generalized Functions in Mathematical Physics
\publaddr   Moscow
\publ       Nauka
\yr         1979
\lang       Russian
\endref

\ref\no 67
\by                   Oleinik~O.A. and Yosifian~G.A.
\paper                On the asymptotic behaviour at infinity of solutions in linear elasticity
\jour                 Arch. Rational Mech. Anal.
\yr                   1983
\vol                  78
\issue                1
\pages                29--53
\endref
%pr

\ref\no 68
   \by          Van Dyke~M.
      \book     Perturbation Methods in Fluid Mechanics
      \publ     Academic
      \publaddr New York and London
      \yr       1964
\finalinfo      Appl. Math. Mech., vol.~8
      \endref
      %pr

\ref\no 69
\by                Il'in~A.M.
       \book       Matching of Asymptotic Expansions of Solutions of Boundary Value Problems
       \publaddr   Providence
       \publ       Amer. Math. Soc.
       \yr         1992
       \finalinfo  Transl. Math. Monogr., vol.~102
\endref

\ref\no 70
\by                Korn~A.
\paper             Solution g\'{e}n\'{e}rale du probl\`{e}me
                   d'\'{e}quilibre dans la th\'{e}orie l'\'{e}lasticit\'{e}, dans le cas
                   o\`{u} les efforts sont donn\'{e}s \`{a} la surface
\jour              Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys.~(2)
\yr                1908
\vol               10
\pages             165--269
\endref

\ref\no 71
\by                Nazarov~S.A.
      \paper       The Korn inequalities which are asymptotically sharp for thin domains
      \jour        Vestnik St. Petersburg University Math.
      \yr          1992
      \vol         2
      \issue       8
      \pages       19--24
\endref

\ref\no 72
\by                Hardy~G.H.
\paper             Note on a~theorem of Hilbert
\jour              Math.~Z.
\yr                1920
\vol               6
 \iftex
\issue             3--4
\else
\issue             3
\fi
\pages             314--317
\endref

\ref\no 73
 \by               Hardy~G.H., Littlewood~J.E., and P\'olya~G.
 \book             Inequalities
 \publ             Cambridge University
 \publaddr         Cambridge
 \yr               1988
\endref

\ref\no 74
 \by                Duvaut~G. and Lions~J.-L.
      \book         Inequalities in Mechanics and Physics
                    %Les inequations en mecanique et physique Dunod, Paris, 1972
       \publ        Springer
      \publaddr     Berlin
      \yr           1976
\endref

\ref\no 75
 \by                      Nazarov~S.A. and Slutskii~A.S.
   \paper                 Korn's inequality for an~arbitrary system of~distorted thin rods
 \jour                    Sib.  Math.~J.
 \yr                      2002
\vol                      43
 \issue                   6
 \pages                   1069--1079
\endref

\ref\no 76
\by                   Mikhlin~S.G.
\book                 The Problem of the Minimum of a~Quadratic Functional
\publaddr             San Francisco, London, and Amsterdam
\publ                 Holden-Day
\yr                   1965
\finalinfo            Holden-Day Series in Mathematical Physics
\endref
\endRefs

\enddocument