Also at ]Center for Advanced Materials Research, Research Institute of
Sciences and Engineering, University of Sharjah, Sharjah, P.O. Box 27272,
United Arab Emirates

# Generalized Distribution Function of Relaxation Times with the Davidson-Cole
Model as a Kernel

Anis Allagui∗<EMAIL_ADDRESS>Dept. of Sustainable and Renewable Energy
Engineering, University of Sharjah, Sharjah, P.O. Box 27272, United Arab
Emirates  [  Dept. of Electrical and Computer Engineering, Florida
International University, Miami, FL33174, United States  Ahmed Elwakil  Dept.
of Electrical Engineering, University of Sharjah, PO Box 27272, Sharjah,
United Arab Emirates

###### Abstract

In this paper we propose a generalized distribution function of relaxation
times (DFRT) considering the Davidson-Cole model as an elementary process
instead of the standard Debye model. The distribution function is retrieved
from the inverse of the generalized Stieltjes transform expressed in terms of
iterated Laplace transforms. We derive computable analytical expressions of
the generalized DFRT for some of the most known normalized impedance (or
admittance) models including the constant phase element, the Davidson-Cole,
Havriliak-Negami and the Kohlrausch-Williams-Watts models.

Keywords: Fox $H$-function; distribution of relaxation time; impedance
spectroscopy.

##  I Introduction

When a linear system is subjected to a frequency-domain input signal
$X(s)\;(s=i\omega$, $\omega$ is the angular frequency) and resulting in a
frequency-domain output signal $Y(s)$, the transfer function describing the
dynamics of the system can be written as the ratio:

$Z(s)=\frac{Y(s)}{X(s)}$ (1)

If $Z(s)$ corresponds to an electrical impedance, then $Y(s)$ is a voltage
signal and $X(s)$ is a current signal. When used for treating capacitive
systems with a single time constant $\tau$, the function $Z(s)$ is given by:

$Z_{D}(s)=R_{s}+\frac{R_{p}}{1+s\tau}$ (2)

where $R_{s}$ and $R_{p}$ are two resistive constants (in units of Ohm). The
fraction:

$Q_{D}(s)=({1+s\tau})^{-1}$ (3)

is the normalized or reduced Debye model, originally used for describing the
frequency-dependent complex susceptibilities of dielectrics. In Nyquist
representation of real vs. imaginary parts of impedance, Eq. 2 shows a
symmetric semi-circular shape, while its time response obtained by inverse
Laplace transform (LT) corresponds to an exponential decay function:

$z_{D}(t)=R_{s}\,\delta(t)+R_{p}\,\tau^{-1}{e^{-t/\tau}}$ (4)

The convolution of the time-domain functions $z_{D}(t)$ with the current
$i(t)$ gives the resulting voltage $v(t)$ as:

$v(t)=(z_{D}\circledast
i)(t)=\int\limits_{-\infty}^{+\infty}z_{D}(t-\xi)i(\xi)d\xi$ (5)

However, if the system under consideration is more complex than the idealized
one, e.g. when the system is porous, amorphous or consists of disordered,
inhomogeneous structures, the Debye model is known to be insufficient for
capturing accurately its frequency behavior. By applying the principle of
superposition, and with the assumption that linearity still holds [1], the
impedance can be expressed by the integral:

$Z_{D}^{*}(s)=R_{s}+R_{p}\int\limits_{0}^{\infty}\frac{g(\tau)}{1+s\tau}d\tau$
(6)

where $g(\tau)$ is a distribution function of relaxation times (DFRT). In
normalized form, we rewrite Eq. 6 as:

$Q_{D}^{*}(s)=\frac{Z_{D}^{*}(s)-R_{s}}{R_{p}}=\int\limits_{0}^{\infty}\frac{g(\tau)}{1+s\tau}d\tau$
(7)

With this approach it is assumed that the relaxing macroscopic system consists
of a large number of subsytems, each of which, upon the application of a step
input, relaxes exponentially to zero with its own single relaxation time
constant [2]. There have been several efforts to construct reliable and robust
numerical algorithms for retrieving the DFRT of a system from its spectral
response (Eq.7), including Fourier transform techniques [3, 4], Tikhonov
regularization [5, 6, 7], Max Entropy [8, 9], genetic algorithm [10], etc.
[11, 12, 13, 2].

In general terms, Eq.7 should be rather viewed as:

$Q_{D}^{*}(s)=\int\limits_{0}^{\infty}{g(\tau)}k({s,\tau})d\tau$ (8)

where $k({s,\tau})$ is the kernel of the integral. In the case of the
elementary Debye response, we have $k({s,\tau})=({1+s\tau})^{-1}$. But there
are several possible extensions of the Debye model that can be considered.
These include for instance the Cole-Cole (CC) model [14],

$Q_{\text{CC}}(s)=({1+(s\tau_{\alpha})^{\alpha}})^{-1}\quad(0<\alpha\leqslant
1)$ (9)

the Davidson-Cole (DC) model [15],

$Q_{\text{DC}}(s)={(1+s\tau_{\beta})^{-\beta}}\quad(0<\beta\leqslant 1)$ (10)

and the Havriliak-Negami (HN) model [16],

$Q_{\text{HN}}(s)={(1+(s\tau_{{\nu}})^{\alpha})^{-\beta}}\quad(0<\alpha,\beta\leqslant
1)$ (11)

which are capable of describing a wide range of depressed and asymmetric semi-
circular shapes in the complex plane with different levels of accuracy. In the
time domain the relaxation processes corresponding to these models are non-
exponential. By inverse LT [17], the response of the CC function to a step
excitation is [18]:

$z_{\text{CC}}(t)=\tau_{\alpha}^{-1}({t}/{\tau_{\alpha}})^{\alpha-1}E_{\alpha,\alpha}^{1}\left[-(t/\tau_{\alpha})^{\alpha}\right]$
(12)

where [17, 19]

${E}_{\alpha,\beta}^{\gamma}(z):=\sum\limits_{k=0}^{\infty}\frac{(\gamma)_{k}}{\Gamma(\alpha
k+\beta)}\frac{z^{k}}{k!}\quad(\alpha,\beta,\gamma\in\mathbb{C},\mathrm{Re}({\alpha})>0)$
(13)

(with
$(\gamma)_{k}=\gamma(\gamma+1)\ldots(\gamma+k-1)=\Gamma(\gamma+k)/\Gamma(\gamma)$)
is the three-parameter Mittag-Leffler (ML) function. The response functions of
the DC and HN models are [18]:

$\displaystyle z_{\text{DC}}(t)$
$\displaystyle=\tau_{\beta}^{-1}({t}/{\tau_{\beta}})^{\beta-1}E_{1,\beta}^{\beta}\left[-t/\tau_{\beta}\right]$
(14)
$\displaystyle=\frac{1}{\Gamma(\beta)}\tau_{\beta}^{-1}({t}/{\tau_{\beta}})^{\beta-1}e^{-t/\tau_{\beta}}$
(15)

and

$z_{\text{HN}}(t)=\tau_{\nu}^{-1}({t}/{\tau_{\nu}})^{\alpha\beta-1}E_{\alpha,\alpha\beta}^{\beta}\left[-(t/\tau_{\nu})^{\alpha}\right]$
(16)

respectively.

Returning to Eq.7, Florsch, Revil, and Camerlynck [20] proposed for instance
the possibility of using the HN model as the kernel of the integral. This
gives for the normalized impedance function:

$Q^{*}_{\text{HN}}(s)=\int\limits_{0}^{\infty}\frac{g(\tau)}{(1+(s\tau_{\nu})^{\alpha})^{\beta}}d\tau\quad(0<\alpha,\gamma\leqslant
1)$ (17)

The problem here is to retrieve $g(\tau)$ assuming that the parameters
$\alpha$ and $\beta$ are known. The goal is that one may want to decompose the
spectral response of the system using for example the Warburg model as a basis
function. This can be obtained by setting $\beta=1$ and $\alpha=0.5$ in Eq.
17. In any case the choice of values for the parameters $\alpha$ and $\beta$
should be justified in advance based on knowledge of the underlying physics.
The authors carried out their analysis for recovering the distribution
function $g(\tau)$ from Eq. 17 by numerical inversion algorithms given the
ill-posedness of the problem.

In the same line of thought, the purpose of this work is to present a general
procedure for obtaining _analytically_ the DFRT associated with an impedance
model when the elementary relaxation process is taken to be the DC model
(Eq.18) instead of the Debye model. The motivation for this generalization
applies for instance to macroscopic systems of self-similar character at a
microscopic level, with the DC type of response in the frequency domain [21,
22].

##  II Theory

Consider the integral transform:

$Q^{*}_{\text{DC}}(s)=\int\limits_{0}^{\infty}\frac{g(\tau)}{(1+s\tau)^{p}}d\tau$
(18)

which is equivalent to Eq. 8 with the kernel $k({s,\tau})=({1+s\tau})^{-p}$.
We will drop both the subscript $"\text{DC}"$ and the superscript $"^{*}"$
from now on for ease of notation. Using the change of variable
$\lambda=\tau^{-1}$, and the distribution function:

$D(\lambda)=\lambda^{p-2}g(\lambda^{-1})$ (19)

Eq. 18 is rewritten as

$Q(s)=\int\limits_{0}^{\infty}\frac{D(\lambda)}{(s+\lambda)^{p}}d\lambda$ (20)

which is the generalized Stieltjes transform (ST) [23]. With $p=1$ we recover
the standard ST [24]. Following the procedure of Schwarz [25], by using the
Mellin transform of a function $f(x)$ defined as
$\mathcal{M}[f(x);s]=\int_{0}^{\infty}x^{s-1}f(x)dx\;(x>0)$, we rewrite the
kernel ${(s+\lambda)^{-p}}$ in Eq. 20 as:

${(s+\lambda)^{-p}}=\frac{1}{\Gamma(p)}\int\limits_{0}^{\infty}x^{p-1}e^{-x(s+\lambda)}dx$
(21)

Plugging it in Eq. 20 makes $Q(s)$ to be:

$Q(s)=\frac{1}{\Gamma(p)}\int\limits_{0}^{\infty}x^{p-1}e^{-xs}\int\limits_{0}^{\infty}D(\lambda)e^{-x\lambda}d\lambda\,dx$
(22)

From the definition of the LT, we recognize that:

$Q(s)=\frac{1}{\Gamma(p)}\mathcal{L}\left[x^{p-1}\mathcal{L}[D(\lambda);x];s\right]$
(23)

and with the inverse LT, we recover $D(\lambda)$ from the iterations:

$D(\lambda)=\Gamma(p)\mathcal{L}^{-1}\left[x^{1-p}\mathcal{L}^{-1}[Q(s);x];\lambda\right]$
(24)

The DFRT $g(\tau)$ is then obtained from Eq. 19 as:

$g(\tau)=\tau^{p-2}D(\tau^{-1})$ (25)

We verify that for $p=1$,
$D(\lambda)=\mathcal{L}^{-1}\left[\mathcal{L}^{-1}[Q(s);x];\lambda\right]$ and
$g(\tau)=\tau^{-1}D(\tau^{-1})$ as they should be [26, 24].

##  III Examples

###  III.1 Constant phase element

Figure 1: Plots of $g_{c}(\tau)$ in Eq. 27 for different combinations of the
parameters $\alpha$ and $p$, and with $\tau_{c}=1$

As a first example we consider the case of the reduced impedance of the
constant phase element given by:

$Q_{c}(s)={(s\tau_{c})^{-\alpha}},\quad 0<\alpha\leqslant 1$ (26)

where $\tau_{c}$ is a positive constant. This is basically the case of the CC
model in Eq. 9 when $s\tau_{\alpha}\gg 1$. Using Eq. 24 followed by Eq. 25 we
obtain:

$g_{c}(\tau)=\frac{\Gamma(p)}{\Gamma(\alpha)\Gamma(p-\alpha)}\tau_{c}^{-\alpha}\tau^{\alpha-1},\;\;p>\alpha$
(27)

Given that for physical reasons the distribution function $g(\tau)$ cannot be
negative, we need to ensure the condition $0<\alpha<p<1$. Otherwise, the term
${\Gamma(p)}/[{\Gamma(\alpha)\Gamma(p-\alpha)}]$ should be positive. We note
that for $p=1$ (i.e. Debye model as a kernel in Eq. 8:
$k({s,\tau})=({1+s\tau})^{-1}$), $g_{c}(\tau)$ simplifies to

$g_{c}(\tau)=\pi^{-1}\sin(\alpha\pi)\tau_{c}^{-\alpha}\tau^{\alpha-1}$ (28)

where we used Euler’s reflection formula for the gamma function,
${\pi}/{\sin(\pi z)}=\Gamma(z)\Gamma(1-z)$ ($z\notin\mathbb{Z}$). This means
that a CPE impedance of parameters $\tau_{c}$ and $\alpha$ can equally be
constructed from the integral:

$Q_{c}(s)={(s\tau_{c})^{-\alpha}}=\frac{\tau_{c}^{-\alpha}}{\Gamma(\alpha)\Gamma(1-\alpha)}\int\limits_{0}^{\infty}\frac{\tau^{\alpha-1}}{1+s\tau}d\tau$
(29)

using the Debye model as a basis function [24], or from the integral:

$Q_{c}(s)={(s\tau_{c})^{-\alpha}}=\frac{\Gamma(p)\,\tau_{c}^{-\alpha}}{\Gamma(\alpha)\Gamma(p-\alpha)}\int\limits_{0}^{\infty}\frac{\tau^{\alpha-1}}{(1+s\tau)^{p}}d\tau$
(30)

using the DC model (of any parameter $p>\alpha$) as a basis function.

In Fig. 1 we show plots of $g_{c}(\tau)$ given by Eq. 27 for different
combinations of the parameters $\alpha$ and $p$, and with $\tau_{c}=1$. We can
see that as the value of $p$ tends to that of $\alpha$, the distribution
function becomes narrower and less disperse, which can be interpreted as a
more efficient way for reconstructing the CPE impedance.

###  III.2 Davidson-Cole model

Figure 2: Plots of $g_{\beta}(\tau)$ in Eq. 32 for different combinations of
the parameters $\beta$ and $p$, and with $\tau_{\beta}=1$

If we consider the impedance function itself to be of a DC type, i.e.

$Q_{\beta}(s)={(1+s\tau_{\beta})^{-\beta}}=\int\limits_{0}^{\infty}\frac{g_{\beta}(\tau)}{(1+s\tau)^{p}}d\tau$
(31)

with $0<\beta,p\leqslant 1$ and $\tau_{\beta}$ a reference time constant, we
find the $g_{\beta}(\tau)$ function to be

$\displaystyle g_{\beta}(\tau)$
$\displaystyle=\frac{\Gamma(p)\tau_{\beta}^{-\beta}}{\Gamma(\beta)\Gamma(p-\beta)}\tau^{p-2}\left({\tau^{-1}}-{\tau_{\beta}^{-1}}\right)^{p-\beta-1},\text{
for }\tau<\tau_{\beta}$ $\displaystyle=0,\text{ for
}\tau\geqslant\tau_{\beta}$ (32)

We verify that for $p=1$, $g_{\beta}(\tau)$ simplifies to [24]:

$\displaystyle g_{\beta}(\tau)$
$\displaystyle=\frac{\sin(\beta\pi)}{\pi}\tau_{\beta}^{-\beta}\tau^{-1}(\tau^{-1}-\tau_{\beta}^{-1})^{-\beta},\text{
for }\tau<\tau_{\beta}$ $\displaystyle=0,\text{ for
}\tau\geqslant\tau_{\beta}$ (33)

and for $p=\beta$, we obtain evidently the Dirac delta function:

$g_{\beta}(\tau)=\delta(\tau-\tau_{\beta})$ (34)

In Fig. 2 we show the highly asymmetric plots of $g_{\beta}(\tau)$ in Eq. 32
for different combinations of the parameters $\beta$ and $p$, and with
$\tau_{\beta}=1$. We have the same remark as for the case of the CPE: a DC-
type of impedance can be reconstructed from the continuous linear
superposition of Debye models, but also from other DC models. Furthermore, Eq.
31, with $g_{\beta}(\tau)$ given by Eq. 32, shows one possible mapping
relationship between two DC models with different parameters.

Figure 3: Schematic of a 1D semi-infinite two-impedance self-similar ladder
network in the $x$-direction, coupled with a semi-infinite bifurcating binary
tree network in the $y$-direction.

###  III.3 Havriliak–Negami model

Now we show the inversion procedure for the the case of the HN model, i.e.:

$Q_{\nu}(s)={(1+(s\tau_{\nu})^{\alpha})^{-\beta}}=\int\limits_{0}^{\infty}\frac{g_{\nu}(\tau)}{(1+s\tau)^{p}}d\tau$
(35)

from which the CC model can be a recovered as a special case by setting
$\beta=1$. For this example, it is more convenient to use the representation
of $Q_{\nu}(s)$ in terms of the Fox $H$-function, as we have shown in more
detail in a separate contribution of ours [24] for the case of
$k({s,\tau})=({1+s\tau})^{-1}$. We recall that the $H$-function of order
$(m,n,p,q)\in\mathbb{N}^{4}$, ($0\leqslant n\leqslant p$, $1\leqslant
m\leqslant q$) and with parameters $A_{j}\in\mathbb{R}_{+}\;(j=1,\ldots,p)$,
$B_{j}\in\mathbb{R}_{+}\;(j=1,\ldots,q)$, $a_{j}\in\mathbb{C}\;(j=1,\ldots,p)$
and $b_{j}\in\mathbb{C}\;(j=1,\ldots,q)$ is defined for
$z\in\mathbb{C},\;z\neq 0$ by the contour integral [27, 28, 29, 30]:

$H^{m,n}_{p,q}\left[z|^{(a_{1},A_{1}),\ldots,(a_{p},A_{p})}_{(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\right]=\frac{1}{2\pi
i}\int_{L}h(s)z^{-s}ds$ (36)

where the integrand $h(s)$ is given by:

$h(s)=\frac{\left\\{\prod\limits_{j=1}^{m}\Gamma(b_{j}+B_{j}s)\right\\}\left\\{\prod\limits_{j=1}^{n}\Gamma(1-a_{j}-A_{j}s)\right\\}}{\left\\{\prod\limits_{j={m+1}}^{q}\Gamma(1-b_{j}-B_{j}s)\right\\}\left\\{\prod\limits_{j={n+1}}^{p}\Gamma(a_{j}+A_{j}s)\right\\}}$
(37)

In Eq. 36, $z^{-s}=\exp\left[-s(\ln|z|+i\arg z)\right]$ and $\arg z$ is not
necessarily the principal value. The contour of integration $L$ is a suitable
contour separating the poles of $\Gamma(b_{j}+B_{j}s)$ ($j=1,\ldots,m$) from
the poles of $\Gamma(1-a_{j}-A_{j}s)$ ($j=1,\ldots,n$). An empty product is
always interpreted as unity. We rewrite $Q_{\nu}(s)$ as

$Q_{\nu}(s)=\frac{1}{\Gamma(\beta)}H^{1,1}_{1,1}\left[(s\tau_{\nu})^{\alpha}\left|\begin{array}[]{c}(1-\beta,1)\\\
(0,1)\hfill\\\ \end{array}\right.\right]$ (38)

using the formula [28]:

$\displaystyle(1-z)^{-\alpha}=\frac{1}{\Gamma(\alpha)}H^{1,1}_{1,1}\left[-z\left|\begin{array}[]{c}(1-\alpha,1)\\\
(0,1)\hfill\\\ \end{array}\right.\right]$ (41)

Knowing that the LT of an $H$-function is again an $H$-function, and is given
by [31, 32, 28]:

$\displaystyle\mathcal{L}$
$\displaystyle\left[t^{\rho-1}H_{p,q}^{m,n}\left[at^{\sigma}\left|\begin{array}[]{c}(a_{p},A_{p})\\\
(b_{q},B_{q})\\\ \end{array}\right.\right];u\right]$ (44)
$\displaystyle=u^{-\rho}H^{m,n+1}_{p+1,q}\left[au^{-\sigma}\left|\begin{array}[]{c}(1-\rho,\sigma),(a_{1},A_{1}),\ldots,(a_{p},A_{p})\\\
(b_{1},B_{1}),\ldots,(b_{q},B_{q})\\\ \end{array}\right.\right]$ (47)

and the inverse LT is given by [28]:

$\displaystyle\mathcal{L}^{-1}$
$\displaystyle\left[u^{-\rho}H_{p,q}^{m,n}\left[au^{\sigma}\left|\begin{array}[]{c}(a_{p},A_{p})\\\
(b_{q},B_{q})\\\ \end{array}\right.\right];t\right]$ (50)
$\displaystyle=t^{\rho-1}H_{p+1,q}^{m,n}\left[at^{-\sigma}\left|\begin{array}[]{c}(a_{p},A_{p}),\ldots,(a_{1},A_{1}),(\rho,\sigma)\\\
(b_{1},B_{1}),\ldots,(b_{q},B_{q})\\\ \end{array}\right.\right]$ (53)

we find, after employing some identities between $H$-functions that [29, 28]:

$g_{\nu}(\tau)=\frac{\Gamma(p)}{\tau_{\nu}\,\Gamma(\beta)}H^{1,1}_{2,2}\left[(\tau/\tau_{\nu})^{\alpha}\left|\begin{array}[]{c}(1-1/\alpha,1),(p-1,\alpha)\\\
(\beta-1/\alpha,1),(0,\alpha)\hfill\\\ \end{array}\right.\right]$ (54)

This general expression includes as special cases the DFRT for the DC and CC
reduced impedances for different values of $p$ ($0<p\leqslant 1$).

As a particular example, let us consider the electrical circuit model shown in
Fig. 3, which represents a 1D semi-infinite two-impedance ($Z_{1}$ and
$Z_{2}$) self-similar ladder network in the $x$-direction, coupled with a
semi-infinite bifurcating binary tree network in the $y$-direction. Its
equivalent impedance function as viewed from the input terminal of the network
is given by (ladder network) [22]:

$2Z_{eq}^{L}=Z_{1}\left(1+\sqrt{1+4Z_{eq}^{T}/Z_{1}}\right)$ (55)

where $Z_{eq}^{T}=\sqrt{Z_{1}Z_{2}}$ (binary tree). This leads to:

$2Z_{eq}^{L}=Z_{1}\left(1+\sqrt{1+4\sqrt{Z_{2}/Z_{1}}}\right)$ (56)

If we take $Z_{1}$ to be a resistance $R$ and $Z_{2}$ to be an inductance of
impedance $sL$, then we obtain a normalized impedance $Z^{L*}_{eq}$ as:

$Z^{L*}_{eq}=\frac{2Z_{eq}^{L}}{R}-1=\sqrt{1+\sqrt{s\tau_{eq}}}$ (57)

where $\tau_{eq}=16{L/R}$. The reciprocal gives a normalized admittance
$Y^{L*}_{eq}=1/Z^{L*}_{eq}$ of the HN type where $\alpha=0.5$ and $\beta=0.5$
(Eq. 35). In Fig. 4(a), we show the plot of $Y^{L*}_{eq}$ ($Y_{eq}$ in the
figure) in Nyquist format, with $\tau_{eq}=1$ over the frequency range 1 mHz
to 10 kHz. Plots of $g_{\nu}(\tau)$ (Eq. 54) with $\alpha=0.5$, $\beta=0.5$,
$\tau_{\nu}=1$, and three different values of $p$ (1.0, 0.5 and 0.3) are given
in Fig. 4(b).

Figure 4: In (a) Nyquist plot of real vs. imaginary parts of
$Y_{eq}=1/\sqrt{1+\sqrt{s\tau_{eq}}}$ for $\tau_{eq}=1$, frequency from 1 mHz
to 10 kHz; In (b) plots of $g_{\nu}(\tau)$ in Eq. 54 for $\alpha=\beta=0.5$,
$\tau_{\nu}=1$, and different values of $p$ (1.0, 0.5 and 0.3)

###  III.4 Kohlrausch-Williams-Watts model

We conclude with the Kohlrausch-Williams-Watts (KWW) model as a fourth example
of our study [33]. The KWW model is actually introduced starting from its
time-domain relaxation function, which is given in normalized form by [18]:

$\rho_{\kappa}(t)=e^{-(t/\tau_{\kappa})^{\kappa}},\;\;t\geqslant
0,\;0<\kappa\leqslant 1$ (58)

This function is also known as the stretched exponential function with an
exponent $\kappa$ and time constant $\tau_{\kappa}$. While the response
function $z(t)$ and the normalized impedance function $Q(s)$ are a LT pair,
$\rho_{\kappa}(t)$ and $Q(s)$ are related via [18]:

$\rho_{\kappa}(t)=1-\mathcal{L}^{-1}[Q_{\kappa}(s)/s;t]=\mathcal{L}^{-1}[(1-Q_{\kappa}(s))/s;t]$
(59)

This comes from $z(t)=-d\rho(t)/dt$. With the $H$-function identity:

$e^{-(t/\tau_{\kappa})^{\kappa}}=H^{1,0}_{0,1}\left[(t/\tau_{\kappa})^{\kappa}\left|\begin{array}[]{c}-\hfill\\\
(0,1)\hfill\\\ \end{array}\right.\right]$ (60)

one can obtain by applying the LT (Eq. 44) to both sides of Eq. 59 [31]:

$Q_{\kappa}(s)=1-H^{1,1}_{1,1}\left[(s\tau_{\kappa})^{\kappa}\left|\begin{array}[]{c}(1,1)\hfill\\\
(1,\kappa)\hfill\\\ \end{array}\right.\right]$ (61)

From Garrappa et al. [18], it can be shown that the LT of the relaxation
function being $(1-Q_{\kappa}(s))/s$ is in fact the ST of its corresponding
spectral function. However, in order for us to apply directly Eq. 24, we
derive from Eq. 58 the response function of the system as:

$z_{\kappa}(t)=-\frac{d\rho_{\kappa}(t)}{dt}=(\kappa/\tau_{\kappa})(t/\tau_{\kappa})^{\kappa-1}e^{-(t/\tau_{\kappa})^{\kappa}}$
(62)

Applying the LT we verify that $Q_{\kappa}(s)$ is also equal to:

$Q_{\kappa}(s)=\kappa\,H^{1,1}_{1,1}\left[(s\tau_{\kappa})^{\kappa}\left|\begin{array}[]{c}(0,1)\hfill\\\
(0,\kappa)\hfill\\\ \end{array}\right.\right]$ (63)

We then find from Eq. 24 and Eq. 25:

$\displaystyle g_{\kappa}(\tau)$
$\displaystyle={\kappa\,\Gamma(p)}{\tau^{-1}}H^{1,1}_{2,2}\left[(\tau/\tau_{\kappa})^{\kappa}\left|\begin{array}[]{c}(1,\kappa),(p,\kappa)\\\
(1,1),(1,\kappa)\hfill\\\ \end{array}\right.\right]$ (66)
$\displaystyle={\kappa\,\Gamma(p)}{\tau^{-1}}H^{1,0}_{1,1}\left[(\tau/\tau_{\kappa})^{\kappa}\left|\begin{array}[]{c}(p,\kappa)\\\
(1,1)\hfill\\\ \end{array}\right.\right]$ (69)

In Fig. 5(a) we show the Nyquist plots of Eq. 63 for $\tau_{\kappa}=1$ and for
different values of $\kappa$ (1.0, 0.7 and 0.5). In Fig. 5(b), we show plots
of $g_{\kappa}(\tau)$ in Eq. 69 for the simple case of $\kappa=0.5$,
$\tau_{\kappa}=1$, and for different values of $p$ (1.0, 0.8 and 0.51).

Figure 5: In (a) Nyquist plot of real vs. imaginary parts of $Q_{\kappa}(s)$
(Eq. 63) for $\tau_{\kappa}=1$ and different values of $\kappa$ (1.0, 0.7 and
0.5); the frequency range is 16 mHz to 100 Hz. In (b) plots of
$g_{\kappa}(\tau)$ in Eq. 69 for $\kappa=0.5$, $\tau_{\nu}=1$, and different
values of $p$ (1.0, 0.8 and 0.51)

##  IV Conclusion

A generalized distribution function of relaxation times based on the
elementary Davidson-Cole model for treating impedance functions is presented.
The inversion procedure involves the generalized Stieltjes transform, which
can be executed relatively easily using the Fox’s $H$-function and its
properties. We tested and validated the procedure for a few cases of well-
known functions.

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§ LUMINOSITY FUNCTION OF AGN1 DERIVED WITH AN MCMC

In this section we report in detail the MCMC applied to derive the new IR LF for AGN1 and AGN2, using simultaneously IR and FUV observations. In particular, we considered the same IR  data used by [Gruppioni et al., 2013] to derive the IR LF. Considering the unification theory of AGN, AGN1 and AGN2 are the same population and differences are due to orientation effects. We therefore expected AGN1 and AGN2 to have similar IR LFs, with some possible differences in the normalisation due to different orientations, i.e. line-of-sight that intercepts or not the dusty torus.  observations do not show any significant difference between the IR LF of AGN1 and AGN2, therefore we decided to consider both populations together to have a better sample of the IR LF. In addition, we considered observations at rest-frame 1450 Å by McGreer et al., 2013, Akiyama et al., 2018 and Schindler et al., 2019, with the addition of the observations by Croom et al., 2009 and Ross et al., 2013 converted to 1450 Å using the correction reported in the last mentioned work.
From the  observations we have also the number of times from which each SED template for AGN1 has been chosen to model observations. From the same SED templates we also derived the conversion between L$_{IR}$ and 1450 Å luminosity, for AGN1 only. In this way we have the possibility to convert every possible AGN1 IR LF to a specific FUV LF and therefore use at the same time the IR and FUV observations to derive the IR AGN1 and AGN2 LFs. As our interested is mainly in reproducing IR observations and we want to compensate for the difference in number between FUV and IR observations, we arbitrary multiply the observational errors of FUV data to have an average value equal to twice the average value of IR errors. In addition, we increase the minimum FUV observational errors to match the minimum IR one. To derive the AGN1 IR LF we adopted a Bayesan approach where the probability $\mathcal{P}$ of the model described by the parameters $\overrightarrow{x}$ given the data $\mathcal{D}$, which contains both IR and FUV, described as:

\begin{equation}
\mathcal{P}(\overrightarrow{x},\mathcal{D})=\mathcal{P}(\mathcal{D},\overrightarrow{x})\mathcal{P}(\overrightarrow{x})
\end{equation}

The list of prior $\mathcal{P}(\overrightarrow{x})$ is listed in Table <ref>.  The likelihood $\mathcal{P}(\mathcal{D},\overrightarrow{x})$ is a combination of the IR and FUV likelihoods, weighted for the number of available points:

\begin{multline}
\mathcal{P}(\mathcal{D}_{\rm IR},\mathcal{D}_{\rm UV}|\overrightarrow{x}_{\rm IR},\overrightarrow{x}_{\rm FUV})\propto \\
\prod\exp \left( -w_{\rm IR}\frac{(\mathcal{D}_{\rm IR}-\overrightarrow{x}_{\rm IR})^{2}}{\sigma_{IR}^{2}}-w_{\rm FUV}\frac{(\mathcal{D}_{\rm FUV}-\overrightarrow{x}_{\rm FUV})^{2}}{\sigma_{FUV}^{2}} \right)
\end{multline}

The weights are derive to compensate the different number of data points available in FUV and IR, i.e. $w_{\rm IR}=\frac{N_{\rm FUV}+N_{\rm IR}}{N_{\rm IR}}$ and $w_{\rm FUV}=\frac{N_{\rm FUV}+N_{\rm IR}}{N_{\rm FUV}}$. We considered the modified Schechter function evolving with redshift, as reported in equation <ref> and <ref>, but we removed the extrapolation at z$>$3, as FUV observations are available at higher redshifts. The [Gruppioni et al., 2013] LF for AGN1 is considered as starting point for the MCMC chains.

Summary of the parameters of the evolving, modified Schechter function and their priors.

Parameter     Prior range      Parameter     Prior range

log$_{10}(L^{*}_{0}/{\rm L}_{\odot})$     ]8,12.5[             k$_{\rho,2}$     ]-12,12[

log$_{10}(\Phi^{*}_{0}/{\rm Mpc}^{-3})$     ]-9,-1[     $\alpha$     ]1,1.6[

k$_{L,1}$     ]-10,10[       $\sigma$     ]0.1,1[

k$_{L,2}$     ]-10,10[       z$_{L}$     ]0,5[

k$_{\rho,1}$     ]-12,12[     z$_{\rho}$     ]0,5[

In Figure <ref> we report the corner plots, showing the correlation between the different parameters, as well as the marginalised histogram of each parameter. All parameters show well defined solutions and we report them in Table <ref>.

In Figure <ref> and <ref>, we show the result of the MCMC compared with the IR and FUV observations and the [Gruppioni et al., 2013] IR LF for AGN2 and AGN1. The previous one well describes the IR LF of AGN1, but overestimates the FUV LF of the same galaxy population. The new IR LF still describes the IR data, of both AGN1 and AGN2, but it also well represents the FUV LF of AGN1, with some discrepancies around z$\sim$2. This discrepancy may be due to the low number of SED templates associated in this work to the AGN1 population. In the future, increasing the number of SED templates may allow to better sample the $L_{\rm IR}-{\rm to}-L_{\rm FUV}$ conversion.

Corner plot of the MCMC performed considering simultaneously the FUV and IR data, contour lines are shown at arbitrary iso-density levels. Marginalised histograms are shown at the top of each column for each parameter. This plot has been derived using the Foreman-Mackey, 2016 code.

MCMC best result (green solid line) compared with AGN1 (black circles) and AGN2 (black squares) IR  data and the AGN1 (blue dashed line) and AGN2 (magenta dot-dashed line) IR LF by [Gruppioni et al., 2013]. Orange lines are random extractions within 1$\sigma$ uncertainty of the MCMC results. On the top left of each panel we report the average redshift and the $\chi^{2}$.

MCMC best result (green solid line) compared with the converted AGN1 IR LF by [Gruppioni et al., 2013] (blue dashed line) and FUV observations (see legend). Orange lines are random extractions within 1$\sigma$ uncertainty of the MCMC results. On the top left of each panel we report the average redshift and the $\chi^{2}$.

§ FILTERS INCLUDED IN

In this Appendix we report the complete list of filters included in the  simulation. In particular, in Table <ref> we report the filter for which we derived absolute magnitudes. In the same table we also report the corresponding reference and the central wavelength of each filter, derived as follows:

\begin{equation}
\lambda_{cen}=\frac{\int \lambda T(\lambda) d\lambda}{\int T(\lambda) d\lambda}
\end{equation}

In Table <ref> we instead report the list of filters included in the  simulation to derive the observed fluxes with their central wavelengths and the reference of each instrument.

Filters included in the  simulation to derive absolute magnitudes, their central wavelength and the corresponding reference.

Filter name     $\lambda_{cen}$     reference

GALEX/FUV     1538.62 Å    Zamojski et al., 2007

GALEX/NUV     2315.66 Å    Zamojski et al., 2007

SDSS/u     3573.89 Å    Gunn et al., 1998

SUBARU/B     4458.32 Å    Miyazaki et al., 2002

SUBARU/V     5477.83 Å    Miyazaki et al., 2002

SDSS/r     6202.46 Å    Gunn et al., 1998

UKIRT/J     1.25 $\mu$m     Casali et al., 2007

WIRCAM/Ks     2.16 $\mu$m     Puget et al., 2004

Spitzer/IRAC/Ch4     8 $\mu$m     Fazio et al., 2004

Filters included in the  simulation to derive observed fluxes, their central wavelength and the corresponding reference.

Filter name     $\lambda_{cen}$     Reference     Filter name     $\lambda_{cen}$     Reference

– continued from previous page

6rContinued on next page

SDSS/u     3561.79 Å    Gunn et al., 1998          Euclid/H     1.78 $\mu$m     Laureijs et al., 2010

ELT/MICADO/U     3605.07 Å    Leschinski et al., 2016     ELT/MICADO/Spec-HK     1.86 $\mu$m     Leschinski et al., 2016

LSST/u     3684.83 $\AA$     Ivezic et al., 2008     JWST/NIRCam/F200W     1.99 $\mu$m     Rieke et al., 2008

VIMOS/U     3720.47 Å    Mieske et al., 2007      ELT/MICADO/K-short     2.06 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/IB427     4263.45 $\AA$     Miyazaki et al., 2002     ELT/MICADO/xK1     2.06 $\mu$m     Leschinski et al., 2016

HST/ACS/F435W     4331.7 Å    Sirianni et al., 2005     ELT/MICADO/He-I     2.06 $\mu$m     Leschinski et al., 2016

ELT/MICADO/B     4412.92 Å    Leschinski et al., 2016     ELT/MICADO/K-mid     2.10 $\mu$m    Leschinski et al., 2016

SUBARU/Suprime-Cam/B     4458.32 $\AA$     Miyazaki et al., 2002     ELT/MICADO/H2-1-0S1     2.13 $\mu$m    Leschinski et al., 2016

SUBARU/Suprime-Cam/IB464     4635.13 $\AA$     Miyazaki et al., 2002     ELT/MICADO/Ks     2.14 $\mu$m    Leschinski et al., 2016

SDSS/g     4718.87 Å    Gunn et al., 1998     ELT/MICADO/Ks2     2.14 $\mu$m    Leschinski et al., 2016

SUBARU/Suprime-Cam/g     4777.07 $\AA$     Miyazaki et al., 2002     WIRCAM/Ks     2.16 $\mu$m     Puget et al., 2004

LSST/g     4802.0 $\AA$     Ivezic et al., 2008     ELT/MICADO/Br$_{\gamma}$     2.17 $\mu$m    Leschinski et al., 2016

SUBARU/Suprime-Cam/IB484     4849.2 $\AA$     Miyazaki et al., 2002     ELT/MICADO/K-cont     2.20 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/IB505     5062.51 $\AA$     Miyazaki et al., 2002     UKIRT/K     2.20 $\mu$m     Casali et al., 2007

SUBARU/Suprime-Cam/IB527     5261.13 $\AA$     Miyazaki et al., 2002     ELT/MICADO/xK2     2.22 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/V     5477.83 Å    Miyazaki et al., 2002     ELT/MICADO/K-long     2.31 $\mu$m     Leschinski et al., 2016

ELT/MICADO/V     5512.0 Å    Leschinski et al., 2016     AKARI/IRC/N2     2.41 $\mu$m     Murakami et al., 2007

SUBARU/Suprime-Cam/IB574     5764.76 $\AA$     Miyazaki et al., 2002     JWST/NIRCam/F277W     2.78 $\mu$m     Rieke et al., 2008

HST/ACS/F606W     5956.83 Å     Sirianni et al., 2005      ELT/METIS/H2Oice     3.10 $\mu$m     Leschinski et al., 2016

SDSS/r     6185.19 Å    Gunn et al., 1998     AKARI/IRC/N3     3.28 $\mu$m      Murakami et al., 2007

LSST/r     6231.2 $\AA$     Ivezic et al., 2008     ELT/METIS/shortL     3.30 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/IB624     6233.09 $\AA$     Miyazaki et al., 2002     ELT/METIS/PAH3.3     3.30 $\mu$m    Leschinski et al., 2016

SUBARU/Suprime-Cam/r     6288.71 $\AA$     Miyazaki et al., 2002     WISE/WISE/W1     3.40 $\mu$m     Wright et al., 2010

ELT/MICADO/R     6592.93 Å    Leschinski et al., 2016      Spitzer/IRAC/Ch1     3.56 $\mu$m     Fazio et al., 2004

SUBARU/Suprime-Cam/IB679     6781.13 $\AA$     Miyazaki et al., 2002     JWST/NIRCam/F356W     3.57 $\mu$m     Rieke et al., 2008

JWST/NIRCam/F070W     7006.11 Å    Rieke et al., 2008     ELT/METIS/HCI-L-short     3.60 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/IB709     7073.63 $\AA$     Miyazaki et al., 2002     ELT/METIS/Lp     3.81 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/NB711     7119.88 $\AA$     Miyazaki et al., 2002     ELT/METIS/HCI-L-long     3.82 $\mu$m     Leschinski et al., 2016

Euclid/VIS     7156.46 Å    Laureijs et al., 2010     ELT/METIS/Bra     4.05 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/IB738     7361.56 $\AA$     Miyazaki et al., 2002     JWST/NIRCam/F444W     4.41 $\mu$m     Rieke et al., 2008

SDSS/i     7499.7 Å    Gunn et al., 1998     AKARI/IRC/N4     4.47 $\mu$m      Murakami et al., 2007

LSST/i     7541.69 $\AA$     Ivezic et al., 2008    Spitzer/IRAC/Ch2     4.51 $\mu$m     Fazio et al., 2004

SUBARU/Suprime-Cam/i     7683.88 $\AA$     Miyazaki et al., 2002    ELT/METIS/CO01ice     4.65 $\mu$m     Leschinski et al., 2016

SUBARU/Suprime-Cam/IB767     7684.89 $\AA$     Miyazaki et al., 2002      WISE/WISE/W2     4.65 $\mu$m     Wright et al., 2010

HST/ACS/F775W     7712.58 Å     Sirianni et al., 2005       ELT/METIS/Mp     4.78 $\mu$m     Leschinski et al., 2016

HST/ACS/F814W     8012.07 Å     Sirianni et al., 2005     JWST/MIRI/F560W     5.65 $\mu$m     Wright et al., 2015

ELT/MICADO/I     8059.97 Å    Leschinski et al., 2016     Spitzer/IRAC/Ch3     5.76 $\mu$m     Fazio et al., 2004

SUBARU/Suprime-Cam/NB816     8149.39 $\AA$     Miyazaki et al., 2002    AKARI/IRC/S7     7.31 $\mu$m      Murakami et al., 2007

SUBARU/Suprime-Cam/IB827     8244.52 $\AA$     Miyazaki et al., 2002     JWST/MIRI/F770W     7.66 $\mu$m     Wright et al., 2015

ELT/MICADO/xI1     8374.01 Å    Leschinski et al., 2016       Spitzer/IRAC/Ch4     7.96 $\mu$m     Fazio et al., 2004

ELT/MICADO/I-long     8680.73 Å    Leschinski et al., 2016     ELT/METIS/PAH8.6     8.60 $\mu$m     Leschinski et al., 2016

LSST/z     8690.47 $\AA$     Ivezic et al., 2008     ELT/METIS/N1     8.65 $\mu$m     Leschinski et al., 2016

SDSS/z     8961.49 Å    Gunn et al., 1998      AKARI/IRC/S9W     9.22 $\mu$m      Murakami et al., 2007

ELT/MICADO/xI2     9022.12 Å    Leschinski et al., 2016       JWST/MIRI/F1000W     9.94 $\mu$m     Wright et al., 2015

SUBARU/Suprime-Cam/z     9036.88 $\AA$     Miyazaki et al., 2002     ELT/METIS/SIV     10.50 $\mu$m    Leschinski et al., 2016

HST/ACS/F850LP     9043.26 Å     Sirianni et al., 2005     AKARI/IRC/S11     10.95 $\mu$m      Murakami et al., 2007

JWST/NIRCam/F090W     9045.79 Å    Rieke et al., 2008      ELT/METIS/PAH11.5     11.20 $\mu$m     Leschinski et al., 2016

LSST/y     9736.41 $\AA$     Ivezic et al., 2008     ELT/METIS/N2     11.63 $\mu$m     Leschinski et al., 2016

HST/WFC3/F098M     9875.26 Å    <www.stsci.edu/hst/wfc3>     WISE/WISE/W3     12.81 $\mu$m      Wright et al., 2010

ELT/MICADO/xY1     1.00 $\mu$m     Leschinski et al., 2016      ELT/METIS/NeII     12.82 $\mu$m     Leschinski et al., 2016

UKIRT/Y     1.02 $\mu$m     Casali et al., 2007      AKARI/IRC/L15     16.16 $\mu$m      Murakami et al., 2007

ELT/MICADO/Y     1.04 $\mu$m     Leschinski et al., 2016      AKARI/IRC/L18W     19.81 $\mu$m      Murakami et al., 2007

HST/WFC3/F105W     1.06 $\mu$m     <www.stsci.edu/hst/wfc3>    WISE/WISE/W4     22.38 $\mu$m      Wright et al., 2010

ELT/MICADO/xY2     1.08 $\mu$m     Leschinski et al., 2016     AKARI/IRC/L24     23.35 $\mu$m      Murakami et al., 2007

Euclid/Y     1.08 $\mu$m     Laureijs et al., 2010     Spitzer/MIPS/24mu     23.84 $\mu$m     Rieke et al., 2004

ELT/MICADO/Spec-IJ     1.11 $\mu$m     Leschinski et al., 2016     OST/OSS/Ch1     34.50 $\mu$m     Bradford et al., 2018

JWST/NIRCam/F115W     1.16 $\mu$m     Rieke et al., 2008     OST/FIP/50     50.00 $\mu$m     Staguhn et al., 2018

ELT/MICADO/J-short     1.19 $\mu$m     Leschinski et al., 2016     OST/OSS/Ch2     58.00 $\mu$m     Bradford et al., 2018

ELT/MICADO/xJ1     1.20 $\mu$m     Leschinski et al., 2016     AKARI/FIS/N60     66.69 $\mu$m      Murakami et al., 2007

UKIRT/J     1.25 $\mu$m     Casali et al., 2007      Herschel/PACS/70     71.93 $\mu$m     Poglitsch et al., 2010

ELT/MICADO/J     1.25 $\mu$m     Leschinski et al., 2016     Spitzer/MIPS/70     72.56 $\mu$m     Rieke et al., 2004

HST/WFC3/F125W     1.25 $\mu$m     <www.stsci.edu/hst/wfc3>     AKARI/FIS/WIDE-S     89.20 $\mu$m      Murakami et al., 2007

ELT/MICADO/J-long     1.27 $\mu$m     Leschinski et al., 2016     OST/OSS/Ch3     97.50 $\mu$m     Bradford et al., 2018

ELT/MICADO/Pa$_{\beta}$     1.29 $\mu$m     Leschinski et al., 2016      Herschel/PACS/100     102.62 $\mu$m     Poglitsch et al., 2010

ELT/MICADO/xJ2     1.30 $\mu$m     Leschinski et al., 2016     AKARI/FIS/WIDE-L     149.94 $\mu$m     Murakami et al., 2007

Euclid/J     1.37 $\mu$m     Laureijs et al., 2010     Spitzer/MIPS/160mu     156.96 $\mu$m     Rieke et al., 2004

JWST/NIRCam/F150W     1.50 $\mu$m     Rieke et al., 2008       AKARI/FIS/N160     163.07 $\mu$m      Murakami et al., 2007

HST/WFC3/F160W     1.54 $\mu$m     <www.stsci.edu/hst/wfc3>     OST/OSS/Ch4     163.50 $\mu$m     Bradford et al., 2018

ELT/MICADO/xH1     1.55 $\mu$m     Leschinski et al., 2016     Herschel/PACS/160     167.14 $\mu$m     Poglitsch et al., 2010

ELT/MICADO/H-cont     1.57 $\mu$m     Leschinski et al., 2016     OST/FIP/250um     250.00 $\mu$m     Staguhn et al., 2018

ELT/MICADO/H-short     1.58 $\mu$m     Leschinski et al., 2016      Herschel/SPIRE/250     251.50 $\mu$m      Griffin et al., 2010

UKIRT/H     1.63 $\mu$m     Casali et al., 2007     OST/OSS/Ch5     275.00 $\mu$m     Bradford et al., 2018

ELT/MICADO/H     1.64 $\mu$m     Leschinski et al., 2016     Herschel/SPIRE/350     352.82 $\mu$m      Griffin et al., 2010

ELT/MICADO/FeII     1.65 $\mu$m     Leschinski et al., 2016     JCMT/SCUBA2/450     450.12 $\mu$m     Holland et al., 2013

ELT/MICADO/H-long     1.69 $\mu$m     Leschinski et al., 2016      OST/OSS/Ch6     462.50 $\mu$m     Bradford et al., 2018

ELT/MICADO/xH2     1.70 $\mu$m     Leschinski et al., 2016     Herschel/SPIRE/500     511.50 $\mu$m     Griffin et al., 2010


[This work is supported by the National
Science Foundation of China, grants no. 12071463.]

Abstract    We derive an infinitesimal  rigidity lemma for the strain tensor of surfaces with their curvatures changing sign. As an application, we obtain the optimal
constant in the first Korn inequality scales like $h^{4/3}$ for such shells of mixed type.

Keywords   surface of mixed type, shell, Korn' inequality, Riemannian geometry

Mathematics Subject Classifications
(2010)   74K20(primary), 74B20(secondary).

§ INTRODUCTION AND THE MAIN RESULTS

The goal of the present paper is twofold to study the rigidity  of the strain tensors of surfaces and then to apply it to obtain the optimal constant in the first Korn inequality  for some shells of mixed type.

The linear strain equations plays a fundamental role in the theory of thin shells, see [11, 12, 13, 14, 22]. The solvability of the strain equation is needed to prove the density of smooth infinitesimal isometries  in the $W^{2,2}(\Om,\R^3)$ infinitesimal isometries and  the matching property of the smooth enough infinitesimal isometries  with higher
order infinitesimal isometries [11, 14, 22]. This matching property is an important tool in
obtaining recovery sequences ($\Ga$-lim sup inequality) for dimensionally-reduced shell theories
in elasticity, when the elastic energy density scales like $h^\b,$ $\b\in(2, 4),$ that is, the intermediate
regime between pure bending ($\b=2$) and the von-Karman regime ($\b=4$). Such results have been obtained for elliptic surfaces [14], the developable surface [11], the  hyperbolic surface  [22], the degenerated surface [1], and the surface with the Gaussian curvature changing sign [2], respectively.
A survey on this topic is presented in [12].

The type of strain tensors depends on the sign of the curvature of the surface: It is elliptic if the curvature is positive; it is parabolic if the curvature is zero but $\Pi\not=0;$  it is hyperbolic for the negative curvature. When the curvature of the surface  changes its sign, the strain tensor is of mixed type.

Moreover, the lower regularity of strain tensors of surfaces represents the rigidity of geometrical shape of surfaces.

Linear strain equations have  been studied in [2] for the surface with its curvature changing sign, where the density of smooth infinitesimal isometries in the $W^{2,2}$ infinitesimal isometries is obtained and the  matching property of infinitesimal
isometries  is proved. In particularly, [2] established the following regularity

$$\|y\|^2_{\WW^{1,2}(S,\R^3)}\leq C\|U\|^2_{\WW^{2,2}(S,T^2_{\sym})},$$
where $y$ is a solution to problem (<ref>) later. However, the above regularity is not enough for establishing the rigidity results of the surface. Here, we derive the $\LL^2$ regularity, that is an infinitesimal  rigidity lemma (Theorem <ref>) for the strain tensors.  This lemma  is one of the key ingredients for the optimal constant in the first Korn inequality  for  shells. The basic tools are  some results on the generalized tensors.

Originally, Korn's inequalities were used to prove existence, uniqueness and well-posedness of
boundary value problems of linear elasticity [3, 16]. The optimal exponential of thickness in Korn's inequalities for thin shells represents the relationship between the rigidity and the thickness of a shell when the small deformations take place since  Korn's inequalities are linearized from the geometric rigidity inequalities under the small deformations [5]. Thus it is the best Korn constant in the Korn inequality that is of
central importance  [4, 15, 17, 18, 19, 20]. Moreover, one has known that the best Korn constant subject to the Gaussian curvature. Under the assumption that the middle surface of the shell is given by a single principal coordinate system, the one for the parabolic shell  scales like $h^{3/2}$ [6, 7], for the hyperbolic shell, $h^{4/3}$ [10] and for the elliptic shell, $h$ [10]. Later the assumption of the single principal coordinate  is removed in [23, 24], where the case of the closed elliptic shell is particularly included.

To the best knowledge of the authors, the present paper for the first time establishes the rigidity results for the surface with the curvature changing sign (Theorems <ref> and <ref>) and thus
yields that the optimal constant in the Korn inequality  scales like $h^{4/3}$ for some shells of mixed type (Theorem <ref>).

Let $S\subset M$ be given by
S={ (t,s) | (t,s)∈[0,a)×(-b_0,b_1) },  a>0,  b_0>0,  b>0,where $\a:$ $[0,a)\times[0,b]\rw M$ is an imbedding map which is a family of regular curves with two parameters $t,$ $s$ such that
Π(_t(t,s),_t(t,s))>0,    (t,s)∈[0,a)×[-b_0,0],where $\a(\cdot,s)$ is a closed curve with  period $a$ for each $s\in[-b_0,b_1].$  Set

$$S=S^+\cup\Ga_0\cup S^-,$$

$$S^+=\{\,\a(t,s)\,|\,(t,s)\in[0,a)\times(0,b_1)\,\},\quad \Ga_0=\{\,\a(t,0)\,|\,t\in[0,a]\,\},$$

$$ S^-=\{\,\a(t,s)\,|\,(t,s)\in[0,a)\times(-b_0,0)\,\}.$$

Curvature assumptions    Let $\kappa$ be the Gaussian curvature function on $M.$ We assume that $S$ satisfies the following curvature conditions:
κ(x)>0x∈S^+∪_b_1;κ=0,  Dκ(x)≠0x∈_0;κ(x)<0x∈S^-∪_-b_0,where

$$\Ga_{b_1}=\{\,\a(t,b_1)\,|\,t\in[0,a)\,\},\quad \Ga_{-b_0}=\{\,\a(t,-b_0)\,|\,t\in[0,a)\,\}.$$

Let $y\in \WW^{1,2}(S,\R^3)$ be a displacement of the middle surface $S.$ We decompose $y$ as

$$y=W+w\n,\quad w=\<y,\n\>,$$
where $\<\cdot,\cdot\>$ denotes the dot metric of the Euclidean space $\R^3.$ The (linear) strain tensor of the middle surface (related to the displacement $y$) is defined by
$$\Upsilon(y)=\sym DW+w\Pi,$$
where $D$ is the Levi-Civita connection of the induced metric $g$ on $S,$

$$\sym DW=\frac12(DW+DW^T),$$

$$\Pi(\a,\b)=\<\nabla_\a\n,\b\>\qfq\a,\,\,\b \in T_xM,\quad x\in S$$
is the second fundamental form of surface $M.$
For given $U\in L^2(S,T^2)$ , consider problem

Let $T^k$ denote the all $k$th-order tensor fields on $S.$ Let $L^2(S,T^k)$ be the space of all $k$th-order tensor fields on $S$ with the norm

$$\<P_1,P_2\>=\sum_{i_1,\cdots,i_k=1}^2P_1(e_{i_1},\cdots,e_{i_k})P_2(e_{i_1},\cdots,e_{i_k})\qfq x\in S,$$
and $e_1,$ $e_2$ is an orthonormal basis of $T_xS.$
Let $[\WW^{m,2}(S,T^k)]'$ be the dual spaces, based on the inner product $(\cdot,\cdot)_{\LL^2(S,T^k)}$ of $\LL^2(S,T^k).$

Our main results are the following, that is, an infinitesimal rigidity lemma on the strain tensor of the middle surface with mixed type.

Suppose that region $S$ of $\CC^5$ satisfies $(\ref{Q0})$-$(\ref{kappa-}).$  Let $y=W+w\n$ solve problem  $(\ref{s}).$ Then there is a constant $C>0,$ independent of $y,$ such that

The proof of Theorem <ref>  will be presented in the end of Section 3.

For the hyperbolic surface $S,$ estimate $(\ref{1.3})$ is given in [24]. If $S$ is elliptic, we then have

$$\|DW\|_{L^2(S,T^2)}^2+\|w\|^2_{\LL^2(S)}\leq C(\|U\|^2_{L^2(S,T^2)}+\|W\|^2_{\LL^2(\pl S,T)}),$$
see [3, 23].

Using Theorem <ref> and by the same argument as in <cit.>, we have the following.

Suppose that region $S$ of $\CC^5$ satisfies $(\ref{Q0})$-$(\ref{kappa-}).$ Then
w^2_L^2(S)≤C(Dw_L^2(S,T)Υ(y)_L^2(S,T^2)+Υ(y)^2_L^2(S,T^2))for all $y=W+w\n\in \WW^{1,2}(S,\R^3)$
with $y|_{\Ga_{-b_0}\cup\Ga_{b_1}}=0.$

Application to elasticity of thin shells.    Suppose that $S$ is the middle surface of the shell with thickness $h>0$

$$\Om=\{\,x+t\n(x)\,|\,x\in S,\,-h/2<t<h/2\,\}.$$

By the same arguments as in <cit.> we have the following results of the optimal exponential of  the first Korn inequality for shells of mixed type.  The details of the proofs are omitted.

Suppose that region $S$ of $\CC^5$ satisfies $(\ref{Q0})$-$(\ref{kappa-}).$
There are $C>0,$ $h_0>0,$ independent of $h>0,$ such that
∇y^2_L^2(,^3×3)≤C/h^4/3∇y^2_L^2(,^3×3)for all $h\in(0,h_0)$ and $y\in \WW^{1,2}(S,\R^3)$ with $y|_{\Sigma}=0,$ where

Moreover, the exponential of the thickness $h$ in $(\ref{1.13})$ is optimal.

Estimates (<ref>) and (<ref>) have been   given in [10] when the middle surface is given by one single principal coordinate, where $S$ is a hyperbolic surface. The assumption that
$S$ consists of one single principal coordinate is removed in [24].

§ GENERALIZED TENSORS

Let $m\geq0$ and $k\geq0$ be  integers. Denote by $\CC_0^m(S,T^k)$  all the $k$ order tensor fields with  continuously mth order derivatives and compact supports on $\ol{S}.$
Denote by $\D_m(S,T^k)$  the  space $\CC^m_0(S,T^k)$ with the following convergence: a sequence $\{\var_n\}\subset\CC_0^m(S,T^k)$ is said to converge to $\var_0\in\CC_0^m(S,T^k)$ if

(1) there exists a subset
$K\subset\subset S$ such that

$$\supp \var_n\subset K\qflq n\geq1;$$

(2) $\sup_{x\in K}|D_{X_1}\cdots D_{X_j}(\var_n-\var_0)|\rw0$ as $n\rw\infty$ for all $X_1,$ $\cdots,$ $X_j\in \CC^m(S,T)$ where $0\leq j\leq m.$

Let $\D'_m(S,T^k)$ consist of all continuously linear functionals on $\D_m(S,T^k).$  An element in $\D_m'(S,T^k)$ is said to be a generalized tensor  of  order $k$.

We define some generalized derivative operations as follows.

For given $w\in\D_m'(S),$ the generalized gradient $Dw\in\D_{m+1}'(S,T)$ is defined by

For given $W\in\D_m'(S,T)$ and $y\in\D_m'(S,\R^3),$ the generalized differentials  $DW\in\D_{m+1}'(S,T^2)$ and $\nabla y\in\D_{m+1}'(S,T^2)$ are defined by

For given $W\in\D_m'(S,T)$ and $U\in\D_m'(S,T^2),$ the generalized divergences $\div W\in\D'_{m+1}(S)$ and $\div U\in\D_{m+1}'(S,T)$ are defined by

For given $y\in\D_m'(S,\R^3)$ and $X\in\D_{m+1}(S,T),$ we define $\nabla_Xy\in\D'_{m+1}(S,T)$ by
∇_Xy(Y)=∇y(Y⊗X)Y∈_m+1(S,T).$\nabla_Xy\in\D_{m+1}'(S,T)$ is said to be the generalized derivative of $y$ along direction $X.$ Similarly, for given $W\in\D_m'(S,T)$ and $X\in\D_{m+1}(S,T),$ $D_XW\in\D_{m+1}'(S,T)$ is defined by

In the above definition the right hand sides of formulas (<ref>)-(<ref>) are linear functionals, respectively. For instant,
if $W\in\LL^2(S,T),$ then we view $W\in\D_m'(S,T)$ as

$$W(Y)=\int_S\<W,Y\>dg\qflq Y\in\D_m(S,T),$$

and so on.

For given $y\in\D_m'(S,\R^3)$ and $X\in\D_{m+1}(S,T),$
∇_Xy(Y)=-y((÷X) Y+∇_XY)Y∈_m+1(S,T).If $y\in\WW^{1,2}(S,\R^3),$ then
∇_Xy(Y)=∫_S∇_Xy,YdgY∈_m+1(S,T).Similarly, if  $W\in \WW^{1,2}(S,T)$ and $X\in\D_{m+1}(S,T),$ then
D_XW(Y)=∫_SD_XW,YdgY∈_m+1(S,T),where $D_XW$ is defined by $(\ref{1.7}).$

Let $x\in S$ be given.  Suppose that $E_1,$ $E_2$ is a frame normal at $x.$ Then
÷(Y⊗X)      =∑_ij=1^2D(Y⊗X)(E_i,E_j,E_j)E_i=∑_ij=1^2E_j(Y,E_iX,E_j)E_i

=D_XY+(÷X)YxX, Y∈_m+1(S,T).Thus (<ref>) follows from definition (<ref>).

Let $y\in\WW^{1,2}(S,\R^3).$ Then $y\in\D_m'(S,\R^3)$ is viewed as a linear functional on $\D_m(S,T)$ by

$$y(Y)=\int_S\<y,Y\>dg\qflq Y\in\D_m(S,T).$$
Noting that

$$\nabla_XY=D_XY-\<Y\otimes X,\Pi\>\n,$$

from (<ref>), (<ref>), and (<ref>), we have
∇_Xy(Y)      =y(-÷(Y⊗X)+Y⊗X,Π)=∫_Sy, -÷(Y⊗X)+Y⊗X,Πdg

=∫_S∇_Xy,YdgY∈_m+1(S,T).A similar argument yields (<ref>).

Let $\E_m(S,T^k)$ be the space $\CC^m(S,T^k)$ with the convergence: A sequence $\{\var_n\}\subset\CC^m(S,T^k)$ is said to converge to some $\var_0\in\CC^m(S,T^k)$ if for any given compact set
$K\subset\subset S$ and any given $X_1,$ $\cdots,$ $X_j\in\CC^m(S,T)$ where $0\leq j\leq m,$ there holds

$$\lim_{n\rw0}\sup_{x\in K}|D_{X_1}\cdots D_{X_j}(\var_n-\var_0)|=0.$$

Denote by $\E_m'(S,T^k)$ all the continuous linear functionals on $\E_m(S,T^k).$

We define some multiplier operations between generalized tensor  as follows.
For given $w\in\D_m'(S)$ and $f\in\E_m(S),$ $fw\in\D_m'(S)$ is defined by
(fw)(z)=w(fz)z∈_m(S);for given $W\in\D_m'(S,T)$ and $f\in\E_m(S),$ $fW\in\D_m'(S,T)$ by
(fW)(Z)=W(fZ)Z∈_m(S,T);for given $U\in\D_m'(S,T^2)$ and $f\in\E_m(S),$ $fU\in\D_m'(S,T^2)$ by
(fU)(P)=U(fP)P∈_m(S,T^2);for given $W\in\D_m'(S,T)$ and $F\in\E_m(S,T),$ $(\<W,F\>)\in\D_m'(S)$ by
(W,F)(z)=W(zF)z∈_m(S);for given $w\in\D_m'(S)$ and $X\in\E_m(S,T),$ $wX\in\D_m'(S,T)$ by
(wX)(Z)=w(X,Z)Z∈_m(S,T);for given $U\in\D_m'(S,T^2)$ and $R\in\E_m(S,T^2),$ $\<U,R\>\in\D_m'(S)$ by
(U,R)(z)=U(zR)z∈_m(S);for given $w\in\D_m'(S)$ and $R\in\E_m(S,T^2),$ $wR\in\D_m'(S,T^2)$  by
(wR)(P)=w(R,P)P∈_m(S,T^2);for given $U\in\D_m'(S,T^2),$ $\tr U\in\D_m'(S)$  by
U(z)=U(z𝕀)z∈_m(S);for given $W\in\D_m'(S,T)$ and $R\in\E_m(S,T^2),$ $RW\in\D_m'(S,T)$  by
(RW)(Z)=W(R^TZ)Z∈_m(S,T);for given $U\in\D_m'(S,T^2)$ and $R_1,$ $R_2\in\E_m(S,T^2),$ $R_1UR_2\in\D_m'(S,T^2)$  by

We need a linear operator $Q$ ([1, 2, 22, 24]) as follows. For each point
$p\in M$, the Riesz representation theorem implies that there
exists an isomorphism $Q:$ $T_pM\rw T_pM$ such that

,Q=(,,̱n⃗(p)), ∈̱T_pM.Let $e_1,$ $e_2$ be an orthonormal basis of $T_pM$ with
positive orientation, that is,

Then $Q$ can be expressed explicitly by
Q=,e_2e_1-,e_1e_2    ∈T_pM.Clearly, $Q$ satisfies

$$ Q^T=-Q,\quad Q^2=-\Id.$$

Operator $Q$ plays an important role in our analysis.

F given or $U\in\D_m'(S,T),$ we define $QU\in\D_m'(S,T^2)$ by

$$(QU)(X\otimes Y)=U(X\otimes Q^TY),\quad (UQ)(X\otimes Y)=U(QX\otimes Y)\qfq X,\,Y\in\D_m(S,T).$$
Moreover, for given $W\in \D'_m(S,T),$ we define $QW\in\D'_m(S,T)$ by

The following formulas hold:
÷(wR)=R^TDw+w÷Rw∈_m'(S),  R∈_m(S,T^2),÷(RW)=R,DW+÷R,WW∈_m'(S,T),  R∈_m(S,T^2),UQ+QU^T=(U)QU∈_m'(S,T^2).

Let $x\in S$ be given.  Suppose that $E_1,$ $E_2$ is a frame normal at $x.$ Then
R,DZ      =∑_ijR(E_i,E_j)D_E_jZ,E_i

=÷(RZ)-÷R,Zx.It follows from (<ref>), (<ref>), (<ref>), and (<ref>) that
÷(wR)(Z)      =-(wR)(DZ)=-w(R,DZ)=-w(÷(RZ))+w(÷R,Z)

=Dw(RZ)+(w÷R)(Z)=(R^TDw+w÷R)(Z)for $Z\in\E_{m+1}(S,T).$

For given $z\in\D_{m+1}(S),$ we have at $x$
R^TDz      =∑_ijE_i(z)R(E_j,E_i)E_j=∑_ij{E_i[zR(E_j,E_i)]-zDR(E_j,E_i,E_i)}E_j

=÷(zR)-z÷R.By (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>),
÷(RW)(z)      =-(RW)(Dz)=-W(R^TDz)

Finally, we shall prove (<ref>). Let $E_1,$ $E_2$ be a local frame with positive orientation. Then

$$QE_1=-E_2,\quad QE_2=E_1,\quad Q^TE_1=E_2,\quad Q^TE_2=-E_1.$$

$$(UQ+QU^T)(E_1\otimes E_2)=U(QE_1\otimes E_2)+U( Q^TE_2\otimes E_1)=(\tr U)\<QE_1,E_2\>,$$

$$(UQ+QU^T)(E_2\otimes E_1)=(\tr U)\<QE_2,E_1\>,\quad(UQ+QU^T)(E_i\otimes E_i)=(\tr U)\<QE_i,E_i\> $$
for $1\leq i\leq2.$

Let $S$ be of $\CC^{m+1}.$ Let $y\in D'_{m}(S,\mathbb{R}^3)$ be given. We define $w\in\D_m'(S),$ $W\in\D_m(S,T),$ $v\in\D_{m+1}(S),$ $V\in\D'_{m+1}(S,T),$ and $U\in\D'_{m+1}(S,T^2)$ by
w(z)=y(z)z∈_m(S),  W(Z)=y(Z)Z∈_m(S,T),v(z)=1/2y(QDz)z∈_m+1(S),V(Z)=y((÷QZ)+∇QZ)Z∈_m+1(S,T),and
U(P)=∇y(P)P∈_m+1(S,T^2),respectively. Then
DW+w∇=U'_m+1(S,T^2)in the sense of generalized tensors.

For given $y\in D'_m(S,\mathbb{R}^3),$ we have

÷V=-(Π)v-Q∇,Ux∈S,.in the sense of generalized tensors.

By (<ref>) and (<ref>) we have

$$ 2v(z)=(W+w\n)(QDz)=-(QW)(Dz)=(\div QW)(z)\qfq z\in D_{m+1}(S).$$

For $Z\in\D_{m+1}(S,T)$ and by (<ref>) and (<ref>)
(Dw-∇W)(Z)      =-w(÷Z)-W(∇^TZ)=-y((÷Z)+∇Z)

By (<ref>), (<ref>), and (<ref>)

$$\nabla y(P)=y(-\div P+\<P,\Pi\>\n)=-W(\div P)+w(\<P,\Pi\>)=(DW+w\Pi)(P)$$

for any $P\in\D_{m+1}(S,T^2).$ Then

from (<ref>), (<ref>), and (<ref>), we obtain
÷W(z)      =-W(Dz)=-W(÷(z𝕀))=DW(z𝕀)

=U(z𝕀)-(Π)w(z)=(U)(z)-(Π)w(z)z∈_m+1(S),that is, the second equation in (<ref>).

Next, we shall prove (<ref>). Let $x\in S$ be given.  Suppose that $E_1,$ $E_2$ is a frame normal at $x$ with positive orientation. Then $E_1=QE_2.$ Let $R$ be the curvature tensor. By the Ricci identity

for $Z,$ $X_1,$ $X_2,$ $X_3\in\D_{m+1}(S,T).$
Using the above formula, we have
÷DZ,E_2      =E_1(DZ)(E_2,E_1)+E_2(DZ)(E_2,E_2)

=E_2(÷Z)+κZ,E_2+1/2E_1(D_E_1Z, E_2-D_E_2Z,E_1)

=D÷Z+κZ-1/2QD÷QZ,  E_2Z∈_m+2(S,T).A similar computation yields

$$\<\div\sym DZ,E_1\>=\<D\div Z+\kappa Z-\frac12QD\div QZ,\,\,E_1\>\qaq x.$$

By (<ref>) and (<ref>)
(÷DW)(Z)      =-DW(DZ)=W(÷DZ)

=(D÷W+κW-QDv)(Z)Z∈_m+2(S,T).Using the relations

$$\nabla\n Q\nabla\n=\ka Q,\quad \nabla\n-(\tr \Pi) \id=Q\nabla\n Q,\quad \div\nabla\n=D(\tr\Pi),$$
we have, by (<ref>), (<ref>), and (<ref>),
∇V(Z)      =V(∇Z)=QV(Q∇Z)=(∇W-Dw)(Q∇Z)

=[Dv-Q÷QUQ](Z)Z∈_m(S,T),where the formula $U-(\tr U)\id=QUQ$ has been used, that is the first equation in (<ref>).

Noting that

$$\<\Pi,Q\nabla\n\>=\<Q,\Pi\>=0,\quad\div QDz=0,\quad \div(zQ\nabla\n)=-\nabla\n QDz,$$

$$\div\Big(z(\tr\Pi)Q\Big)=-QD(z\tr\Pi)\qfq z\in\D_{m+1}(S),$$
we compute, by (<ref>), (<ref>), (<ref>), and (<ref>),
÷V(z)      =-V(Dz)=-y((÷QDz)+∇QDz)=-W(∇QDz)

=-(U,Q∇)(z)-v(zΠ)=-(U,Q∇)(z)-((Π)v)(z)for $z\in\D_{m+1}(Z),$ where the formula $Q\nabla\n=(Q\nabla\n)^T+(\tr\Pi)Q$ has been used.
Thus the second equation in (<ref>) is true.

§ PROOF OF THEOREM <REF>

Let $\Om\subset M$ be  a bounded Lipschitz region with boundary $\Ga.$ Consider operator $\B:$ $\LL^2(\Om,T)\rw\LL^2(\Om,T)$ given by

$$\B W=\div DW,\quad D(\B)=\WW^{2,2}(\Om,T)\cap\WW^{1,2}_0(\Om,T).$$

Let $W\in D(\B)$ and $x\in\Om$ be fixed. Let $x\in\Om$ be fixed. Let $E_1,$ $E_2$ be a frame field normal at $x.$ By <cit.>

$$\<\B W,E_i\>=\sum_{j=1}^2D(DW)(E_i,E_j,E_j)=\<\sum_jD_{E_j}^2W, E_i\>=-\<{\bf \Delta}W,E_i\>+\Ric(W,E_i)$$
at $x$ for $i=1,$  $2,$
where ${\bf\Delta}$ is the Hodge-Laplacian and $\Ric$ is the Ricci tensor. Thus

$$\B W=-{\bf \Delta}W+\ii_W\Ric.$$
By <cit.> $\B$ is a self-adjoint operator with the compact resolvent. We then have a direct sum

$$\LL^2(\Om,T)=\R(\B)\oplus \N(\B),$$
where $\R(\B)$ and $\N(\B)$ are the valued field and the null space of $\B,$ respectively. Moreover, $\dim\N<\infty,$ and for any $Y\in\R(\B),$ there exists a unique
$Z\in D(\B)\cap\R(\B)$ that solves problem

$$\div DZ=Y,\quad Z|_\Ga=0$$
and satisfies

$$\|Z\|_{\WW^{2,2}(\Om,T)}\leq C\|Y\|_{\LL^2(\Om,T)}.$$

There exists constant $C>0$ such that

Step 1.   First, we claim that there is a constant $C>0$ such that
W^2_^2(,T)≤C(DW^2_^-1,2(,T^2)+W^2_^2(,T))W∈().Let $W\in\R(\B)$ be given. For any $Y\in\WW^{1,2}(\Om,T),$ consider the direct sum

$$ Y=Y_0+Y_1,\quad Y_0\in\N(\B),\quad Y_1\in D(\B).$$
Then there is $Z\in D(\B)\cap\R(\B)$ such that $Y_1=\div DZ.$ We have
W(Y)      =W(Y_1)=∫_W,÷DZdg=-∫_DW,DZdg+∫_DZ(W,ν)d

≤C(DW_^-1,2(,T)+W_^2(,T))Y_^2(,T)Y∈^2(,T),that is, (<ref>).

Step 2.   Let $\Phi_1,$ $\cdots,$ $\Phi_k$ be an orthonormal basis of $\N(\B),$ where $k=\dim\N(\B).$ Now we claim there are constants $\si>0$ and $C>0$ such that

≤C[(∑_i=1^k_k^2)^1/2+DW_^-1,2(,T^2)+W_^2(,T)]for all $(\a_1,\cdots,\a_k)\in\R^k$ and $W\in\R(\B).$ Clearly, the right hand side of (<ref>) holds true. We prove the left hand side by contradiction. Suppose the left hand side of (<ref>) is not true. Then there exist $\{\,(\a_{j1},\cdots,\a_{jk})\,\}\subset\R^k$ and $W_j\in\R(\B)$ such that

$$\|\sum_{i=1}^k\a_{ji}D\Phi_i+DW_j\|_{\WW^{-1,2}(\Om,T^2)}+\|W_j\|_{\LL^2(\Ga,T)}\leq\frac1j\qflq j\geq1.$$

We assume that $(\a_{j1},\cdots,\a_{jk})\rw(\a_{1},\cdots,\a_{k}).$ Then

$$W_j\rw 0\qiq\LL^2(\Ga,T),\quad DW_j\rw U\qiq \WW^{-1,2}(\Om,T^2).$$
By Step 1

$$W_j\rw W_0\qiq\LL^2(\Om,T)$$
with $W_0|_\Ga=0.$ Thus

$$(\sum_{i=1}^k\a_{i}^2)^{1/2}+\|DW_0\|_{\WW^{-1,2}(\Om,T^2)}=1,\quad D(\sum_i\a_i\Phi_i+W_0)=0,\quad (\sum_i\a_i\Phi_i+W_0)|_\Ga=0.$$

Then $\sum_i\a_i\Phi_i+W_0=0$ and, so

$$(\a_{1},\cdots,\a_{k})=0,\quad W_0=0.$$
That is a contradiction.

Step 3.   Let $W\in\LL^2(\Om,T)$ be given. Let

$$W=\sum_i\a_i\Phi_i+W_0,\quad \a_i=(W,\Phi_i)_{\LL^2(\Om,T)},\quad W_0\in\R(\B).$$

For given $Y\in\WW^{1,2}(\Om,T),$ we have by Steps 1 and 2
W(Y)      =∑_i_i_̱i+W_0(Y)≤||||̱+C(DW_0_^-1,2(,T^2)+W_0_^2(,T))Y_^2(,T)

$$\a=(\a_1,\cdots,\a_k),\quad\b=(\b_1,\cdots,\b_k),\quad \b_i=(Y,\Phi_i)_{\LL^2(\Om,T)}.$$

Finally, (<ref>) follows from (<ref>).

The following estimate holds.

For given $f\in\WW^{1,2}(\Om),$ we solve problem

$$-\Delta z=f\qfq x\in\Om,\quad z|_\Ga=0.$$

$$\|z\|_{\WW^{3,2}(\Om)}\leq C\|f\|_{\WW^{1,2}(\Om)}.$$

Thus we have
(v,f)_^2()      =-∫_vDz,νd+(Dv,Dz)_^2(,T)

≤C(Dv_[^2,2(,T)]'+v_[^3/2,2()]'))f_^1,2().Thus (<ref>) follows.

Let $\Om\subset M$ be a region and  boundary $\Ga$ consist of finite many closed curves. Then

Assume $P\in \WW^{2,2}(\Om,T^2).$  From <cit.>, we have
÷(PZ)=P,DZ+÷P, Zx∈,  Z∈^1,2(,T).For given $z\in\WW^{2,2}(\Om),$ it follows from (<ref>) that
(÷÷P, z)_^2()      =∫_[÷(z÷P)-÷P,Dz]dg

≤CP_^2(,T^2)z_^2,2().In addition, the assumption that boundary $\Ga$ consists of finite many closed curves implies
[^,2(,T^k)]'=^-,2(,T^k),where $\lam>0$ is a real number and $k\geq0$ is an integer.

Since $\WW^{2,2}(\Om,T^2)$ is dense in $[\WW^{2,2}(\Om,T^2)]',$ (<ref>) follows from (<ref>) and (<ref>).

Suppose that for given $\varepsilon>0$ small, $\eta_0\in\CC^\infty_0(-\infty,\infty)$  is given satisfying

$$0\leq\eta_0\leq1;\quad \eta_0=0\qfq s\leq \varepsilon/2;\quad \eta_0=1\qfq s\geq\varepsilon.$$

$$\eta(\a(t,s))=s\eta_0(s)\qfq (t,s)\in(0,a)\times(-b_0,b_1).$$

For given $\varepsilon>0$ small, let

is given in <cit.>.  We define
=e^-κ[(÷Q∇V-ηV,QX)QX+(÷V-ηV,∇X)∇X],Ł_0^*V      =∇Q D(e^-κQX,V)-D(e^-κ∇X,V)

-ηe^-κ(V,QXQX+V,∇X∇X),for $V\in\LL^2(\Om_{-\varepsilon},T).$ Let $m\geq0$ be an integer. By <cit.> we have
(W,Ł_0V)_^2(,T)      =(V,Ł_0^*W)_^2(,T)

+∫_(_1V,νW,ν-_2V,QXW,QX)dfor $V,$ $W\in T\Om,$ where
_1=e^-κ/X,νΠ(X,X),  _2=e^-κ/X,νΠ(Qν,Qν) x∈.

Consider problems

V,QX|__-ε=p,  V,ν|__b_1=q.and

V,ν|__-ε=p,  V,QX|__b_1=q,.respectively, where $F\in\WW^{m,2}(\Om_{-\varepsilon},T),$ $p\in\WW^{m,2}(\Ga_{-\varepsilon}),$ and
$q\in\WW^{m,2}(\Ga_{b_1})$ are given.
It follows from <cit.> that
Let $S$ be of $\CC^{m+3}.$ There exists a unique solution $W\in\WW^{m,2}(\Om_{-\varepsilon},T)$ to problem $(\ref{3x.9}),$ or $(\ref{3x.10})$ satisfying
V_^m,2(_-ε,T)+V_^m,2(_-ε, T)

≤C( F_^m,2(_-ε,T)+ q_^m,2(_b_1)+ p_^m,2(_-ε)).

We define linear operators $\L_0:$ $\WW^{m,2}(\Om_{-\varepsilon},T)\rw\WW^{m,2}(\Om_{-\varepsilon},T)$ by
=e^-κ[(÷Q∇V)QX+(÷V)∇X]-V,D(Ł_0)={ V∈^m,2(S,T)|       ÷V∈^m,2(_-ε), ÷Q∇V∈^m,2(_-ε)

V,QX|__-ε=V,ν|__b_1=0 },where
V=ηe^-κ(V,QXQX+V,∇X∇X),  η(x)=sη_0(s)for $x=\a(t,s)\in\Om_{-\varepsilon}.$
Moreover, set
Ł_0^*V=∇Q D(e^-κQX,V)-D(e^-κ∇X,V)-V.
$$ D(\L_0^*)=\{\,V\in \WW^{m,2}(S,T)\big|\,\L_0^*V\in\WW^{m,2}(\Om_{-\varepsilon},T),\,\<V,\nu\>|_{\Ga_{-\varepsilon}}=\<V,QX\>|_{\Ga_{b_1}}=0\,\}.$$

By Theorem <ref> and <cit.> we have the following.

$(i)$   Operators $\L_0$ and $\L_0^*$ have bounded inverses on $\WW^{m,2}(\Om_{-\varepsilon},T)$ with the estimates
Ł_0^-1V_^m,2(_-ε,T)+Ł_0^-1V_^m,2(_-ε, T)

Ł_0^*^-1V_^m,2(_-ε,T)+Ł_0^*^-1V_^m,2(_-ε, T)≤CV_^m,2(_-ε,T)for $V\in\WW^{m,2}(\Om_{-\varepsilon},T).$

$(ii)$   Operator $\L^{-1}_0\C$ and ${\L_0^*}^{-1}\C:$ $\WW^{m,2}(\Om_{-\varepsilon},T)\rw\WW^{m+1,2}(\Om_{-\varepsilon},T)$ are compact.

Define operators $\L$ and $\L^*:$ $\D'_m(S,T)\rightarrow \D'_{m+1}(S,T)$ by

$$\L V=\L_0 V+\C V\quad\mbox{and}\quad \L^*V=\L_0^*V+\C V,$$
Given $Z_0\in \E_m(\Om_{-\varepsilon},T)$, consider problem

We define operator $\A:$ $\LL^2(\Om,T)\rw\WW^{1,2}(\Om,T)$ by

$$\A Z=\nabla\n Dz\qfq Z\in\LL^2(\Om,T),$$
where $z\in \WW^{2,2}(\Om,T)$ is given by

$$\left\{\begin{array}{l}-\Delta z=e^{-\g\kappa}\<Z,\nabla\n X\>\tr\Pi\qfq x\in\Om,\\
\pl_\nu z=0\quad\mbox{on}\quad x\in\Ga.\end{array}\right.$$

$$\A:\quad \LL^2(\Om,T)\rw\LL^2(\Om_{-\varepsilon},T)$$
is compact. Thus by Theorem <ref>

$${\L_0^*}^{-1}(\C+\A):\quad \LL^2(\Om,T)\rw\LL^2(\Om_{-\varepsilon},T)$$
is also compact. Set

$$\dim \N<\infty,\quad \dim \N_*<\infty.$$

Let $S$ be of $\CC^{m+3}.$
Problem $(\ref{1.33})$ admits a unique solution $Z\perp \N$ in $\LL^2(\Om_{-\varepsilon},T)$ if and only if  $Z_0\perp \N_*$ in $\LL^2(\Om_{-\varepsilon},T).$ Moreover, there exists $C>0$ such that if
$Z_0\in\WW^{2,2}(\Om_{-\varepsilon},T)$ and $(Z,z)$ solves problem $(\ref{1.33})$ with $Z\perp \N,$  then $(Z,z)\in\WW^{m,2}(\Om_{-\varepsilon},T)\times\WW^{m+2,2}(\Om_{-\varepsilon})$ satisfying

For given $Z_0\in\LL^2(\Om_{-\varepsilon},T),$ it is easy to check that $Z\in\LL^2(\Om_{-\varepsilon},T)$ solves problem (<ref>) if and only if $Z$ satisfies

By Fredholm's theorem the above problem has a solution $Z\in\LL^2(\Om_{-\varepsilon},T)$ if and only if $Z_0\perp \N_*$ in $\LL^2(\Om_{-\varepsilon},T).$

Let $Z_0\in\WW^{m,2}(\Om_{-\varepsilon},T).$ By <cit.>  $Z\in\WW^{m,2}(\Om_{-\varepsilon},T).$ Thus by the regularity of the elliptic problem $z\in\WW^{m+2,2}(\Om_{-\varepsilon})$ and the estimates

$$\|z\|^2_{\WW^{m+2,2}(\Om_{-\varepsilon})}\leq C\|Z\|^2_{\WW^{m,2}(\Om_{-\varepsilon},T)}$$

follows. Moreover, using <cit.>, we have

$$\|Z\|^2_{\WW^{m,2}(\Om_{-\varepsilon},T)}+\|Z\|^2_{\WW^{m,2}(\pl\Om_{-\varepsilon},T)}\leq C\|Z_0\|^2_{\WW^{m,2}(\Om_{-\varepsilon},T)}.$$
Thus (<ref>) follows.

Let $\Phi_1,$ $\cdots,$ $\Phi_k$ be an orthonormal basis of $\N,$ where $k=\dim\N.$ Define the projection operator ${\P}:$ $[\WW^{m,2}(\Om_{-\varepsilon},T)]'\rw\N$ by

$${\P}V=\sum_{j=1}^k(V,\Phi_j)_{\LL^2(\Om_{-\varepsilon})}\Phi_k\qfq V\in[\WW^{m,2}(\Om_{-\varepsilon},T)]'.$$

Consider problem

V,QX|__-ε=p,  V,ν|__b_1=q..

We have the following.

Let $S$ be of $\CC^5.$ Suppose that $(v,V)$ solves problem $(\ref{3.11n}).$ Then there exists a $C>0$ such that
V_[^2,2(_-ε,T)]'+v_[^1,2(_-ε)]'≤C(V_[^2,2(_-ε,T)]'+I(F,f,p,q)),  where
I(F,f,p,q)      =÷QF_[^2,2(_-ε)]'+F_[^2,2(_-ε,T)]'+f_[^2,2(_-ε)]'

From (<ref>) we have

÷V=-ρv+fx∈_-ε,.where $\rho=\tr\Pi.$
It follows from (<ref>), (<ref>),  and (<ref>) that

$$G=-e^{-\g\kappa}[(\div QF)QX+(\rho v-f)\nabla\n X].$$

Ξ={  Z∈^2,2(_-ε,T) | Z⊥_* }.For given $Z_0\in\Xi,$ we solve problem (<ref>) to have the solution $(Z, z)\in\WW^{2,2}(\Om_{-\varepsilon},T)\times\WW^{4,2}(\Om_{-\varepsilon}).$ Let

$$z_1=e^{-\g\kappa}\<Z,QX\>,\quad z_2=\rho e^{-\g\kappa}\<Z,\nabla\n X\>,\quad z_3=e^{-\g\kappa}\<Z,\nabla\n X\>.$$
By (<ref>), (<ref>), and the first equation in
(<ref>), we  have
(V,Z_0)_^2(_-ε,T)      =(V,Ł_0^*Z+Z+∇Dz)_^2(_-ε,T)=(G, Z)_^2(_-ε,T)

=-(÷QF,z_1)_^2(_-ε)+(f, z_3)_^2(_-ε)

+∫__-ε_2pZ,QXd-∫__b_1_1qZ,νd.Noting that

by (<ref>) with $m=2$ and Lemma <ref>  we obtain

Since $\Xi$ is a closed subspace of $\WW^{2,2}(\Om_{-\varepsilon},T),$ (<ref>) shows that there exists a function in $\Xi,$ denoted by ${\B}V,$ such that

$$(V,Z_0)_{\LL^2(\Om_{-\varepsilon},T)}=({\B}V,Z_0)_{\WW^{2,2}(\Om_{-\varepsilon},T)}\qflq Z_0\in\Xi.$$

Let $\II$ be the canonical map from $[\WW^{2,2}(\Om_{-\varepsilon},T)]'\rw\WW^{2,2}(\Om_{-\varepsilon},T).$ Let $V_0={\II}^{-1}{\B}V.$ Then $V_0\in[\WW^{2,2}(\Om_{-\varepsilon},T)]'$ satisfies

$$(V-V_0,Z_0)_{\LL^2(\Om_{-\varepsilon},T)}=0\qflq Z_0\in\Xi.$$

$$V={\P} V+V_0\in[\WW^{2,2}(\Om_{-\varepsilon},T)]',$$
where ${\P}V=V-V_0.$ By (<ref>) we obtain
V_[^2,2(_-ε,T)]'      ≤ V_[^2,2(_-ε,T)]'+V_0_[^2,2(_-ε,T)]'

= V_[^2,2(_-ε,T)]'+sup_Z_0∈^2,2(_-ε,T), Z_0_^2,2(_-ε,T)=1(V,Z_0)_^2(_-ε,T)

≤ V_[^2,2(_-ε,T)]'+CI(F,f,p,q).

For given $z_0\in\WW^{1,2}(\Om_{-\varepsilon}),$ we solve problem

$$\Delta w=z_0\qfq x\in\Om_{-\varepsilon},\quad \<Dw,\nu\>|_{\pl\Om_{-\varepsilon}}=0.$$

$$\|w\|_{\WW^{3,2}(\Om_{-\varepsilon})}\leq C\|z_0\|_{\WW^{1,2}(\Om_{-\varepsilon})}\qflq z_0\in\WW^{1,2}(\Om_{-\varepsilon}).$$

By the first equation in (<ref>) we have
(v,z_0)_^2(_-ε)      =-(Dv,Dw)_^2(_-ε,T)=-(∇V+F,Dw)_^2(_-ε,T)

≤C(V_[^2,2(_-ε,T)]'+F_[^2,2(_-ε,T)]')z_0_^1,2(_-ε)for all $z_0\in\WW^{1,2}(\Om_{-\varepsilon}),$ that yields

Thus (<ref>) follows from (<ref>) and (<ref>).

$$U=\sym DW+w\Pi\qfq y=W+w\n.$$
Then there exists $C>0,$ independent of $y=W+w\n,$ such that

Let $V$ be given by (<ref>). By Theorem <ref> $(W,w)$ and $(v,V)$ satisfy problems

$$\div Q\nabla\n W=-\div V=\rho v-\<Q\nabla\n,U\>\qfq x\in\Om_{-\varepsilon}.$$

Applying Theorem <ref> to problem (<ref>) with

$$F=-QV,\quad f=\tr U,\quad p=\<W,QX\>|_{\Ga_{-\varepsilon}},\quad q=\<W,\nu\>|_{\Ga_{b_1}},$$
we have

+W,QX_^-2,2(_-ε)+W,ν_^-2,2(_b_1)).Then applying Theorem <ref> to problem (<ref>) with

$$F=Q\div QUQ,\quad f=-\<Q\nabla\n,U\>,\quad p=\<V,QX\>,\quad q=\<V,\nu\>$$

+V,QX_^-2,2(_-ε)+V,ν_^-2,2(_b_1)),where Lemma <ref> has been used. In addition, it follows from the first equation in (<ref>) that

Using (<ref>) and (<ref>) in (<ref>) and by Lemma <ref> below, we have

+Q(Dw-∇W)_^-2,2(_-ε,T)).  We claim that the terms $\|{\P}W\|_{[\WW^{2,2}(\Om_{-\varepsilon},T)]'}$ and $\|{\P}Q(Dw-\nabla\n W)\|_{[\WW^{2,2}(\Om_{-\varepsilon},T)]'}$ can be removed in (<ref>) to have
W_[^2,2(_-ε,T)]'+w_[^1,2(_-ε)]'≤C(U_^2(_-ε,T^2)+W_^2(_-ε,T))  by a compactness-uniqueness argument as follows.

By contradiction. Suppose that (<ref>) does not hold true. Then there are $w_k,$ $W_k,$ $v_k,$  $V_k,$ and $U_k,$ which satisfy (<ref>)-(<ref>), respectively, such that
1=W_k_^2(_-ε,T)+w_k_[^1,2(_-ε)]'≥k(U_k_^2(_-ε,T^2)+W_k_^2(_-ε,T))for all $k\geq1.$

$$U_k\rw0\qiq\LL^2(\Om_{-\varepsilon},T^2),\quad W_k\rw0\qiq\LL^2(\pl\Om_{-\varepsilon},T).$$
By Lemma <ref> below

$$V_k\rw0\qiq\WW^{-2,2}(\pl\Om_{-\varepsilon}),\quad w_k\rw0\qiq\WW^{-1,2}(\pl\Om_{-\varepsilon}).$$
Noting that $\div V_k=-\rho v_k-\<Q\nabla\n, U_k\>,$ by (<ref>) there is a subsequence, denoted stll by $(v_k,V_k),$ that converges to some
$(v_0,V_0)\in[\WW^{2,2}(\Om_{-\varepsilon})]'\times[\WW^{2,2}(\Om_{-\varepsilon},T)]'$ in $[\WW^{2,2}(\Om_{-\varepsilon})]'\times[\WW^{2,2}(\Om_{-\varepsilon},T)]'.$
It follows from (<ref>) that

V_0|__-ε=0..By (<ref>) there exists a subsequence $(w_k,W_k)$ which satisfies

$$(w_k,W_k)\rw (w_0,W_0)\qiq [\WW^{1,2}(\Om_{-\varepsilon})]'\times[\WW^{2,2}(\Om_{-\varepsilon},T)]'$$
for some $(w_0,W_0)\in[\WW^{1,2}(\Om_{-\varepsilon})]'\times[\WW^{2,2}(\Om_{-\varepsilon},T)]'.$
It is easy to check that $(w_0,W_0)$ solves problem

w_0|__-ε=0,  W_0|__-ε=0,.
$$\sym DW_0+w_0\Pi=0\qfq x\in\Om_{-\varepsilon},$$

$$v_0=\frac12\div QW_0\qfq x\in\Om_{-\varepsilon},$$

Consider problem (<ref>). Let $\tau=\a_t/|\a_t|$ for $x\in\Ga_{b_1}.$ Suppose  $\nu=Q\tau.$  Then $\nu,$ $\tau$ has positive orientation. We have
2v_0      =÷QW_0=D_νW_0,τ-D_τW_0,ν=D_νW_0,τ+D_τW_0,ν

=-2w_0Π(τ,ν)=0x∈_b_1.It follows from (<ref>) that

v_0=Dv_0=0x∈_b_1..By the uniqueness of the elliptic problem

$$v_0=0\qfq x\in\Om_{-\varepsilon},\quad\kappa(x)>0.$$

Moreover,  a similar argument yields

$$v_0=0\qfq x\in\Ga_{-\varepsilon}.$$
Then we have a hyperbolic problem

$$\left\{\begin{array}{l}\div(\nabla\n)^{-1}Dv_0=-\rho v_0\qfq x\in \Om_{-\varepsilon},\quad\kappa(x)<0,\\
v_0=Dv_0=0\qfq x\in\Ga_{-\varepsilon}.\end{array}\right.$$

Since the curves

$$\a(\cdot,s)\qfq -\varepsilon\leq s\leq 0,$$
are not characteristic, the uniqueness of the hyperbolic problem implies

$$v_0=0\qfq x\in\Om_{-\varepsilon},\quad \kappa(x)<0.$$

Thus we obtain

$$v_0=V_0=0\qfq x\in\Om_{-\varepsilon}.$$
Moreover, by (<ref>) we have

$$\left\{\begin{array}{l}\div(\nabla\n)^{-1}Dw_0=-\rho w_0\qfq x\in \Om_{-\varepsilon},\\
w_0=Dw_0=0\qfq x\in\pl\Om_{-\varepsilon}.\end{array}\right.$$

By a similar argument as for $(v_0,V_0)$ we obtain

$$w_0=W_0=0\qfq x\in\Om_{-\varepsilon},$$
which contradicts the first equality in (<ref>).

Finally, noting that

$$\|DW\|_{[\WW^{1,2}(\Om_{-\varepsilon},T^2)]'}=\|\sym DW\|_{[\WW^{1,2}(\Om_{-\varepsilon},T^2)]'},$$
by  Lemma <ref> and (<ref>) we have
W_^2(_-ε,T)      ≤C(DW_[^1,2(_-ε,T^2)]'+W_^2(_-ε,T))

≤(U_^2(_-ε,T^2)+W_^2(_-ε,T)).Thus (<ref>) follows from (<ref>) and (<ref>).

Suppose that $(w,W)$ and $(v,V)$ solve problem $(\ref{3n.34})$ and $(\ref{n3x.34}),$ respectively. Then

Let $\tau=\a_t/|\a_t|$ for $x\in\Ga_{b_1},$ Then $\tau,$ $\nu$ forms an orthonormal basis along $\Ga_{b_1}.$ We have
wΠ(τ,τ)=U(τ,τ)-DW(τ,τ)x∈_b_1.Since $\Pi(\tau,\tau)\not=0$ for all $x\in\Ga_{B_1},$ it follows from (<ref>) that
w_^-1,2(_-ε)      ≤C(U_^-1,2(_-ε,T)+τW,τ_^-1,2(_-ε)+W,D_ττ_^-1,2(_-ε))

Moreover, by (<ref>) we obtain
ν(w)Π(τ,τ)      =ν(U(τ,τ))-D^2W(τ,τ,ν)-D_ττ,ν[DW(ν,τ)+DW(τ,ν)]-wν(Π(τ,τ))

ν(w)_^-2,2(_b_1)      ≤C(U_^2(_-ε,T)+w_^-1,2(_b_1)

+D^2W(τ,τ,ν)_^-2,2(_b_1)).Next, we shall estimate $\|D^2W(\tau,\tau,\nu)\|_{\WW^{-2,2}(\Ga_{b_1})}.$ By Ricci's identity, we have
D^2W(τ,τ,ν)      =D^2W(τ,ν,τ)+R(τ,ν,W,τ)=τ(DW(τ,ν))

-D_ττ,ν[U(ν,ν)-wΠ(ν,ν)]-D_τν,τ)[U(τ,τ)-wΠ(τ,τ)],which yields

$$ \|D^2W(\tau,\tau,\nu)\|_{\WW^{-2,2}(\Ga_{b_1})}\leq C(\|U\|_{\LL^2(\Om_{-\varepsilon},T)}+\|W\|_{\LL^2(\Ga_{b_1})}+\|w\|_{\WW^{-1,2}(\Ga_{b_1})}),$$
where $R(\cdot,\cdot,\cdot,\cdot)$ is the Riemmannian curvature tensor.
It follows from (<ref>) that

$$\|\nu(w)\|_{\WW^{-2,2}(\Ga_{b_1})}\leq C(\|U\|_{\LL^2(\Om_{-\varepsilon},T)}+\|W\|_{\LL^2(\Ga_{b_1})}+\|w\|_{\WW^{-1,2}(\Ga_{b_1})}).$$

By the first equation in (<ref>) we obtain
V_^-2,2(_b_1,T)      ≤C(W_^-2,2(_b_1,T)+ν(w)_^-2,2(_b_1)+τ(w)_^-2,2(_b_1))

Combing (<ref>) and (<ref>), we have

$$\|V\|_{\WW^{-2,2}(\Ga_{b_1},T)}+\|w\|_{\WW^{-1,2}(\Ga_{-\varepsilon})}\leq C(\|U\|_{\LL^2(\Om_{-\varepsilon},T)}+\|W\|_{\LL^2(\Ga_{-\varepsilon})}).$$

A similar argument shows that the above estimates hold when $\Ga_{b_1}$ is replaced by $\Ga_{-\varepsilon}.$
Thus (<ref>) follows.

Proof of Theorem <ref>.   Let $y=W+w\n$ satisfy problem (<ref>) for given $U\in\LL^2(S,T^2).$ Let

$$S({-b_0},-\varepsilon)=\{\,\a(t,s)\in S\,|\,(t,s)\in(0,a)\times(-b_0,-\varepsilon)\,\}.$$
By (<ref>) $S({-b_0},-\varepsilon)$ is a  non-characteristic region. By <cit.> there is a $C>0,$ independent of $y,$ such that
W_^2(S(-b_0,-ε),T)≤C(U_^2(S(-b_0,-ε),T^2)+W_^2(_-b_0,T)),from that we obtain
w_[^1,2(S(-b_0,-ε))]'      ≤C(U_[^1,2(S(-b_0,-ε),T^2)]'+DW_[^1,2(S(-b_0,-ε),T^2)]')

≤C(U_^2(S(-b_0,-ε),T^2)+W_^2(_-b_0,T)),where the assumption

$$\Pi(\a_t,\a_t)>0\qflq x\in\overline{S({-b_0},-\varepsilon)}$$
is used. Moreover, by <cit.>
W_^2(_-ε,T)≤C(U_^2(S(-b_0,-ε),T^2)+W_^2(_-b_0,T)).Thus estimate (<ref>) follows from Theorem <ref> and (<ref>)-(<ref>). $\Box$

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.

Declaration of competing interest

The author declares that there is no conflict of interest.

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