# Neutrinos : from the r-process to the diffuse supernova neutrino background

Maria Cristina Volpe  Astro-Particule et Cosmologie (APC), CNRS UMR 7164,
Université Denis Diderot,
10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
<EMAIL_ADDRESS>
###### Abstract

Neutrinos from dense environments are connected to the longstanding open
questions of how massive stars explode and what are the sites where
$r$-process elements are made. Flavor evolution and neutrino properties can
influence nucleosynthetic abundances. GW170817 has given indirect evidence for
r-process elements in binary neutron star mergers. We discuss the impact of
non-standard interactions in such sites. Nearby compact objects, strong
gravitational fields are present. We discuss their influence upon neutrino
decoherence in a wave packet treatment of neutrino propagation. We conclude by
mentioning the upcoming measurement of the diffuse supernova neutrino
background.

##  1 Introduction

###  1.1 Core-collapse supernovae and binary neutron star mergers

There are powerful sources of low energy neutrinos in the Universe. Iron core-
collapse supernovae are massive stars (M $>$ 8 Msun) that produce $10^{58}$
$\nu$ in 10 seconds. Neutrinos of all flavors takes away $99\%$ of the
gravitational binding energy (a few $10^{53}$ ergs). With comparable
luminosities, binary neutron star merger (BNS) remnants produce $\nu_{e}$,
$\nu_{\mu}$ and $\nu_{\tau}$ as well as antineutrinos of all flavors in the
tens of MeV energy range.

Evidence for supernova neutrinos was brought by Sanduleak 69∘202 that exploded
at 50 kpc (in the Large Magellanic Cloud), giving SN1987A. This is a unique
event to date where neutrinos from the inner core of an exploding massive star
were observed. A total number of 24 neutrino events were recorded in
Kamiokande [1], IMB [2] and Baksan [3] (LVD events [4] remain debated), with
energy spectra and fluences in good agreement with expectations [5]. This
exceptional observation refuted the prompt explosion model and favored the
delayed neutrino heating mechanism by Bethe and Wilson [6] that includes an
accretion phase [7]. The latter is currently considered the mechanism under
which the majority of core-collapse supernovae explode. SN1987A events brought
a wealth of information on astrophysics, on neutrino properties (e.g. neutrino
speed and magnetic moment), non-standard particles like axions, and
interactions. Moreover the observations of asymmetric ejecta and strong mixing
gave momentum to multidimensional simulations that nowadays (over)explode in
2D and seem to be close to successfull explosions in 3D [8, 9, 10].

GW170817 [11] is a unique event where gravitational waves from binary neutron
star mergers were measured, in coincidence with a short gamma ray burst and a
kilonova. The electromagnetic signal, in comparison with models, shows
presence of lanthanides-free ejecta (blue component) and ejecta with
lanthanides (red component). This constitutes indirect evidence for the
production of $r$-process elements in binary neutron star mergers which are
candidate sites, as core-collapse supernovae, for $r$-process
nucleosynthesis111The rapid neutron capture process, responsible for the
production of about half of the elements heavier than iron in our solar
system, occurs on timescales of seconds and requires a neutron rich
environment.. Neutrino properties and flavor evolution impact $r$-process
nucleosynthetic abundances.

An established flavor mechanism is the Mikheev-Smirnov-Wolfenstein (MSW)
effect [12, 13], due to neutrino coupling with matter, that is responsible for
the suppression of solar 8B neutrinos. Important progress has been done in the
last two decades in the understanding of how neutrinos change flavor in dense
astrophysical environments. A wealth of theoretical investigation has clearly
shown that neutrinos change their flavor in astrophysical and cosmological
environments due to standard and non-standard interactions, to neutrino self-
interactions, shock waves and turbulence (in core-collapse supernovae). Flavor
mechanisms are intertwined with neutrino properties. The inclusion of neutrino
mixings and of the full collision term for cosmological neutrinos as well as
of plasma QED corrections has given a very precise value of the effective
number of degrees of freedom $N_{\rm eff}=3.0440$ [14]. Nearby compact objects
(black holes or neutron stars), neutrinos evolve in presence of strong
gravitational fields. These influence neutrino evolution through the redshift,
the bending of the trajectories (see e.g. [16, 15]) and by modifying
interference and flavor phenomena due to decoherence in curved spacetime [17].

These theoretical studies are essential to unravel the sites and conditions
for $r$-process nucleosynthesis, to determine the core-collapse supernova
mechanism and for future observations of supernova neutrinos and of the
diffuse supernova neutrino background.

##  2 Non-standard interactions and flavor evolution in dense media

Non-standard interactions are present in theories beyond the standard model.
Constraints on non standard neutrino-matter interactions (NSI) are given by
solar, oscillations and scattering experiments (see e.g. [18, 19]). Further
limits have been obtained by neutrino nucleus coherent scattering [20], using
both time and energy information for example in [21]. Non standard neutrino-
neutrino interactions constraints are loose [22]. Apart from giving different
values of the mixing parameters, if NSI exist they strongly influence neutrino
flavor evolution in dense environments.

Figure 1: Neutrino flavor evolution in an a BNS merger in presence of non-
standard neutrino matter interactions. Left : Both the (standard and non-
standard) matter potential $V_{M}$ and the effective spins precession
frequency $\omega_{sync}$, inversely proportional to the self-interaction
coupling, are shown. The crossing of the two curves corresponds to the
synchronized I-resonance condition. The results are for
$\epsilon_{0}=10^{-4},\delta\epsilon^{n}=-0.88$. Middle : Electron
antineutrino survival probability for $\epsilon_{0}=10^{-4}$,
$\delta\epsilon^{n}=-0.9$. Neutrinos undergo four I-resonances, the first is a
”normal” one, whereas the other three are synchronized. At about 100 km a MNR
resonance takes place. Right : The curves show locations where the I-resonance
condition is met. Note that the $Y_{e}$ values here are from the BNS
simulations, without flavor evolution. Adapted from [27].

In the core-collapse supernova context, NSI have been studied for example in
[23, 24]. Ref.[25] studied the influence on the MSW H- and L-resonances. Ref.
[23] investigated the interplay of non-standard interactions and neutrino
self-interactions222In the so called bulb model, assuming spherical symmetry..
A new MSW-like phenomenon, the I-resonance333I stands for inner takes place,
in the inner deleptonized stellar layers, due to a cancellation of the
standard and non-standard terms. Ref.[24] has performed an extensive
investigation of the NSI in presence of self-interactions covering a wide
range of NSI parameters, showing the appearance of an MSW-like resonance, the
matter neutrino resonance (MNR), previously seen in BNS444The MNR occurs
because of a cancellation between the matter and the neutrino self-interaction
terms. [26].

Ref.[27] has performed the first study of NSI effects (in two flavors) in BNS
based on the multidimensional simulations of BNS remnants (from Ref.[28]).
Contrarily to previous findings, this study uncovered that the I-resonance can
be triggered by self-interactions. By using the formalism of effective spins,
the effective neutrino and antineutrino spins are shown to go through the
resonance simultaneously and independently of their energy. The spin
precession frequency $\omega_{sync}$, inversely proportional to the neutrino
self-interaction coupling, gets small for sizeable self-interactions and meets
the smallness of the (standard and non-standard) matter term (Figure 1).

The investigation of a large number of trajectories (with NSI coupling values
below current bounds) has uncovered a complex interplay of flavor mechanisms,
producing neutrino spectral modifications. Interestingly significant flavor
modification is found even for very small values of the NSI couplings. An
example is given in Figure 1, which also shows that the I-resonance condition
is fullfilled both in the BNS funnel and in the polar regions. While a self-
consistent calculation of a neutrino driven wind following the matter
composition and flavor evolution is still missing, such studies show that NSI
can clear impact r-process nucleosynthetic abundances in kilonovae [27] and in
core-collapse supernovae [24].

##  3 Decoherence by wave packet separation due to strong gravitational fields

In most studies, neutrinos are treated as plane waves. Neutrinos being
localized particles a wave packets (WPs) description would be necessary,
although this increases the complexity of the theoretical description of
neutrinos in dense environments even further. In such a framework, if the WPs
associated with each mass eigenstate overlap at a production point, they
separate during propagation and their decoherence can suppress flavor
modification.

WPs decoherence can be quantified by the decoherence length $L_{coh}$,
distance at which the separation between the two WPs centroids (in two
neutrino flavors) satisfies $\Delta x=\sigma_{x}$, with $\sigma_{x}$ the WP
width that depends on the production process (see e.g. [29]). Neutrino
decoherence by WPs separation has been studied in flat spacetime (see e.g.
[29, 30, 31]). In a density matrix description of neutrino propagatioin, for
Gaussian WPs, one can show that the off-diagonal elements $\rho_{jk}$ of the
neutrino density matrix are exponentially damped with
$L^{jk}_{coh}=4\sqrt{2}E^{2}\sigma_{x}/|\Delta m^{2}_{jk}|$, $E$ being the
neutrino energy and $\Delta m^{2}_{jk}$ the squared mass differences between
the $j,k$ mass eigenstates.

Nearby compact objects, the presence of strong gravitational fields can modify
WPs decoherence. This was studied for the first time in Ref. [17] that has
extended the density matrix formalism to curved spacetime (Figure 2). The case
of a static gravitational field with spherical symmetry, described by the
Schwarzschild metric, was considered. It was shown that for Gaussian WPs, the
off-diagonal elements of the density matrix have a similar factorisation as in
the flat spacetime case, with an exponential suppression whose damping factor
also has analogous expression as in flat spacetime. However, it depends now on
the (unphysical) coherence coordinate distance $r^{PD}_{coh}$ and the neutrino
energy for an observer at infinity.

Using kinematical arguments, one can relate the $r^{PD}_{coh}$ to a coherence
proper time $\tau_{coh}$ at which the difference in the coherence times at a
”detection” point D satisfies $\tau_{D}^{jk}=\sigma_{t}\sqrt{B(r_{D})}$ with
$\sigma_{x}\approx\sigma_{t}$, $B(r)=1-r_{s}/r$, $r_{s}=2M$ being the
Schwarzschild radius, $M$ the compact object mass. WPs decoherence, with
respect to flat spacetime, i.e. $\eta=(\tau_{coh}-L_{coh})/L_{coh}$ (in
percentage) is shown in Figure 2. One can see a modification of several tens
of a percent for typical neutron star masses, showing that strong
gravitational fields strongly influence WPs decoherence.

Ref.[32] has considered the implications of these results for an inertial
observer. Further studies should address this question, by including the
coupling with matter and neutrinos, outside the compact object.

Figure 2: Left : Artistic drawing of neutrino WPs propagating from a
production point P to ”detection” point D in presence of strong gravitational
fields. Each mass eigenstate WPs follows a trajectory close to null-geodesics.
The coloured widths picture the distribution of trajectories due to the WP
finite extension. Right : Relative difference between the coherence proper
time (curved) and length (flat spacetime), as a function of the Schwarzschild
mass $M$. Adapted from [17].

##  4 Future observation of the diffuse supernova neutrino background

Neutrinos from past supernova explosions constitute the relic, or diffuse,
supernova neutrino background (DSNB). The Super-Kamiokande Collaboration has
started taking data with the addition of Gadolinium that should improve the
signal-to-background ratio by improving neutron tagging [33]. The JUNO
experiment is currently under construction [34], while the Hyper-Kamiokande
has been approved [35]. Available predictions are close to the current upper
bounds [36, 37] indicating the DSNB discovery should be close.

The DSNB is sensitive to the core-collapse supernova rate which is still
uncertain, the debated fraction of failed supernovae [38], non-standard
neutrino properties (see e.g. [39]), the contribution from binary systems [40,
41] and to redshifted supernova neutrino fluxes. The latter are not only
influenced by the MSW effect (usually implemented in predictions), but also by
other flavor mechanisms.

Ref. [42] has investigated the combined effect of shock waves and neutrino
self-interactions (in the so called ”bulb” model). The results have shown that
the spectral swapping produced by $\nu\nu$ interactions is counterbalanced by
the phase effects due to shock waves. This interplay is shown to impact the
expected DSNB rates in water Cherenkov and argon based detectors such as DUNE
by as much as 10-20$\%$, depending on the (inverted or normal) mass ordering,
which is as sizeable as the MSW effect.

Clearly, the upcoming discovery of the DSNB will bring invaluable information
on supernova neutrino fluxes, neutrino properties, the core-collapse supernova
rate and the fraction of failed supernovae. And maybe, we will be lucky enough
to detect supernova neutrinos from a(n) (extra)galactic supernova one day.

## References

## References

  * [1] Hirata K et al., 1987 Phys. Rev. Lett. 58 1490\.
  * [2] Bionta R M, et al., 1987 Phys. Rev. Lett. 58 1494\.
  * [3] Alekseev E N, Alekseeva L N, Krivosheina I V and Volchenko V I, 1988 Phys. Lett. B 205 209\.
  * [4] Aglietta M et al., 1987 Europhys. Lett. 3 1315\.
  * [5] Vissani F,, 2015 J. Phys. G 42 013001 (Preprint astro-ph/1409.4710).
  * [6] Bethe H A and Wilson J R, 1985, Astrophys. J. 295 14\.
  * [7] Loredo T J and Lamb D Q, 2002 Phys. Rev. D 65 063002 (Preprint astro-ph/0107260).
  * [8] Janka H T, (Preprint astro-ph/1702.08825).
  * [9] Radice D R et al. , 2018 J. Phys. G 45 053003 (Preprint astro-ph/1710.01282).
  * [10] Bruenn S W et al., 2020 Astrophys. J. Suppl. 248 11 (Preprint astro-ph/1809.05608).
  * [11] Abbott B P et al. [LIGO Scientific and Virgo], 2017 Phys. Rev. Lett. 119 161101 (Preprint gr-qc/1710.05832).
  * [12] Wolfenstein L, 1978 Phys. Rev. D 17 2369
  * [13] Mikheev S P and Smirnov A Y, 1985 Sov. J. Nucl. Phys. 42 913 [1985 Yad. Fiz. 42 1441].
  * [14] Froustey J, Pitrou C and Volpe M C, 2020 JCAP 12 015 (Preprint hep-ph/2008.01074).
  * [15] Caballero O L, McLaughlin G C and Surman R, 2012 Astrophys. J. 745 170 (Preprint astro-ph/1105.6371).
  * [16] Yang Y and Kneller J P, 2017 Phys. Rev. D 96 023009 (Preprint astro-ph/1705.09723).
  * [17] Chatelain A and Volpe M C, 2020 Phys. Lett. B 801 135150 (Preprint hep-ph]/1906.12152).
  * [18] Davidson S, Pena-Garay C, Rius N and Santamaria A, 2003 JHEP 03 011 (Preprint hep-ph/0302093).
  * [19] Farzan Y and Tortola M, 2018 Front. in Phys. 6 10 (Preprint hep-ph/1710.09360).
  * [20] Akimov D et al. [COHERENT], 2017 Science 357 1123 (Preprint nucl-ex/1708.01294).
  * [21] Giunti C, 2020 Phys. Rev. D 101 035039 (Preprint hep-ph/1909.00466).
  * [22] Bilenky M S and Santamaria A, Preprint hep-ph/9908272.
  * [23] Esteban-Pretel A, Tomas R and Valle J W F, 2007 Phys. Rev. D 76 053001 (Preprint hep-ph/0704.0032).
  * [24] Stapleford C J et al., 2016 Phys. Rev. D 94 093007 (Preprint hep-ph/1605.04903).
  * [25] Fogli G L, Lisi E, Mirizzi A and Montanino D, 2002 Phys. Rev. D 66 013009 (Preprint hep-ph/0202269).
  * [26] Malkus A et al., 2012 Phys. Rev. D 86 085015 (Preprint hep-ph/207.6648).
  * [27] Chatelain A and Volpe M C, 2018 Phys. Rev. D 97 023014 (Preprint hep-ph/1710.11518).
  * [28] Perego A et al., 2014 Mon. Not. Roy. Astron. Soc. 443 3134 (Preprint astro-ph/405.6730)].
  * [29] Kersten J and Smirnov A Y, 2016 Eur. Phys. J. C 76 339 Preprint hep-ph/1512.09068.
  * [30] Akhmedov E, Kopp J and Lindner M, 2017 JCAP 1709 017 (Preprint 1702.08338).
  * [31] Giunti C, 2004 Found. Phys. Lett. 17 103 (Preprint hep-ph/0302026).
  * [32] Petruzziello L, 2020 Phys. Lett. B 809 135784 (Preprint gr-qc/2009.06044).
  * [33] Beacom J F and Vagins M R, 2004 Phys. Rev. Lett. 93 171101 (Preprint hep-ph/0309300).
  * [34] An F et al. [JUNO], 2016 J. Phys. G 43 030401 (Preprint physics.ins-det/1507.05613).
  * [35] Abe K, T et al. Preprint hep-ex/1109.3262.
  * [36] Zhang H et al. [Super-Kamiokande], 2015 Astropart. Phys. 60 46 (Preprint hep-ex/1311.3738).
  * [37] Abe K et al. [Super-Kamiokande], (Preprint astro-ph/2109.11174).
  * [38] Lunardini C, 2009 Phys. Rev. Lett. 102 231101 (Preprint astro-ph/0901.0568).
  * [39] De Gouvêa A et al., 2020 Phys. Rev. D 102 123012 (Preprint hep-ph/2007.13748).
  * [40] Kresse D, Ertl T and Janka H T, 2021 Astrophys. J. 909 169 (Preprint astro-ph/2010.04728).
  * [41] Horiuchi S et al., 2021 Phys. Rev. D 103 043003 (Preprint astro-ph/2012.08524).
  * [42] Galais S, Kneller J, Volpe C and Gava J, 2010 Phys. Rev. D 81 053002 (Preprint hep-ph/0906.5294).


# A Hölder Stability Estimate for a 3D Coefficient Inverse Problem for a
Hyperbolic Equation With a Plane Wave

Michael V. Klibanov${}^{\ast\text{ }}$and Vladimir G. Romanov∗∗ ∗Department of
Mathematics and Statistics, University of North  Carolina at Charlotte,
Charlotte, NC 28223, USA  ∗∗Sobolev Institute of Mathematics, Novosibirsk
630090  Russian Federation<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A 3D coefficient inverse problem for a hyperbolic equation with non-
overdetermined data is considered. The forward problem is the Cauchy problems
with the initial condition the delta function concentrated at a single plane
(i.e. the plane wave). A certain associated operator is written in finite
differences with respect to two out of three spatial variables, i.e. “partial
finite differences”. The grid step size is bounded from the below by a fixed
number. A Carleman estimate is applied to obtain, for the first time, a Hölder
stability estimates for this problem. Another new result is an estimate from
the below of the amplitude of the first term of the expansion of the solution
of the forward problem near the characteristic wedge.

Key Words: coefficient inverse problem, hyperbolic equation, geodesic lines,
Carleman estimate, Hölder stability estimate.

2010 Mathematics Subject Classification: 35R30.

##  1 Introduction

We consider a forward Cauchy problem for a wave-like PDE in $\mathbb{R}^{3}.$
In this problem, the source is the $\delta-$function concentrated at a plane,
which models the incident plane wave. For this problems, we consider a
Coefficient Inverse Problem (CIP). Applications of this CIP is discussed in
section 2. We obtain a Hölder stability estimate for our CIP. Uniqueness
theorem follows immediately from this estimate.

We assume that a certain $2\times 2$ system of non local PDEs associated with
the original Cauchy problem is written in partial finite differences, i.e.
finite differences with respect to two out of three spatial variables. The
derivatives with respect to both the third spatial variable and time are
written in the conventional manner. The grid step size of finite differences
is assumed to be bounded from the below by a certain small positive constant.
We point out that such a bound always takes place in computations. Thus, this
assumption has an applied meaning.

The data for our CIP are non-overdetermined ones, i.e. the number $m=3$ of
free variables in the data equals the number $n=3$ of free variables in the
unknown coefficient. It is well known that uniqueness and stability results
for multidimensional CIPs with non-overdetermined data are currently proven
only by the method of [6], which is based on Carleman estimates, see, e.g. [3,
4, 11, 17, 30] and references cited therein for some samples of publications,
which use this method.

Applications of the idea of [6] to the developments of the so-called
“convexification” globally convergent numerical method for CIPs can be found
in, e.g. [7, 8, 13, 16, 17]. The originating publications about the
convexification are [9, 10]. Numerical studies by the convexification method
of a similar CIP for the same hyperbolic PDE as the one of this paper and with
the point source in the initial condition can be found in [13] and [17,
Chapter 8]. The second generation of the convexification method was developed
in [1, 2, 19].

However, in terms of uniqueness theorems and stability estimates, the method
of [6] works only under the assumption that one of initial conditions of a
corresponding PDE does not equal zero in the entire domain of interest. The
only exceptions are two recent works of Rakesh and Salo [22, 23] as well as
some follows up publications of these authors. In [22] a stability result was
proven for a CIP for the PDE $u_{tt}=\Delta
u+q\left(\mathbf{x}\right)u,\mathbf{x}\in\mathbb{R}^{3}$ with two initializing
plane waves and with non-overdetermined data. In [23] uniqueness is proven for
a CIP with non-overdetermined data for the analog of this equation in the
frequency domain.

In this paper, we consider the case of the hyperbolic PDE with a non constant
unknown coefficient in the principal part of the hyperbolic operator with a
single incident plane wave. Uniqueness and stability results were not proven
for this CIP in the past. Our case is more complicated than the one in [22].
This is basically because the geodesic lines in our case are curves rather
than straight lines of the above PDE. The above mentioned assumption of
partial finite differences for boundary value problems for $2\times 2$ systems
of non-local PDEs is imposed to avoid the assumption of [6] of the non-
vanishing initial condition.

To prove our target result, we use the expansion of the solution of our
forward problem near the characteristic wedge. Such an expansion is known only
for the case of the point source [24, Theorem 4.1], [25, Lemma 2.2.1]. Then we
combine this expansion with the Carleman estimate for the above mentioned
$2\times 2$ systems of non-local PDEs.

There are two new results of this paper:

  1. 1.

An estimate from the below of the amplitude of the first term of the expansion
of the solution of the forward problem near the characteristic wedge (Theorem
3.1).

  2. 2.

A Hölder stability estimate for the above mentioned CIP (Theorem 6.1).

Remark 1.1. _It seems to be convenient to formulate our main theorems in the
next section 2. However, prior to these formulations, we need to apply some
transformations to the solution of our Forward Problem. Thus, we postpone
those formulations until sections 3, 5 and 6._

The paper is organized as follows. In section 2 we pose forward and inverse
problems and describe some applications of our CIP. In section 3, we derive
the structure of the solution for the forward problem. In section 4, we change
variables and reduce our CIP to a boundary value problem for a $2\times 2$
system of non-local PDEs and reformulate in partial finite differences a
boundary value problem, which was derived in this section. In section 5 we
formulate and prove two Carleman estimates. In section 6 we prove a stability
estimate for the CIP.

##  2 Statements of Forward and Inverse Problems

Below $\mathbf{x}=\left(x,y,z\right)\in\mathbb{R}^{3}.$ Denote

$\mathbb{R}_{+}^{3}=\left\\{\left(x,y,z\right):z>0\right\\},\text{
}\mathbb{R}_{-}^{3}=\left\\{\left(x,y,z\right):z<0\right\\},\text{
}\mathbb{R}_{T}^{4}=\mathbb{R}^{3}\times\left(0,T\right).$

Let $X,T>0$ be two numbers. We define our domain of interest
$\Omega\subset\mathbb{R}^{3}$ and related surfaces as:

$\Omega=\left\\{\mathbf{x}:-X<x,y<X,~{}z\in\left(0,1\right)\right\\},$ (2.1)
$\Gamma=\left\\{\mathbf{x=}\left(x,y,z\right):-X<x,y<X,z=0\right\\}\subset\partial\Omega,~{}\Gamma_{T}=\Gamma\times\left(0,T\right),$
(2.2)
$\Gamma^{\prime}=\left\\{\mathbf{x=}\left(x,y,z\right):-X<x,y<X,z=1\right\\}\subset\partial\Omega,~{}\Gamma_{T}^{\prime}=\Gamma^{\prime}\times\left(0,T\right),$
(2.3) $\Theta=\left\\{x,y=\pm
X,z\in\left(0,1\right)\right\\}\subset\partial\Omega,\text{
}\Theta_{T}=\Theta\times\left(0,T\right).$ (2.4)

We assume that the function $n\left(\mathbf{x}\right)$ be defined in
$\mathbb{R}^{3}$ and

$n\left(\mathbf{x}\right)\in C^{16}\left(\mathbb{R}^{3}\right).$ (2.5)

Even though the smoothness requirement (2.5) seems to be excessive, we are
unaware how to decrease the smoothness. Indeed, we pose the Forward Problem in
section 2. Next, we prove in section 3 a certain representation of the
solution of this problem. This representation uses (2.5). In addition, it is
worth to point out that the issue of the minimal smoothness is traditionally a
minor concern in the field of Coefficient Inverse Problems, see, e.g. [20,
21], [24, Theorem 4.1].

Consider two arbitrary numbers $n_{0},n_{00}$ such that $1<n_{0}<n_{00}.$ We
assume everywhere below that

$1\leq n(\mathbf{x})\leq
n_{0},~{}\mathbf{x}\in\mathbb{R}^{3},\|n\|_{C^{2}(\mathbb{R}^{3})}\leq
n_{00},$ (2.6) $n\left(\mathbf{x}\right)=1\text{ in
}\mathbb{R}_{-}^{3}\cup\\{~{}|x|\geq X,~{}|y|\geq X\\}.$ (2.7)

In the case of an electric wave field propagation, $n\left(\mathbf{x}\right)$
is the refractive index, $n^{2}\left(\mathbf{x}\right)$ is the spatially
distributed dielectric constant of the medium, and the function
$u\left(\mathbf{x},t\right)$, which we introduce below, is a component of that
electric field. Therefore, the CIP, which we study here, has an application in
the problem of the determination of the dielectric constant of the medium
using the data of the scattered electric wave. These data are measured at a
part $\left(\Gamma\cup\Theta\right)\subset\partial\Omega$ of the boundary
$\partial\Omega$ of the domain $\Omega$. It was shown numerically on microwave
experimental data in [7, 8, 17] and analytically in [28] that in some
applications, one wave-like PDE can govern the process of electric waves
propagations equally well with the full system of Maxwell’s equations. In
fact, this is also claimed heuristically in the classic textbook of Born and
Wolf [5, pages 695,696]. Another area of applications of the CIP of this paper
is acoustics, in which case the function $1/n\left(\mathbf{x}\right)$ is the
speed of sound waves propagation in the medium.

Forward Problem. _Solve the following Cauchy problem:_

$n^{2}\left(\mathbf{x}\right)u_{tt}-\Delta
u=\delta\left(z\right)\delta\left(t\right),\text{
}\left(\mathbf{x},t\right)\in\mathbb{R}^{4},$ (2.8) $u\mathbf{\mid}_{t<0}=0.$
(2.9)

Since by (2.7) $n\left(\mathbf{x}\right)=1$ in $\mathbb{R}_{-}^{3},$ then the
incident wave in (2.8), (2.9) is the plane wave, which is incident at the
plane $\Sigma,$

$\Sigma=\left\\{\mathbf{x}\in\mathbb{R}^{3}\,:z=0\right\\}$ (2.10)

and propagates in the direction parallel to the $z-$axis. The function
$u\left(\mathbf{x},t\right)$ in the CIP is the solution of the Forward
Problem.

Coefficient Inverse Problem (CIP). _Suppose that the following functions_

__ $f_{0},f_{1},f_{2}$_are given:_

$u\mid_{\Gamma_{T}}=f_{0}\left(\mathbf{x},t\right),u_{z}\mid_{\Gamma_{T}}=f_{1}\left(\mathbf{x},t\right),u\mid_{\Theta_{T}}=f_{2}\left(\mathbf{x},t\right),$
(2.11)

_where_ $u\left(\mathbf{x},t\right)$ _is the solution of Forward Problem (
2.8), (2.9). Determine the function _$n\left(\mathbf{x}\right)$ _for_
$\mathbf{x}\in\Omega,$ _assuming that this function satisfies conditions (
2.6), (2.7)._

It follows from (2.1), (2.2) that the data at $\Gamma_{T}$ in (2.11) are the
backscattering data.

Remark 2.1. _Let_ $S_{T}=\partial\Omega\times\left(0,T\right).$ _Note that we
do not need the data at_ $S_{T}\diagdown\left(\Gamma_{T}\cup\Theta_{T}\right)$
_in our CIP, i.e. we do not need the data on the transmitted side of the
rectangular prism_ $\Omega.$

##  3 The Structure of the Solution of the Forward Problem

###  3.1 Geodesic lines

Let $\tau(\mathbf{x})$ be the travel time, which the above plane wave needs to
travel from the plane (2.10) to the point $\mathbf{x}\in\mathbb{R}_{+}^{3}.$
The function $\tau(\mathbf{x})$ satisfies the eikonal equation and it equals
zero at the plane $\Sigma$, i.e.

$\left|\nabla\tau(\mathbf{x})\right|^{2}=n^{2}\left(\mathbf{x}\right),~{}\mathbf{x}\in\mathbb{R}_{+}^{3};\text{
}\tau|_{z=0}=0.$ (3.1)

We obtain from (3.1)

$\tau(\mathbf{x})=\pm\mathop{\displaystyle\int}\limits_{0}^{z}\sqrt{n^{2}\left(x,y,s\right)-\left(\tau_{x}^{2}+\tau_{y}^{2}\right)\left(x,y,s\right)}\,ds.$

However, since Physics tells us that $\tau(\mathbf{x})>0,$ then the correct
formula for $\tau(\mathbf{x})$ is

$\tau(\mathbf{x})=\mathop{\displaystyle\int}\limits_{0}^{z}\sqrt{n^{2}\left(x,y,s\right)-\left(\tau_{x}^{2}+\tau_{y}^{2}\right)\left(x,y,s\right)}\,ds>0,\text{
}\mathbf{x}\in\mathbb{R}_{+}^{3}.$ (3.2)

In particular,

$\partial_{z}\tau(\mathbf{x})>0,\text{ }\mathbf{x}\in\mathbb{R}_{+}^{3}.$
(3.3)

The function $\tau(\mathbf{x})$ is generated by the conformal Riemannian
metric with the element of its length

$d\tau=n(\mathbf{x})|d\mathbf{x}|.$ (3.4)

Geodesic lines of this metric are orthogonal at any point $\mathbf{x}$ to the
corresponding wave front passing through this point. We assume everywhere
below that the following assumption holds:

Assumption 3.1. _Geodesic lines of metric_ (3.4) _satisfy the regularity
condition in_ $\mathbb{R}_{+}^{3}$_, i.e. for each point_
$\mathbf{x}\in\mathbb{R}_{+}^{3}$_there exists a single geodesic line_
$L(\mathbf{x})$_connecting_ $\mathbf{x}$_with the plane_ $\Sigma$_in ( 2.10)
and such that _$L(\mathbf{x})$_intersects the plane_ $\Sigma$_orthogonally._

Below $L(\mathbf{x})$ denotes geodesic lines specified in Assumption 3.1. The
function $\tau(\mathbf{x})$ has the form

$\tau(\mathbf{x})=\mathop{\displaystyle\int}\limits_{L(\mathbf{x})}n(\mathbf{x}^{\prime})d\sigma,\text{
}\mathbf{x\in}\mathbb{R}_{+}^{3},$ (3.5)

where $\mathbf{x}^{\prime}=\mathbf{x}^{\prime}(\sigma)$ and $d\sigma$ is the
element of the Euclidean length. Denote

$\mathbf{p}(\mathbf{x})=\nabla\tau(\mathbf{x}).$ (3.6)

To find $L(\mathbf{x})$ we need to solve the Cauchy problem for the following
system of ordinary differential equations:

$\frac{d\mathbf{x}}{ds}=\frac{\mathbf{p}(\mathbf{x})}{n^{2}(\mathbf{x})},\quad\frac{d\mathbf{p}(\mathbf{x})}{ds}=\nabla\ln
n(\mathbf{x}),\quad\frac{d\tau(\mathbf{x})}{ds}=1,~{}s>0,$ (3.7)
$\mathbf{x}|_{s=0}=\mathbf{x}^{0},~{}\mathbf{p}|_{s=0}=\mathbf{p}^{0},~{}\tau|_{s=0}=0,$
(3.8)

where $\mathbf{x}^{0}=(x^{0},y^{0},0)\in\Sigma$. Since the vector
$\mathbf{p}^{0}$ is orthogonal to the plane $\Sigma$ and $|\mathbf{p}^{0}|=1$
on $\Sigma$, then $\mathbf{p}^{0}=(0,0,1).$

For each point $\mathbf{x}^{0}\in\Sigma,$ the solution of Cauchy problem
(3.7), (3.8) defines the geodesic line of the Riemannian metric (3.4), which
is orthogonal to the plane $\Sigma$ at the point $\mathbf{x}^{0}$. By
Assumption 3.1, there exists a one-to-one correspondence between the points
$\mathbf{x\in}\mathbb{R}_{+}^{3}$ and the pairs
$\left(\mathbf{x}^{0},s\right)\in\Sigma\times\left(0,\infty\right)$. The
equation of $L(\mathbf{x})$ is given in the form
$\mathbf{x}=\mathbf{\xi}(s,\mathbf{x}^{0})$. It follows from the last equation
(3.7) and initial conditions (3.8) that, to find the function
$\tau(\mathbf{x}),$ we need to solve the equation
$\mathbf{x}=\mathbf{\xi}(s,\mathbf{x}^{0})$ with respect to the vector
$\left(s,x^{0},y^{0}\right)$. Then we find $s=s(\mathbf{x}).$ Next, recalling
that by (3.7) and (3.8) the parameter $s$ coincides with $\tau(\mathbf{x}),$
we set $\tau(\mathbf{x})=s(\mathbf{x})$. The smoothness of functions
$\mathbf{\xi}(s,\mathbf{x}^{0})$, $\mathbf{p}(s,\mathbf{x}^{0})$,
$s(\mathbf{x})$ is determined by the smoothness of the function
$n(\mathbf{x})$. Since $n\in C^{16}\left(\mathbb{R}^{3}\right)$ by (2.5), then
the function $\tau\in C^{16}\left(\mathbb{R}_{+}^{3}\right)$ (see, for
instance, [[25], pp. 26-27], for the similar derivation).

Introduce the non-negative function $\varphi(\mathbf{x})$ as

$\varphi(\mathbf{x})=\left\\{\begin{array}[]{c}-z\text{ for }z\leq 0,\\\
\tau(\mathbf{x})\text{ for }z>0.\end{array}\right.$ (3.9)

Then the function $\varphi\in
C^{16}\left(\overline{\mathbb{R}_{+}^{3}}\right).$ The equation
$t=\varphi(\mathbf{x})$ defines the characteristic wedge in $\mathbb{R}^{4}$
for the plane wave originated on the plane $\left\\{z=0\right\\}$ while it
travels inside space $\mathbb{R}^{3}$.

Let $T>0$ be a number. Define the domain $D_{T}$ as

$D_{T}=\\{(\mathbf{x},t):\,0\leq\varphi(\mathbf{x})\leq t\leq T\\}.$ (3.10)

By (2.6) the speed $1/n(\mathbf{x})\leq 1.$ Hence, (3.2) and (3.9) imply that

$D_{T}\subset\left\\{(x,y)\in\mathbb{R}^{2},z\in[-T,T\right\\}\times\left[0,T\right].$
(3.11)

###  3.2 The structure of the solution of the forward problem

Lemma 3.1. _Let conditions ( 2.1)-(2.7) be in place and let_
$L(\mathbf{x})$_be the geodesic line corresponding to_
$\tau(\mathbf{x})$._Then the following inequality holds along this geodesic
line:_

$\frac{d}{ds}\Delta\tau(\mathbf{x})\leq
6n_{00}^{2},~{}\mathbf{x}\in\mathbb{R}_{+}^{3},$ (3.12)

_where constant $n_{00}$ is defined in (2.6)._

Proof. Only in this proof we introduce the following notations:

$\mathbf{x}=(x_{1},x_{2},x_{3}),~{}\mathbf{p}(\mathbf{x})=\nabla\tau(\mathbf{x})\text{,
}\tau_{x_{i}x_{j}}(\mathbf{x})=\kappa_{ij}(\mathbf{x}),i,j=1,2,3,$
$\Delta\tau(\mathbf{x})=\kappa(\mathbf{x})=\sum\limits_{i=1}^{3}\kappa_{ii}(\mathbf{x}).$

Denote $p_{i}=\tau_{x_{i}}$, $i=1,2,3$ and use the eikonal equation (3.1)
$\left|\nabla\tau(\mathbf{x})\right|^{2}=n^{2}(\mathbf{x}).$ Hence,

$\sum_{i=1}^{3}p_{i}^{2}=n^{2}(\mathbf{x}).$

Differentiate this equation with respect to $x_{j}$ and use
$(p_{i})_{x_{j}}=(p_{j})_{x_{i}}=\kappa_{ij}$. We obtain

$\sum_{i=1}^{3}p_{i}\kappa_{ij}=nn_{x_{j}},~{}j=1,2,3.$

Differentiating this equality with respect to $x_{\ell}$, we get

$\sum_{i=1}^{3}p_{i}(\kappa_{j\ell})_{x_{i}}+\sum_{i=1}^{3}\kappa_{ij}\kappa_{i\ell}=n_{x_{j}}n_{x_{\ell}}+nn_{x_{\ell}x_{j}},~{}j,\ell=1,2,3.$

Along the geodesic line $\Gamma(\mathbf{x})$ we can rewrite the latter
equation as

$\frac{d\kappa_{j\ell}}{ds}+\frac{1}{n^{2}}\sum_{i=1}^{3}\kappa_{ij}\kappa_{i\ell}=\frac{1}{n^{2}}\left(n_{x_{j}}n_{x_{\ell}}+nn_{x_{\ell}x_{j}}\right),~{}j,\ell=\overline{1,3}.$

Take here $\ell=j$ and consider the summation with respect to $j$ from $1$ to
$3$. We obtain

$\frac{d\kappa}{ds}+\frac{1}{n^{2}(\mathbf{x})}\sum\limits_{i,j=1}^{3}\kappa_{ij}^{2}=\frac{1}{n^{2}(\mathbf{x})}|\nabla
n(\mathbf{x})|^{2}+\frac{1}{n(\mathbf{x})}\Delta n(\mathbf{x}).$ (3.13)

It follows from (2.6) that

$\left|\frac{1}{n^{2}(\mathbf{x})}|\nabla
n(\mathbf{x})|^{2}+\frac{1}{n(\mathbf{x})}\Delta n(\mathbf{x})\right|\leq
6n_{00}^{2},~{}\mathbf{x}\in\mathbb{R}_{+}^{3}.$ (3.14)

The target estimate (3.12) of Lemma 3.1 follows from (3.13) and (3.14).
$\square$

Let $H\left(t\right)$ be the Heaviside function,

$H\left(t\right)=\left\\{\begin{array}[]{c}1\text{ for }t\geq 0,\\\ 0\text{
for }t<0.\end{array}\right.$

Theorem 3.1. _Let conditions ( 2.1)-(2.7) hold. Then:_

_1._ _There exists unique solution of problem ( 2.8), (2.9), which can be
represented as _

$u(\mathbf{x},t)=H\left(t-\varphi(\mathbf{x})\right)\left[A(\mathbf{x})+\widehat{u}(\mathbf{x},t)\right],~{}\mathbf{x}\in\mathbb{R}^{3},~{}t\in\left(0,T\right],$
(3.15)

where the function $\widehat{u}\in C^{2}\left(D_{T}\right)$, is compactly
supported in the domain $D_{T}$ and

$\lim_{t\rightarrow\varphi(\mathbf{x})^{+}}\widehat{u}(\mathbf{x},t)=0,$
(3.16) $A\left(\mathbf{x}\right)>0\text{ in }\mathbb{R}^{3}.$ (3.17)

_The function_ $A(\mathbf{x})\in C^{14}\left(\mathbb{R}^{3}\right)$  _and has
the form:_

$A(\mathbf{x})=\frac{1}{2}\left\\{\begin{array}[]{c}1\text{ for }z\leq 0,\\\
\exp\left(-\frac{1}{2}\int\limits_{L(\mathbf{x})}\frac{\Delta\tau(\mathbf{x}^{\prime})}{n^{2}(\mathbf{x}^{\prime})}ds\right)\text{
for }z>0,\end{array}\right.$ (3.18)

_where_ $\mathbf{x}^{\prime}$ _is the variable point along_ $L(\mathbf{x})$_._

2\. _The inequality_ $A\left(\mathbf{x}\right)>0$ _in ( 3.17) can be replaced
with the following stronger estimate from the below:_

$A\left(\mathbf{x}\right)\geq
A_{0}=\frac{1}{2}\exp\left(-3n_{00}^{2}n_{0}^{2}/2\right),~{}\mathbf{x}\in\Omega.$
(3.19)

__ Proof.  Problem (2.8), (2.9) is equivalent with the following one:

$\mathcal{L}u\equiv n^{2}\left(\mathbf{x}\right)u_{tt}-\Delta
u=0,~{}\left(\mathbf{x},t\right)\in\mathbb{R}^{4},~{}z\neq
0,~{}u\mathbf{\mid}_{t<0}=0,$ (3.20)
$u_{z}|_{z=+0}-u_{z}|_{z=-0}=-\delta\left(t\right),~{}u|_{z=+0}-u|_{z=-0}=0,~{}(x,y,t)\in\mathbb{R}^{3}.$
(3.21)

Let $r>1$ be an integer, which will be chosen later. We represent the solution
of problem (3.20), (3.21) in the form

$u(\mathbf{x},t)=\sum\limits_{k=0}^{r}\alpha_{k}(\mathbf{x})H_{k}(t-\varphi(\mathbf{x}))+u_{r}(\mathbf{x},t),$
(3.22)

where

$H_{k}(t)=\frac{t^{k}}{k!}H(t),~{}k=0,1,2,\ldots,r.$ (3.23)

Also, denote $H_{-1}(t)=\delta(t)$ and $H_{-2}(t)=\delta^{\prime}(t)$.

Recall that the function $\varphi(\mathbf{x})$ is continuous in
$\mathbb{R}^{3}$, $\varphi(\mathbf{x})=-z$ for $z\leq 0$ and $\varphi\in
C^{16}(\mathbb{R}_{+}^{3})$. But the derivative $\varphi_{z}(\mathbf{x})$ is
discontinuous across $z=0$, namely, $\varphi_{z}(\mathbf{x})=-1$ for $z\leq 0$
and $\varphi_{z}(\mathbf{x})=1$ for $z=+0$. Thus,

$\varphi|_{z=0}=0,~{}\varphi_{z}|_{z=+0}=1,~{}\varphi_{z}|_{z=-0}=-1.$ (3.24)

Taking into account (3.24), we need to consider representation (3.22)
separately for $z<0$ and $z\geq 0$.

We seek functions $\alpha_{k}(\mathbf{x})$ in the form:

$\alpha_{k}(\mathbf{x})=\left\\{\begin{array}[]{ll}\alpha_{k}^{-}(\mathbf{x}),&z<0,\\\
\alpha_{k}^{+}(\mathbf{x}),&z\geq 0.\end{array}\right.$ (3.25)

Substituting representation (3.22) in (3.21) and equating coefficients at
$H_{k}(t)$, we obtain

$\left\\{\begin{array}[]{l}-(\alpha_{k}\varphi_{z})_{z=+0}+(\alpha_{k}\varphi_{z})_{z=-0}=-\delta_{k0},\\\
\alpha_{k}|_{z=+0}-\alpha_{k}|_{z=-0}=0,\end{array}\right.$ (3.26)

where $\delta_{k0}$ is the Kronecker’s delta. Then function
$u_{r}(\mathbf{x},t)$ satisfies the following conjugate conditions

$(u_{r})_{z}|_{z=+0}-(u_{r})_{z}|_{z=-0}=0,~{}u_{r}|_{z=+0}-u_{r}|_{z=-0}=0.$
(3.27)

Using (3.24), we find from equations (3.26)

$\alpha_{k}^{-}|_{z=-0}=\alpha_{k}^{+}|_{z=0}=\frac{\delta_{k0}}{2},~{}k=0,1,\ldots,r.$
(3.28)

Apply the operator $\mathcal{L}u$ for $z\neq 0$ to both sides of (3.22).
First,

$\partial_{t}^{2}\left[n^{2}\left(\mathbf{x}\right)\alpha_{k}(\mathbf{x})H_{k}(t-\varphi(\mathbf{x}))\right]=\alpha_{k}(\mathbf{x})n^{2}\left(\mathbf{x}\right)H_{k-2}(t-\varphi(\mathbf{x}).$
(3.29)

Second,

$-\Delta\left[\alpha_{k}(\mathbf{x})H_{k}(t-\varphi(\mathbf{x}))\right]=-H_{k}(t-\varphi(\mathbf{x}))\Delta\alpha_{k}(\mathbf{x})-2\nabla\alpha_{k}(\mathbf{x})\cdot\nabla
H_{k}(t-\varphi(\mathbf{x}))$
$-\alpha_{k}(\mathbf{x})\Delta\left[H_{k}(t-\varphi(\mathbf{x}))\right]$
$=-H_{k}(t-\varphi(\mathbf{x}))\Delta\alpha_{k}(\mathbf{x})+2\nabla\alpha_{k}(\mathbf{x})\cdot\nabla\varphi(\mathbf{x})H_{k-1}(t-\varphi(\mathbf{x}))$
(3.30)
$+\alpha_{k}(\mathbf{x})\Delta\varphi\left(\mathbf{x}\right)H_{k-1}(t-\varphi(\mathbf{x}))-\alpha_{k}(\mathbf{x})\left|\nabla\varphi\left(\mathbf{x}\right)\right|^{2}H_{k-2}(t-\varphi(\mathbf{x})).$

Since by (3.1)
$n^{2}\left(\mathbf{x}\right)-\left|\nabla\varphi\left(\mathbf{x}\right)\right|^{2}=0,$
then (3.29) and (3.30) imply:

$\left(n^{2}\left(\mathbf{x}\right)\partial_{t}^{2}-\Delta\right)\left[\alpha_{k}(\mathbf{x})H_{k}(t-\varphi(\mathbf{x}))\right]=$
$-H_{k}(t-\varphi(\mathbf{x}))\Delta\alpha_{k}(\mathbf{x})+2\nabla\alpha_{k}(\mathbf{x})\nabla\varphi(\mathbf{x})H_{k-1}(t-\varphi(\mathbf{x}))+\alpha_{k}(\mathbf{x})\Delta\varphi\left(\mathbf{x}\right)H_{k-1}(t-\varphi(\mathbf{x})).$

Hence,

$\sum\limits_{k=0}^{r}\alpha_{k}(\mathbf{x})H_{k}(t-\varphi(\mathbf{x}))=\sum\limits_{k=0}^{r}\left[2\nabla\alpha_{k}(\mathbf{x})\nabla\varphi(\mathbf{x})+\alpha_{k}(\mathbf{x})\Delta\varphi\left(\mathbf{x}\right)\right]H_{k-1}(t-\varphi(\mathbf{x}))$
$-\sum\limits_{k=0}^{r}[\Delta\alpha_{k}(\mathbf{x})]H_{k}(t-\varphi(\mathbf{x})).$

Next,

$-\sum\limits_{k=0}^{r}[\Delta\alpha_{k}(\mathbf{x})]H_{k}(t-\varphi(\mathbf{x}))=-\sum\limits_{k=1}^{r+1}[\Delta\alpha_{k-1}(\mathbf{x})]H_{k-1}(t-\varphi(\mathbf{x}))=$
$=-\sum\limits_{k=0}^{r}[\Delta\alpha_{k-1}(\mathbf{x})]H_{k-1}(t-\varphi(\mathbf{x}))-[\Delta\alpha_{r}(\mathbf{x})]H_{r}(t-\varphi(\mathbf{x})),$

where we formally set $\alpha_{-1}(\mathbf{x})\equiv 0$. Hence, we obtain

$\mathcal{L}u=\sum\limits_{k=0}^{r}[2\nabla\alpha_{k}(\mathbf{x})\cdot\nabla\varphi(\mathbf{x})+\alpha_{k}(\mathbf{x})\Delta\varphi(\mathbf{x})-\Delta\alpha_{k-1}(\mathbf{x})]H_{k-1}(t-\varphi(\mathbf{x}))$
$-[\Delta\alpha_{r}(\mathbf{x})]H_{r}(t-\varphi(\mathbf{x}))+\mathcal{L}u_{r}(\mathbf{x},t).~{}$

Equating here to zero terms at $H_{k-1}(t-\varphi(\mathbf{x}))$ and taking
into account conditions (3.28), we obtain equations for $\alpha_{k}^{-}$ and
$\alpha_{k}^{+}$, $k=0,1,\ldots,r$,

$2\nabla\alpha_{k}^{-}(\mathbf{x})\cdot\nabla\varphi(\mathbf{x})+\alpha_{k}^{-}(\mathbf{x})\Delta\varphi(\mathbf{x})=\Delta\alpha_{k-1}^{-}(\mathbf{x}),~{}z<0,~{}\alpha_{k}^{-}|_{z=-0}=\frac{1}{2}\delta_{k0},$
(3.31)
$2\nabla\alpha_{k}^{+}(\mathbf{x})\cdot\nabla\tau(\mathbf{x})+\alpha_{k}^{+}(\mathbf{x})\Delta\tau(\mathbf{x})=\Delta\alpha_{k-1}^{+}(\mathbf{x}),~{}z>0,~{}\alpha_{k}^{+}|_{z=0}=\frac{1}{2}\delta_{k0},$
(3.32)

where $\alpha_{-1}^{-}(\mathbf{x})=\alpha_{-1}^{+}(\mathbf{x})=0$.

Note that in equations (3.31) $\varphi(\mathbf{x})=-z$. Therefore it can be
written as follows

$-2(\alpha_{k}^{-})_{z}(\mathbf{x})=\Delta\alpha_{k-1}^{-}(\mathbf{x}),~{}z<0,~{}\alpha_{k}^{-}|_{z=0}=\frac{1}{2}\delta_{k0},~{}k=0,1,\ldots,r.$
(3.33)

It follows from (3.33) that

$\alpha_{k}^{-}(\mathbf{x})=\frac{1}{2}\delta_{k0},~{}k=0,1,\ldots,r.$ (3.34)

Since $\mathcal{L}u(\mathbf{x},t)=0$ for $z\neq 0$ and conditions (3.27) hold,
the equation for the function $u_{r}(\mathbf{x},t)$ is:

$n^{2}(\mathbf{x})(u_{r})_{tt}-\Delta
u_{r}=F_{r}(\mathbf{x},t),~{}(\mathbf{x},t)\in\mathbb{R}^{4},\quad
u_{r}|_{t<0}=0,$ (3.35)

where

$F_{r}(\mathbf{x},t)=H_{r}(t-\varphi(\mathbf{x}))\Delta\alpha_{r}(\mathbf{x}).$
(3.36)

Integrate now equation (3.32) along the geodesic line $L(\mathbf{x})$. By
(3.6) and (3.7) we have along this line
$\nabla\tau(\mathbf{x})=\mathbf{p}(\mathbf{x})=n^{2}(\mathbf{x})d\mathbf{x}/ds$.
Therefore,

$\nabla\alpha_{k}^{+}(\mathbf{x})\cdot\nabla\tau(\mathbf{x})=n^{2}(\mathbf{x})\nabla\alpha_{k}^{+}(\mathbf{x})\cdot\frac{d\mathbf{x}}{ds}=n^{2}(\mathbf{x})\frac{d}{ds}\alpha_{k}^{+}(\mathbf{x}).$

Hence, (3.32) is equivalent with:

$2n^{2}(\mathbf{x})\frac{d}{ds}\alpha_{k}^{+}(\mathbf{x})+\alpha_{k}^{+}(\mathbf{x})\Delta\tau(\mathbf{x})=\Delta\alpha_{k-1}^{+}(\mathbf{x}),~{}\alpha_{k}^{+}|_{z=0}=\frac{1}{2}\delta_{0k}.$
(3.37)

The solution of the Cauchy problem

$2n^{2}(\mathbf{x})\frac{d}{ds}\alpha_{0}^{+}(\mathbf{x})+\alpha_{0}^{+}(\mathbf{x})\Delta\tau(\mathbf{x})=0,~{}\alpha_{0}^{+}|_{z=0}=\frac{1}{2}$

is given by the formula

$\alpha_{0}^{+}(\mathbf{x})=\frac{1}{2}\exp\left(-\frac{1}{2}\int\limits_{L(\mathbf{x})}\frac{\Delta\tau(\mathbf{x}^{\prime})}{n^{2}(\mathbf{x}^{\prime})}ds\right),$
(3.38)

where $\mathbf{x}^{\prime}$ is a variable point along $L(\mathbf{x})$. Next,
dividing both sides of equation (3.37) by $2n^{2}(\mathbf{x})$ we can rewrite
(3.37) in the form

$\exp\left(-\frac{1}{2}\int\limits_{L(\mathbf{x})}\frac{\Delta\tau(\mathbf{x}^{\prime})}{n^{2}(\mathbf{x}^{\prime})}ds\right)\frac{d}{ds}\left[\alpha_{k}^{+}(\mathbf{x})\exp\left(\frac{1}{2}\int\limits_{L(\mathbf{x})}\frac{\Delta\tau(\mathbf{x}^{\prime})}{n^{2}(\mathbf{x}^{\prime})}ds\right)\right]=$
(3.39) $=\frac{\Delta\alpha_{k-1}^{+}(\mathbf{x})}{2n^{2}(\mathbf{x})}.$

It follows from (3.37) and (3.38) that (3.39) is equivalent with

$\frac{d}{ds}\left(\frac{\alpha_{k}^{+}(\mathbf{x})}{\alpha_{0}^{+}(\mathbf{x})}\right)=\frac{\Delta\alpha_{k-1}^{+}(\mathbf{x})}{2n^{2}(\mathbf{x})\alpha_{0}^{+}(\mathbf{x})},~{}\alpha_{k}^{+}|_{z=0}=0,~{}k=1,\ldots,r.$
(3.40)

Integrating (3.40) with respect to $s$ along $L(\mathbf{x})$, we obtain

$\alpha_{k}^{+}(\mathbf{x})=\frac{1}{2}\alpha_{0}^{+}(\mathbf{x})\int\limits_{L(\mathbf{x})}\frac{\Delta\alpha_{k-1}^{+}(\mathbf{x}^{\prime})}{n^{2}(\mathbf{x}^{\prime})\alpha_{0}^{+}(\mathbf{x}^{\prime})}ds,~{}k=1,\ldots,r.$
(3.41)

Let $m$ be a sufficiently large integer, which we will choose below. If the
function $n\in C^{m}(\mathbb{R}^{3})$, then

$\tau\in C^{m}(\mathbb{R}_{+}^{3}),~{}\alpha_{k}^{+}\in
C^{m-2k-2}(\mathbb{R}_{+}^{3}),~{}\Delta\alpha_{r}^{+}\in
C^{m-2r-4}(\mathbb{R}_{+}^{3}).$

Define the domain $G_{X}$ as

$G_{X}=\\{\mathbf{x}\in\mathbb{R}_{+}^{3}:\,|x|\geq X,|y|\geq X,z\geq 0\\}.$
(3.42)

It follows from (3.38) and (3.41) that $\alpha_{0}^{+}(\mathbf{x})=1/2$ and
functions $\alpha_{k}^{+}(\mathbf{x})=0$, $k=1,\ldots,r$, for $\mathbf{x}\in
G_{X}$. Indeed, since by (2.7) $n\left(\mathbf{x}\right)=1$ in $G_{X},$ then
Assumption 3.1 implies that if $\mathbf{x}\in G_{X},$ then the geodesic line
$L(\mathbf{x})$ is a segment of the straight line orthogonal to the plane
$z=0,$ and this line does not intersect $\Omega$.

Setting $A(\mathbf{x})=\alpha_{0}(\mathbf{x})$ and using (3.22), (3.25),
(3.34) and (3.38), we obtain (3.18).

It follows from (3.36) and the above arguments that the function
$F_{r}(\mathbf{x},t)$ possesses the following properties:

1\. For any $T>0$

$support\left(F_{r}(\mathbf{x},t)\right)\subset D_{T}^{+}=\\{(\mathbf{x},t)\in
D_{T}:\,z\geq 0\\},$

where the set $D_{T}$ is defined in (3.10).

2\. $F_{r}(\mathbf{x},t)=0$ for $\mathbf{x}\in G_{X}$.

Furthermore, the projection of the set $D_{T}^{+}$ on the space
$\mathbb{R}^{3}$ coincides with the set
$Y_{T}=\\{\mathbf{x}\in\mathbb{R}^{3}:\,0\leq\tau(\mathbf{x})\leq T\\}$. It
follows from (3.11) that this set is bounded with respect to $z$, i.e.
$Y_{T}\subset\left\\{(x,y)\in\mathbb{R}^{2},z\in[0,T]\right\\}$. Hence, the
set $\left(Y_{T}\setminus
G_{X}\right)\subset\\{\left|x\right|,\left|y\right|<X,0\leq z\leq T\\}$ is
bounded, where $G_{X}$ is the set defined in (3.42). Thus, the function
$F_{r}(\mathbf{x},t)=0$ outside of the finite domain
$\\{\left|x\right|,\left|y\right|<X,0\leq z\leq T\\}$, i.e. it is compactly
supported in $\mathbb{R}_{T}^{4}$. Moreover, $F_{r}\in
H^{d}(\mathbb{R}_{T}^{4})$, where $d=\min(m-2r-4,r)$. Using the general theory
of hyperbolic equations [18, Chapter 4], we conclude that the unique solution
$u_{r}\in H^{d+1}(\mathbb{R}_{T}^{4})$ of the Cauchy problem (3.35), (3.36)
exists and this solution is also compactly supported in $\mathbb{R}_{T}^{4}$.
Moreover, since $F_{r}(\mathbf{x},t)=0$ for $(\mathbf{x},t)\notin D_{T}$, then
$u_{r}(\mathbf{x},t)=0$ for $t<\varphi(\mathbf{x}).$ Embedding theorem implies
$u_{r}\in C^{2}\left(\overline{\mathbb{R}_{T}^{4}}\right)$ if $d+1>4$. Choose
$d=r=4$ and $m=16$. Then $u_{r}\in
C^{2}\left(\overline{\mathbb{R}_{T}^{4}}\right)$. Since
$u_{r}(\mathbf{x},t)=0$ for $t$$<$$\varphi(\mathbf{x})$, we conclude that
$u_{r}\in C^{2}\left(\overline{D_{T}}\right)$ and $u_{r}(\mathbf{x},t)=0$ for
$t=\varphi(\mathbf{x})$ together with derivatives up to the second order. This
explains our smoothness condition (2.5).

Finally, setting in (3.22)

$A(\mathbf{x})=\alpha_{0}(\mathbf{x}),~{}\widehat{u}(\mathbf{x},t)=\sum\limits_{k=1}^{r}\alpha_{k}(\mathbf{x})\frac{(t-\varphi(\mathbf{x}))^{k}}{k!}+u_{r}(\mathbf{x},t),$

we finish the proof of (3.15) and (3.16).

We now want to prove that $A(\mathbf{x})=\alpha_{0}(\mathbf{x})\geq A_{0}$
with the positive number $A_{0}$ defined in (3.19). We use Lemma 3.1 for this
purpose.

The function $\alpha_{0}(\mathbf{x})=1/2$ for $z\leq 0$. Using (3.12) and
(3.38), estimate now the function $\alpha_{0}^{+}(\mathbf{x})$. Note first
that

$\Delta\tau(\mathbf{x})|_{z=0}=0.$ (3.43)

Indeed, eikonal equation (3.1) implies that

$\nabla\partial_{z}\tau(\mathbf{x})\cdot\nabla\tau(\mathbf{x})=n(\mathbf{x})n_{z}(\mathbf{x}).$
(3.44)

It follows from the condition $\tau(\mathbf{x})|_{z=0}=0$ in (3.1) that, at
$z=0$, $(\tau)_{xx}=(\tau)_{yy}=0$, $(\tau)_{z}=1$ and
$(\tau)_{xz}=(\tau)_{yz}=0$. Therefore (3.44) at $z=0$ becomes

$(\tau)_{zz}(x,y,0)=n(x,y,0)n_{z}(x,y,0)=0.$

Here the equality $n_{z}(x,y,0)=0$ follows from (2.5) and (2.7). Thus, (3.43)
holds. Hence, integrating inequality (3.12) of Lemma 3.1 with respect to
$s\in\left(0,s^{\prime}\right)$, we conclude that along $L(\mathbf{x})$

$\frac{\Delta\tau(\mathbf{x}^{\prime})}{2n^{2}(\mathbf{x}^{\prime})}\leq
3n_{00}^{2}s^{\prime},~{}\mathbf{x}^{\prime}\in\mathbb{R}_{+}^{3},$ (3.45)

Formulae (3.38) and (3.45) imply:

$\alpha_{0}^{+}(\mathbf{x})\geq\frac{1}{2}\exp\left(-\int\limits_{\Gamma(\mathbf{x})}3n_{00}^{2}s^{\prime}ds^{\prime}\right)=\frac{1}{2}\exp\left(-\frac{3n_{00}^{2}}{2}\tau^{2}(\mathbf{x})\right),~{}\mathbf{x}\in\mathbb{R}_{+}^{3},$

It follows from (2.6) and eikonal equation (3.1) that

$\partial_{z}\tau\leq n_{0}\text{ \ in }\mathbb{R}_{+}^{3}.$ (3.46)

Since $\tau(x,y,0)=0,$ then

$\tau(\mathbf{x})=\int\limits_{0}^{z}\partial_{r}\tau(x,y,r)dr\leq
n_{0},~{}\mathbf{x}\in\overline{\Omega}.$

Hence,

$A(\mathbf{x})\geq\alpha_{0}^{+}(\mathbf{x})\geq\
\frac{1}{2}\exp\left(-\frac{3n_{00}^{2}}{2}n_{0}^{2}\right)=A_{0},~{}\mathbf{x}\in\overline{\Omega}.\text{
}~{}~{}\square$

##  4 A Boundary Value Problem in Partial Finite Differences

###  4.1 The boundary value problem

Consider the function $v\left(\mathbf{x},t\right),$

$v\left(\mathbf{x},t\right)=\mathop{\displaystyle\int}\limits_{0}^{t}u(\mathbf{x},s)ds.$
(4.1)

Recall that by (3.9) $\varphi(\mathbf{x})=\tau(\mathbf{x})$ for
$\mathbf{x}\in\Omega.$ Hence, by (3.15), (3.16) and (4.1)

$v(\mathbf{x},t)=\left(t-\tau(\mathbf{x})\right)H\left(t-\tau(\mathbf{x})\right)\left[A(\mathbf{x})+\widehat{v}(\mathbf{x},t)\right],~{}\mathbf{x}\in\Omega,$
(4.2)

where $\widehat{v}(\mathbf{x},\tau(\mathbf{x})+0)=0$.

Estimate $\max_{\mathbf{x}\in\overline{\Omega}}\tau(\mathbf{x})$. Since
$\tau|_{z=0}=0,$ then, using (3.46), we obtain

$\tau\left(x,y,z\right)=\mathop{\displaystyle\int}\limits_{0}^{z}\partial_{r}\tau\left(x,y,r\right)dr\leq
n_{0}z\leq n_{0},\text{ }\mathbf{x=}\left(x,y,z\right)\in\Omega.$

Hence,

$\max_{\mathbf{x}\in\overline{\Omega}}\tau(\mathbf{x})\leq n_{0}$ (4.3)

We assume that $T>n_{0}$. Denote

$T_{1}=T-n_{0}>0.$ (4.4)

Also, denote $Q_{T_{1}}=\Omega\times\left(0,T_{1}\right).$ Let

$P\left(\mathbf{x},t\right)=v(\mathbf{x},t+\tau(\mathbf{x}))\text{ for
}(\mathbf{x},t)\in Q_{T_{1}}.$ (4.5)

This function is defined for all $(\mathbf{x},t)\in Q_{T_{1}}$ since by (4.3)

$0<t+\tau(\mathbf{x})<T_{1}+\max_{\mathbf{x}\in\overline{\Omega}}\tau(\mathbf{x})\leq
T_{1}+n_{0}=T\text{ \ in }Q_{T_{1}}.$

Hence, by (4.2), (4.4) and (4.5)

$P\left(\mathbf{x},t\right)=t\left[A(\mathbf{x})+\widehat{P}(\mathbf{x},t)\right],\text{
}\left(\mathbf{x},t\right)\in Q_{T_{1}},$ (4.6) $\lim_{t\rightarrow
0^{+}}\widehat{P}(\mathbf{x},t)=0.$ (4.7)

Using (2.8), (3.1), (4.1) and (4.5), we obtain

$\Delta P-2\nabla_{\mathbf{x}}P_{t}\cdot\nabla\tau-P_{t}\Delta\tau=0,\text{
}\left(\mathbf{x,}t\right)\in Q_{T_{1}}.$ (4.8)

Denote

$w(\mathbf{x},t)=P_{t}\left(\mathbf{x},t\right).$ (4.9)

Note that by (3.19), (4.6) and (4.7)

$w(\mathbf{x},0)=A\left(\mathbf{x}\right)\geq
A_{0}=\frac{1}{2}\exp\left(-3n_{00}^{2}n_{0}^{2}/2\right)>0,~{}\mathbf{x}\in\Omega.$
(4.10)

Setting in (4.8) $t=0$, using the fact that by (4.6) and (4.7) $\Delta
P\left(x,0\right)=0$ and also using (4.9) and (4.10), we obtain
$\Delta\tau+2\left(\nabla\ln
w\left(\mathbf{x},0\right)\right)\cdot\nabla\tau=0$ for $\mathbf{x}\in\Omega.$
Hence, using this equation and (4.6), we obtain the $2\times 2$ system of non
local nonlinear PDEs:

$\left\\{\begin{array}[]{c}\Delta
w-2\nabla_{\mathbf{x}}w_{t}\cdot\nabla\tau+2w_{t}\left(\nabla\ln
w\left(\mathbf{x},0\right)\right)\cdot\nabla\tau=0,\text{
}\left(\mathbf{x,}t\right)\in Q_{T_{1}},\\\ \Delta\tau+2\left(\nabla\ln
w\left(\mathbf{x},0\right)\right)\cdot\nabla\tau=0,\text{
}\mathbf{x}\in\Omega.\end{array}\right.$ (4.11)

To find boundary conditions for system (4.11), we use functions
$f_{0}\left(\boldsymbol{x},t\right),f_{1}\left(\mathbf{x},t\right),f_{2}\left(\mathbf{x},t\right)$
in (2.11). Let

$g_{0}\left(\mathbf{x},t\right)=f_{0}\left(\mathbf{x},\tau\left(\mathbf{x}\right)+t\right),\text{
}\left(\mathbf{x},t\right)\in\Gamma_{T_{1}},$
$g_{1}\left(\mathbf{x},t\right)=f_{1}\left(\mathbf{x},\tau\left(\mathbf{x}\right)+t\right)+\partial_{t}g_{0}\left(\mathbf{x},t\right)\partial_{z}\tau\left(\mathbf{x}\right),~{}\left(\mathbf{x},t\right)\in\Gamma_{T_{1}},$
$g_{2}\left(\mathbf{x},t\right)=f_{2}\left(\mathbf{x},\tau\left(\mathbf{x}\right)+t\right),~{}\left(\mathbf{x},t\right)\in\Theta_{T_{1}}.$

Since by (2.2) $\Gamma\subset\left\\{z=0\right\\}$ and since by (2.7)
$n\left(\mathbf{x}\right)\mid_{\Gamma}=1$ and also since by the second
condition in (3.1) $\tau\mid_{\Gamma}=0,$ then
$\partial_{x}\tau\mid_{\Gamma}=\partial_{y}\tau\mid_{\Gamma}=0.$ Hence, (3.1)
and (3.3) imply
$\partial_{z}\tau\mid_{\Gamma}=n\left(\mathbf{x}\right)\mid_{\Gamma}=1.$ Also,
it obviously follows from (2.4) and (2.7) that
$\tau\left(\mathbf{x}\right)\mid_{\Theta}=z.$ Thus, the boundary conditions
for system (4.11) are:

$w\mid_{\Gamma_{T_{1}}}=g_{0}\left(\mathbf{x},t\right),\text{
}w_{z}\mid_{\Gamma_{T_{1}}}=g_{1}\left(\mathbf{x},t\right),w\mid_{\Theta_{T_{1}}}=g_{2}\left(\mathbf{x},t\right),$
(4.12)
$\tau\mid_{\Gamma}=0,~{}\partial_{z}\tau\mid_{\Gamma}=1,~{}\tau\mid_{\Theta}=z.$
(4.13)

Therefore, we arrive at the following Boundary Value Problem for the $2\times
2$ system (4.11) of nonlinear and non local PDEs:

Boundary Value Problem  (BVP): _Find the pair of functions_
$\left(w,\tau\right)\in C^{2}\left(\overline{Q}_{T_{1}}\right)\times
C^{2}\left(\overline{\Omega}\right)$_satisfying equations ( 4.11) and boundary
conditions (4.12), (4.13), assuming that (4.10) holds. _

As it was pointed out in Introduction, we cannot prove stability estimates for
this BVP. However, we can prove the desired stability estimates if we rewrite
this BVP in the form of partial finite differences, in which the derivatives
with respect to $x$ and $y$ are written in finite differences, whereas the
derivatives with $z$ and $t$ are written in the conventional continuous way.
In doing so, we assume that the step size $h$ of the finite difference scheme
is bounded from the below by a fixed positive number. The latter assumption is
a quite natural one in computations. Thus, we rewrite in subsection 4.2 the
above BVP in partial finite differences.

###  4.2 Partial finite differences

For brevity, we use the same grid step size $h$ in both $x$ and $y$
directions. Choose a small number $h_{0}\in\left(0,1\right).$ We assume
everywhere below in this paper that

$h\geq h_{0}.$ (4.14)

Consider two partitions of the interval $\left[-X,X\right],$

$\displaystyle-X=x_{0}<x_{1}<\ldots<x_{N}<x_{N+1}=X,\quad x_{i}-x_{i-1}=h,$
$\displaystyle-X=y_{0}<y_{1}<\ldots<y_{N}<y_{N+1}=X,\quad y_{i}-y_{i-1}=h.$

Thus, the interior grid points are
$\left\\{\left(x_{i},y_{j}\right)\right\\}_{i,j=1}^{N}$ and other grid points
are located on the part of the boundary $\Theta\subset\partial\Omega,$ see
(2.4). Let $\mathbf{x}^{h}=\left(x_{i},y_{j},z\right)_{i,j=0}^{N+1}$ denotes
semi-discrete points. For any function $s\left(\mathbf{x},t\right)$ defined on
the set $\overline{Q_{T_{1}}},$ we denote by $s\left(\mathbf{x}^{h},t\right)$
the corresponding semi-discrete vector function defined at points
$\left(\mathbf{x}^{h},t\right)\in\overline{\Omega}\times\left[0,T_{1}\right].$
We also introduce the following notations:

$\Omega_{h}=\left\\{\mathbf{x}^{h}=\left(x_{i},y_{j},z\right)_{i,j=1}^{N},z\in\left(0,1\right)\right\\},$
$\overline{\Omega}_{h}=\left\\{\mathbf{x}^{h}=\left(x_{i},y_{j},z\right)_{i,j=0}^{N+1},z\in\left[0,1\right]\right\\},$
$Q_{h,T_{1}}=\Omega_{h}\times\left(0,T_{1}\right),\text{
}\overline{Q_{h,T_{1}}}=\overline{\Omega}_{h}\times\left[0,T_{1}\right],$
$\Gamma_{h}=\left\\{\mathbf{x}^{h}=\left\\{\left(x_{i},y_{j},0\right)\right\\}_{i,j=1}^{N}\right\\},\text{
}\Gamma_{h,T_{1}}=\Gamma_{h}\times\left(0,T_{1}\right),$
$\Gamma_{h}^{\prime}=\left\\{\mathbf{x}^{h}=\left\\{\left(x_{i},y_{j},1\right)\right\\}_{i,j=1}^{N}\right\\},\text{
}\Gamma_{h,T_{1}}^{\prime}=\Gamma_{h}^{\prime}\times\left(0,T_{1}\right),$
$\Theta_{h}=\left\\{\mathbf{x}^{h}\in\Theta\right\\},\text{
}\Theta_{h,T_{1}}=\Theta_{h}\times\left(0,T_{1}\right).$

The derivatives in finite differences with respect to $x$ are defined as:

$\partial_{x}s_{i,j}^{h}\left(z\right)=\frac{s_{i-1,j}^{h}\left(z,t\right)-s_{i+1,j}^{h}\left(z,t\right)}{2h},\text{
}i,j=1,...,N,$ (4.15)
$\partial_{x}s^{h}\left(\mathbf{x}^{h},t\right)=s_{x}^{h}\left(\mathbf{x}^{h},t\right)=\left\\{\partial_{x}s_{i,j}^{h}\left(z,t\right)\right\\}_{i,j=1}^{N},$
(4.16)
$\partial_{x}^{2}s_{i,j}^{h}\left(z,t\right)=\frac{s_{i+1,j}^{h}\left(z,t\right)-2s_{i,j}^{h}\left(z,t\right)+s_{i-1,j}^{h}\left(z,t\right)}{h^{2}},\text{
}i,j=1,...,N,$ (4.17)
$\partial_{x}^{2}s^{h}\left(\mathbf{x}^{h},t\right)=s_{xx}^{h}\left(\mathbf{x}^{h},t\right)=\left\\{\partial_{x}^{2}s_{i,j}^{h}\left(z,t\right)\right\\}_{i,j=1}^{N}.$
(4.18)

Derivatives $s_{y}^{h},s_{yy}^{h}$ are defined completely similarly. Formulas
(4.15)-(4.18) as well as their analogs for $s_{y}^{h},s_{yy}^{h}$ fully define
finite difference versions of $x,y-$derivatives involved in equations (4.11).
Everywhere below the corresponding Laplace operator in the partial finite
differences as well as the gradient vector are given by:

$\Delta^{h}s^{h}=s_{xx}^{h}+s_{yy}^{h}+s_{zz}^{h},~{}\nabla^{h}s^{h}=\left(s_{x}^{h},s_{y}^{h},s_{z}^{h}\right),$
(4.19)

where the $z-$derivatives are understood in the regular manner, and the same
for the $t-$derivatives in follow up formulas.

We introduce the semi-discrete analogs of conventional function spaces of real
valued functions as:

$H^{2,h}\left(Q_{h,T_{1}}\right)=\left\\{\begin{array}[]{c}s^{h}\left(\mathbf{x}^{h},t\right):\left\|s^{h}\right\|_{H^{2,h}\left(Q_{h,T_{1}}\right)}^{2}=\\\
=\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\mathop{\displaystyle\sum}\limits_{m=0}^{2}\left(\partial_{z}^{m}s^{h}\right)^{2}+\left(\partial_{zt}^{2}s^{h}\right)^{2}\right]\left(x_{i},y_{j},z,t\right)dzdt\\\
+\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\partial_{t}s^{h}\right)^{2}+\left(s^{h}\right)^{2}\right]\left(x_{i},y_{j},z,t\right)dzdt<\infty,\end{array}\right.$
$H^{1,h}\left(\Omega_{h}\right)=$
$=\left\\{s^{h}\left(\mathbf{x}^{h}\right):\left\|s^{h}\right\|_{H^{1,h}\left(\Omega_{h}\right)}^{2}=\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}\mathop{\displaystyle\sum}\limits_{m=0}^{n}h^{2}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\partial_{z}s^{h}\right)^{2}+\left(s^{h}\right)^{2}\right]\left(x_{i},y_{j},z\right)dz<\infty\right\\},$
$L_{2}^{h}\left(\Omega_{h}\right)=\left\\{s^{h}\left(\mathbf{x}^{h}\right):\left\|s^{h}\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}^{2}=\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{1}\left(s^{h}\right)^{2}\left(x_{i},y_{j},z\right)dz\right\\}<\infty,$
$H^{1,h}\left(\Gamma_{h,T_{1}}\right)=\left\\{\begin{array}[]{c}s^{h}\left(\mathbf{x}^{h},t\right):\left\|s^{h}\right\|_{H^{1,h}\left(\Gamma_{rh,T_{1}}\right)}^{2}\\\
=\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\left[\left(s_{t}^{h}\right)^{2}+\left(s^{h}\right)^{2}\right]\left(x_{i},y_{j},0,t\right)dt<\infty\end{array}\right.,$
$L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)=\left\\{s^{h}\left(\mathbf{x}^{h},t\right):\left\|s^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}=\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\left(s^{h}\right)^{2}\left(x_{i},y_{j},0,t\right)dt<\infty\right\\},$
$L_{2}^{h}\left(\Theta_{h,T_{1}}\right)=\left\\{\begin{array}[]{c}s^{h}\left(\mathbf{x}^{h},t\right):\left\|s^{h}\right\|_{L_{2}^{h}\left(\Theta_{h,T_{1}}\right)}^{2}=\\\
=\mathop{\displaystyle\sum}\limits_{j=0}^{N+1}h\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(s^{h}\right)^{2}\left(x_{0},y_{j},z,t\right)+\left(s^{h}\right)^{2}\left(x_{N+1},y_{j},z,t\right)\right]dzdt\\\
+\mathop{\displaystyle\sum}\limits_{i=0}^{N+1}h\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(s^{h}\right)^{2}\left(x_{i},y_{0},z,t\right)+\left(s^{h}\right)^{2}\left(x_{i},y_{N+1},z,t\right)\right]dzdt<\infty,\end{array}\right.$
$C^{2,h}\left(\overline{Q_{h,T_{1}}}\right)=\left\\{\begin{array}[]{c}s^{h}\left(\mathbf{x}^{h},t\right):\left\|s^{h}\right\|_{C^{2,h}\left(\overline{Q_{h,T_{1}}}\right)}=\\\
=\max_{m=0,1,2}\left(\max_{\overline{Q_{h,T_{1}}}}\left(\left|\partial_{z}^{m}s^{h}\right|\right),\max_{\overline{Q_{h,T_{1}}}}\left|\partial_{zt}^{2}s^{h}\right|\right)<\infty,\end{array}\right.$
$C^{n,h}\left(\overline{\Omega_{h}}\right)=\left\\{\begin{array}[]{c}s^{h}\left(\mathbf{x}^{h}\right):\left\|s^{h}\right\|_{C^{n,h}\left(\overline{\Omega_{h}}\right)}\\\
=\max_{m\in\left[0,n\right]}\left(\max_{\overline{\Omega_{h}}}\left|\partial_{z}^{m}s^{h}\right|,\max_{\overline{\Omega_{h}}}\left|s^{h}\right|\right)<\infty,\\\
n\in\left[0,2\right].\end{array}\right.$

###  4.3 The Boundary Value Problem in Partial Finite Differences

We now rewrite the BVP (4.11)-(4.13) as the BVP in partial finite differences
with respect to the vector functions
$\left(w^{h}\left(\mathbf{x}^{h},t\right),\tau^{h}\left(\mathbf{x}^{h}\right)\right)$:

$\Delta^{h}w^{h}-2\tau_{z}^{h}w_{zt}^{h}-2w_{tx}^{h}\tau_{x}^{h}-2w_{ty}^{h}\tau_{y}^{h}+$
(4.20) $+2w_{t}^{h}\left(\nabla^{h}\ln
w^{h}\left(\mathbf{x}^{h},0\right)\right)\cdot\nabla^{h}\tau^{h}=0,\text{
}\left(\mathbf{x}^{h}\mathbf{,}t\right)\in Q_{T_{1}}^{h},$
$\Delta^{h}\tau^{h}+2\left(\nabla^{h}\ln
w^{h}\left(\mathbf{x}^{h},0\right)\right)\cdot\nabla^{h}\tau^{h}=0,\text{
}\mathbf{x}^{h}\in\Omega^{h}.$ (4.21)

In addition, (4.12) and (4.13) lead to the following boundary conditions for
the $2\times 2$ system (4.20), (4.21)

$w^{h}\mid_{\Gamma_{h,T_{1}}}=g_{0}^{h},\text{
}w_{z}^{h}\mid_{\Gamma_{h,T_{1}}}=g_{1}^{h},\text{
}w^{h}\mid_{\Theta_{h,T_{1}}}=g_{2}^{h},$ (4.22)
$\tau^{h}\mid_{\Gamma_{h}}=0,~{}\partial_{z}\tau^{h}\mid_{\Gamma_{h}}=1,\tau^{h}\mid_{\Theta_{h}}=z.$
(4.23)

Also, using (4.10), we impose the following condition on the function
$w^{h}\left(\mathbf{x}^{h},0\right):$

$w^{h}\left(\mathbf{x}^{h},0\right)=A^{h}\left(\mathbf{x}^{h}\right)\geq
A_{0}=\frac{1}{2}\exp\left(-3n_{00}^{2}n_{0}^{2}/2\right)>0\text{ for
}\mathbf{x}^{h}\in\overline{\Omega}^{h}.$ (4.24)

Boundary Value Problemh (BVPh). _Find the pair of functions_

$\left(w^{h},\tau_{s}^{h}\right)\in
C^{2,h}\left(\overline{Q_{h,T_{1}}}\right)\times
C^{2,h}\left(\overline{\Omega_{h}}\right)$_satisfying conditions (
4.20)-(4.23), assuming that (4.24) holds. _

##  5 Two Carleman Estimates

In this section, we prove two Carleman estimates for operators written in the
above partial finite differences. Let the function
$\xi\left(\mathbf{x}^{h}\right)\in C^{1,h}\left(\overline{\Omega_{h}}\right)$.
Consider three numbers $\xi_{0},\xi_{1},\xi_{2}$ such that $0<\xi_{0}<\xi_{1}$
and $\xi_{2}>0.$ We assume that

$0<\xi_{0}\leq\xi^{h}\left(\mathbf{x}^{h}\right)\leq\xi_{1},\quad\mbox{ for
all }\mathbf{x}^{h}\in\overline{\Omega_{h}},$ (5.1)
$\xi_{2}=\max_{\overline{\Omega_{h}}}\left|\xi_{z}^{h}\left(\mathbf{x}^{h}\right)\right|.$
(5.2)

For functions $v^{h}\in H^{2,h}(Q_{h,T_{1}})$, we define the linear operator
$L^{h}$ as:

$L^{h}v^{h}=\Delta^{h}v^{h}-\xi^{h}\left(\mathbf{x}^{h}\right)v_{zt}^{h},\text{
in }Q_{h,T_{1}},$ (5.3)

where the operator $\Delta^{h}$ is defined in (4.19).

Theorem 5.1 (the first Carleman estimate). _Let the function_
$\xi\left(\mathbf{x}^{h}\right)\in C^{1,h}\left(\overline{\Omega_{h}}\right)$
_satisfies conditions ( 5.1), (5.2). Consider the number _$\alpha_{0},$__

$\alpha_{0}=\alpha_{0}\left(\xi_{1}\right)=\frac{2}{3\xi_{1}}>0.$ (5.4)

_Then there exists a sufficiently large number_
$\lambda_{0}=\lambda_{0}\left(T_{1},\xi_{0},\xi_{1},\xi_{2},h_{0},X\right)\geq
1$ _depending only on listed parameters, such that for all_
$\alpha\in\left(0,\alpha_{0}\right],$_all_ $\lambda\geq\lambda_{0}$_and for
all functions_ $v\in H^{2,h}(Q_{h,T_{1}})$_the following Carleman estimate is
valid:_

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(L^{h}v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\geq
C\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(v_{z}^{h}\right)^{2}+\left(v_{t}^{h}\right)^{2}+\lambda^{2}\left(v^{h}\right)^{2}\right]\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$+C\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(v_{z}^{h}\right)^{2}+\lambda^{2}\left(v^{h}\right)^{2}\right]\left(x_{i},y_{j},z,0\right)e^{-2\lambda
z}dz$ (5.5) $-C\lambda e^{-2\lambda\alpha
T_{1}}\left(\left\|v_{z}^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}^{2}+\lambda^{2}\left\|v^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}^{2}\right)$
$-C\left\|v^{h}\right\|_{L_{2}^{h}\left(\Theta_{h,T_{1}}\right)}^{2}-C\lambda\left(\left\|v_{z}^{h}\right\|_{H^{1,h}\left(\Gamma_{h,T_{1}}\right)}^{2}+\lambda^{2}\left\|v^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}\right),$

_where the constant_
$C=C\left(T_{1},\xi_{0},\xi_{1},\xi_{2},h_{0},X,\alpha\right)>0$_depends only
on listed parameters._

Proof. Here and below in this paper
$C=C\left(T_{1},\xi_{0},\xi_{1},\xi_{2},h_{0},X,\alpha\right)>0$ denotes
different constants depending only on listed parameters. By (4.17)-(4.19),
(5.3) and Young’s inequality

$\left(L^{h}v^{h}\right)^{2}\left(\mathbf{x}^{h},t\right)\geq\frac{1}{2}\left(\partial_{z}^{2}v^{h}-\xi^{h}v_{zt}^{h}\right)^{2}\left(\mathbf{x}^{h},t\right)-\left(\partial_{x}^{2}v^{h}+\partial_{y}^{2}v^{h}\right)^{2}\left(\mathbf{x}^{h},t\right).$
(5.6)

Obviously

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\partial_{x}^{2}v^{h}+\partial_{y}^{2}v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\leq
C\left\|v^{h}\right\|_{L_{2}^{h}\left(\Theta_{h,T_{1}}\right)}^{2}+C\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt.$

Hence, (5.6) implies

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(L^{h}v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$\geq\frac{1}{2}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\partial_{z}^{2}v^{h}-\left(n^{h}\right)^{2}v_{zt}^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (5.7)
$-C\left\|v^{h}\right\|_{L_{2}^{h}\left(\Theta_{h,T_{1}}\right)}^{2}-C\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt.$

Theorem 3.1 of [16, Theorem 3.1] implies that, given (5.1) and (5.4), the
following Carleman estimate holds:

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\partial_{z}^{2}v^{h}-\xi^{h}v_{zt}^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\geq
C\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\left(v_{z}^{h}\right)^{2}+\left(v_{t}^{h}\right)^{2}+\lambda^{2}\left(v^{h}\right)^{2}\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (5.8) $-C\lambda e^{-2\lambda\alpha
T_{1}}\left(\left\|v_{z}^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}^{2}+\lambda^{2}\left\|v^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}^{2}\right)$
$-C\lambda\left(\left\|v_{z}^{h}\right\|_{H^{1,h}\left(\Gamma_{h,T_{1}}\right)}^{2}+\lambda^{2}\left\|v^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}\right).$

Choosing a sufficiently large
$\lambda_{0}=\lambda_{0}\left(T_{1},\xi_{0},\xi_{1},\xi_{2},h_{0},X\right)\geq
1,$ setting $\lambda\geq\lambda_{0}$ and combining (5.7) and (5.8), we obtain
(5.5), which is the target estimate of this theorem. $\square$

Remarks 5.1:

1\. _We now explain why the terms reflecting boundary conditions at_
$\Gamma_{h,T_{1}}^{\prime}$_are absent in the right hand side of ( 5.5) and
why the condition _$\alpha\in\left(0,\alpha_{0}\right],$_with_ $\alpha_{0}$_as
in ( 5.4) is imposed. The point here is that the condition
_$\alpha\in\left(0,\alpha_{0}\right]$ _ensures that those terms are non-
negative. This follows immediately from the combination of ( 5.1) with the
formula (3.1) of [16] as well as with the following formulas in the proof of
Theorem 3.1 of [16]: the formula (3.14) (third and fourth lines), the
inequality just below (3.14) and the formula (3.15)._

2._We also note that Carleman estimate ( 5.5) is valid for any value
_$T_{1}>0.$_This is because of the presence of the negative term in ( 5.5),_

$-C\lambda e^{-2\lambda\alpha
T_{1}}\left(\left\|v_{z}^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}^{2}+\lambda^{2}\left\|v^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}^{2}\right).$

Theorem 5.2 (the second Carleman estimate). _Let the parameter_ $\alpha$_be
the same as in Theorem 5.1. Then there exists a sufficiently large number_
$\lambda_{1}=\lambda_{1}\left(h_{0},X\right)\geq 1$ _such that_ _for all_
$\lambda\geq\lambda_{1}$_and for all functions_ $v^{h}\in
H^{2,h}(Q_{h,T_{1}})$_the following Carleman estimate is valid_

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\Delta^{h}v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\geq
C_{1}\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(v_{z}^{h}\right)^{2}+\lambda^{2}\left(v^{h}\right)^{2}\right]\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (5.9)
$-C_{1}\left\|v^{h}\right\|_{L_{2}^{h}\left(\Theta_{h,T_{1}}\right)}^{2}-C_{1}\lambda\left[\left\|v_{z}^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}+\lambda^{2}\left\|v^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}\right].$

_Both constants_ $\lambda_{1}\left(h_{0},X\right)\geq 1$_and_
$C_{1}=C_{1}\left(h_{0},X\right)>0$_depend only on listed parameters._

Proof. __ Here and below in this paper $C=C\left(h_{0},X\right)>0$ denotes
different constants depending only on listed parameters. We obtain similarly
with (5.7)

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\Delta^{h}v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$\geq\frac{1}{2}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(v_{zz}^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (5.10)
$-C\left\|v^{h}\right\|_{L_{2}^{h}\left(\Theta_{h,T_{1}}\right)}^{2}-C\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(v^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt.$

It follows from [15, Theorem 7.1] that the following Carleman estimate holds
for sufficiently large $\lambda\geq\lambda_{0}\geq 1:$

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(v_{zz}^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\geq
C\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(v_{z}^{h}\right)^{2}+\lambda^{2}\left(v^{h}\right)^{2}\right]\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (5.11)
$-C\lambda\left[\left\|v_{z}^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}+\lambda^{2}\left\|v^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}\right].$

Combining (5.10) and (5.11), we obtain (5.9), which is the target estimate of
this theorem. $\square$

##  6 The stability Estimate for the CIP

Consider the set $\mathcal{N}$ of functions $n\left(\mathbf{x}\right)$ defined
as:

$\mathcal{N}=\left\\{n\left(\mathbf{x}\right)\text{ satisfies
(\ref{3.1})-(\ref{2.3}) and }n_{z}\left(\mathbf{x}\right)\geq 0\text{ in
}\mathbb{R}^{3}\right\\}.$ (6.1)

Lemma 6.1. _The following inequality holds:_

$\partial_{z}\tau\left(\mathbf{x}\right)\geq 1,\text{
}\mathbf{x}\in{\Omega},~{}\forall n\left(\mathbf{x}\right)\in\mathcal{N}.$

Proof. It follows from (3.6)-(3.8) that, along the geodesic line
$L\left(\mathbf{x}\right),$

$\frac{d}{ds}\left(\partial_{z}\tau\right)=\frac{n_{z}}{n},\text{ for
}s>0\text{ }~{}$

and $\tau\mid_{s=0}=0.$ At $s=0$ the geodesic line $L\left(\mathbf{x}\right)$
intersects with the plane $\Sigma=\left\\{z=0\right\\}$ defined in (2.10).
Hence, $\partial_{x}\tau\mid_{s=0}=\partial_{y}\tau\mid_{s=0}=0.$ By (2.7)
$n\left(\mathbf{x}\right)\mid_{\Sigma}=1.$ Hence, (3.2) implies that
$\partial_{z}\tau\mid_{s=0}=1.$ Hence, using (6.1), we obtain

$\partial_{z}\tau\left(\mathbf{x}\right)=1+\mathop{\displaystyle\int}\limits_{L\left(\mathbf{x}\right)}\frac{n_{z}}{n}\left(\mathbf{\xi}\left(s\right)\right)ds\geq
1.\text{ }\square$

Our CIP is an ill-posed problem. Therefore, to prove the desired stability
estimate, it is necessary to assume, in accordance with the well known
Tikhonov’s concept of conditional correctness for ill-posed problems [29],
that some _a priori_ known bounds are imposed on the functions
$w^{h},\tau^{h}$.

Thus, let $M>0$ be a positive number. Introduce the set of semi-discrete
functions $S^{h}=S^{h}\left(M,X,n_{0},n_{00}\right)$ as:

$S^{h}=S^{h}\left(M,n_{0},n_{00}\right)=$
$=\left\\{\begin{array}[]{c}\left(w^{h},\tau^{h}\right)\in
C^{2,h}\left(\overline{Q_{h,T_{1}}}\right)\times
C^{2,h}\left(\overline{\Omega_{h}}\right):\\\
\left|\nabla^{h}\tau^{h}\left(\mathbf{x}^{h}\right)\right|\leq n_{0},\text{
}\mathbf{x}^{h}\in\overline{\Omega_{h}},\\\
\left\|w^{h}\right\|_{C^{2,h}\left(\overline{Q_{h,T_{1}}}\right)},\left\|\tau^{h}\right\|_{C^{2,h}\left(\overline{\Omega_{h}}\right)}\leq
M,\\\ 1\leq\partial_{z}\tau^{h}\left(\mathbf{x}\right)\leq n_{0},\text{
}\mathbf{x}^{h}\in\overline{\Omega}_{h},\\\
w^{h}\left(\mathbf{x}^{h},0\right)\geq A_{0},\\\ \text{where the vector
function }\left(w^{h},\tau^{h}\right)\\\ \text{ is generated by the above
procedure}\\\ \text{ and the number }A_{0}\text{ is defined in
(\ref{3.262}).}\end{array}\right.$ (6.2)

Conditions in the second, fourth and fifth lines of (6.2) are imposed due to
(2.6), (3.1), Lemma 6.1 and Theorem 3.1, respectively. We have

$S^{h}=S_{w}^{h}\times
S_{\tau}^{h}=\left\\{\left(w^{h}\left(\mathbf{x}^{h},t\right),\tau^{h}\left(\mathbf{x}^{h}\right)\right)\right\\}.$
(6.3)

Thus, by (6.2) and (6.3)

$w^{h}\left(\mathbf{x}^{h},0\right)\geq
A_{0},\mathbf{x}^{h}\in\overline{\Omega}_{h},\text{ }\forall w^{h}\in
S_{w}^{h}.$ (6.4) $\text{
}1\leq\partial_{z}\tau^{h}\left(\mathbf{x}^{h}\right)\leq n_{0},\text{
}\mathbf{x}^{h}\in\overline{\Omega}_{h},\forall\tau^{h}\in S_{\tau}^{h}.$
(6.5)

Using (3.1), define the function $n^{h}\left(\mathbf{x}^{h}\right)$ as:

$\left|\nabla_{\mathbf{x}}\tau^{h}\left(\mathbf{x}^{h}\right)\right|^{2}=\left(n^{h}\right)^{2}\left(\mathbf{x}^{h}\right),\text{
}\forall\tau^{h}\in S_{\tau}^{h}.$ (6.6)

Theorem 6.1 (Hölder stability estimate). _Let two vector functions_
$\left(w_{1}^{h},\tau_{1}^{h}\right)$,
$\left(w_{2}^{h},,\tau_{2}^{h}\right)\in S^{h}$_be solutions of BVP_ h _with
two sets of boundary data at_
$\Gamma_{h,T_{1}},\Gamma_{h},\Theta_{h,T_{1}}$_and_ $\Theta_{h}$,

$w_{j}^{h}\mid_{\Gamma_{h,T_{1}}}=g_{j,0}^{h},\text{
}\partial_{z}w_{j}^{h}\mid_{\Gamma_{h,T_{1}}}=g_{j,1}^{h},~{}w_{j}^{h}\mid_{\Theta_{h,T_{1}}}=g_{j,2}^{h},\text{
}j=1,2,$ (6.7)
$\tau_{j}^{h}\mid_{\Gamma_{h}}=0,~{}\partial_{z}\tau_{j}^{h}\mid_{\Gamma_{h}}=1,~{}\tau_{j}^{h}\mid_{\Theta_{h}}=z,\text{
}j=1,2.$ (6.8)

_Denote_

$\left\\{\begin{array}[]{c}\widetilde{w}^{h}=w_{1}^{h}-w_{2}^{h},~{}\widetilde{\tau}^{h}=\tau_{1}^{h}-\tau_{2}^{h},\\\
\widetilde{g}_{0}^{h}=g_{1,0}^{h}-g_{2,0}^{h},~{}\widetilde{g}_{1}^{h}=g_{1,1}^{h}-g_{2,1}^{h},~{}\widetilde{g}_{2}^{h}=g_{1,2}^{h}-g_{2,2}^{h},\\\
\widetilde{n}^{h}=n_{1}^{h}-n_{2}^{h},\end{array}\right.$ (6.9)

_where functions_ $n_{1}^{h}$_and_ $n_{2}^{h}$_are obtained from functions_
$\tau_{1}^{h}$_and_ $\tau_{2}^{h}$_respectively via ( 6.6). Assume that_

$\left\|\widetilde{g}_{0}^{h}\right\|_{H^{1,h}\left(\Gamma_{h,T_{1}}\right)},~{}\left\|\widetilde{g}_{1}^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)},\left\|\widetilde{g_{2}}^{h}\right\|_{L_{2}^{h}\left(\Theta_{h,T_{1}}\right)}<\delta,$
(6.10)

_where_ $\delta\in\left(0,1\right)$_is a number. Define the number_
$\alpha_{0}=\alpha_{0}\left(n_{0}\right)=2/\left(3n_{0}\right).$ _Consider an
arbitrary number_ $\alpha\in\left(0,\alpha_{0}\right]$_and set_
$T_{1}=3/\alpha$. _Then there exists a sufficiently small number_
$\delta_{0}=\delta_{0}\left(M,n_{0},n_{00},h_{0},X,\alpha\right)\in\left(0,1\right)$_and
a number_ $C_{2}=C_{2}\left(M,n_{0},n_{00},h_{0},X,\alpha\right)>0,$_both
numbers depending only on listed parameters, such that if_
$\delta\in\left(0,\delta_{0}\right),$_then the following stability estimates
are valid for the functions_
$\widetilde{w}^{h},\widetilde{\tau}^{h},\widetilde{n}^{h}:$

$\left\|\widetilde{w}^{h}\right\|_{H^{1,h}\left(Q_{h,1/\alpha}\right)},~{}\left\|\widetilde{\tau}^{h}\right\|_{H^{1,h}\left(\Omega_{h}\right)},~{}\left\|\widetilde{n}^{h}\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}\leq
C_{2}\delta^{1/3}\ln\left(\delta^{-1}\right).$ (6.11)

Proof. Below $C_{2}=C_{2}\left(M,n_{0},n_{00},h_{0},X,\alpha\right)>0$ denotes
different constants depending only on listed parameters. Let
$a_{1},a_{2},b_{1},b_{2}\in\mathbb{R}$ be arbitrary numbers. Let
$\widetilde{a}=a_{1}-a_{2}$ and $\widetilde{b}=b_{1}-b_{2}.$ Then

$a_{1}b_{1}-a_{2}b_{2}=\widetilde{a}b_{1}+\widetilde{b}a_{2}.$ (6.12)

Using (3.19), (6.2), (6.9) and (6.12), we obtain

$\left|\nabla^{h}\ln w_{1}^{h}\left(\mathbf{x}^{h},0\right)-\nabla^{h}\ln
w_{2}^{h}\left(\mathbf{x}^{h},0\right)\right|\leq
C_{2}\left(\left|\nabla^{h}\widetilde{w}\left(\mathbf{x}^{h},0\right)\right|+\left|\widetilde{w}\left(\mathbf{x}^{h},0\right)\right|\right).$
(6.13)

It is well known that, when applying a Carleman estimate, one can replace
differential equations with appropriate differential inequalities, see, e.g.
[17]. Hence, subtract two equations (4.20), (4.21) for the pair
$\left(w_{2}^{h},,\tau_{2}^{h}\right)$ from two equations (4.20), (4.21) for
the pair $\left(w_{1}^{h},\tau_{1}^{h}\right),$ use the first line of (6.9)
and (6.12). Then, using (6.13), turn resulting equations in inequalities with
respect to functions $\widetilde{w}^{h}$ and $\widetilde{\tau}_{s}^{h}.$ We
obtain two differential inequalities:

$\left|\Delta^{h}\widetilde{w}^{h}-\partial_{z}\tau_{1}^{h}\widetilde{w}_{zt}^{h}\right|$
$\leq
C_{2}\left(\left|\widetilde{w}_{t}^{h}\right|+\left|\nabla^{h}\widetilde{w}^{h}\left(\mathbf{x}^{h}\mathbf{,}0\right)\right|+\left|\widetilde{w}^{h}\left(\mathbf{x}^{h}\mathbf{,}0\right)\right|+\left|\nabla^{h}\widetilde{\tau}^{h}\right|\right),\text{
}\left(\mathbf{x}^{h}\mathbf{,}t\right)\in Q_{T_{1}}^{h},$ (6.14)
$\left|\Delta^{h}\widetilde{\tau}^{h}\right|\leq
C_{2}\left(\left|\nabla^{h}\widetilde{w}^{h}\left(\mathbf{x}^{h}\mathbf{,}0\right)\right|+\left|\widetilde{w}^{h}\left(\mathbf{x}^{h}\mathbf{,}0\right)\right|+\left|\nabla^{h}\widetilde{\tau}^{h}\right|\right),\text{
}\mathbf{x}^{h}\in\Omega_{h}.$ (6.15)

By (6.8) and the second line of (6.9) the boundary conditions are:

$\widetilde{w}^{h}\mid_{\Gamma_{h,T_{1}}}=\widetilde{g}_{0}^{h},\text{
}\widetilde{w}_{z}^{h}\mid_{\Gamma_{h,T_{1}}}=\widetilde{g}_{1}^{h},\text{
}\widetilde{w}^{h}\mid_{\Theta_{h,T_{1}}}=\widetilde{g}_{2}^{h},$ (6.16)
$\widetilde{\tau}^{h}\mid_{\Gamma_{h}}=0,~{}\partial_{z}\widetilde{\tau}^{h}\mid_{\Gamma_{h}}=0,~{}\widetilde{\tau}^{h}\mid_{\Theta_{h}}=0.$
(6.17)

Denote

$I\left(x_{i},y_{j},z,t\right)=$
$\left[\left|\widetilde{w}_{t}^{h}\left(x_{i},y_{j},z,t\right)\right|+\left(\left|\nabla^{h}\widetilde{w}^{h}\right|+\left|\widetilde{w}^{h}\right|\right)\left(x_{i},y_{j},z,0\right)+\left|\nabla^{h}\widetilde{\tau}\right|\left(x_{i},y_{j},z\right)\right]^{2}.$
(6.18)

Using Cauchy-Schwarz inequality, we obtain

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}I\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\leq
C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\widetilde{w}_{t}^{h}\left(x_{i},y_{j},z,t\right)\right)^{2}e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$+C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\left(w_{z}^{h}\right)^{2}+\left(w^{h}\right)^{2}\right)\left(x_{i},y_{j},z,0\right)\right]e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (6.19)
$+C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\left(\widetilde{\tau}^{h}\right)^{2}\right]\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt.$

And also

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\left|\nabla^{h}\widetilde{w}^{h}\right|+\left|\widetilde{w}^{h}\right|\right)\left(x_{i},y_{j},z,0\right)+\left|\nabla^{h}\widetilde{\tau}\right|\left(x_{i},y_{j},z\right)\right]^{2}e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\leq
C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\left(\widetilde{\tau}^{h}\right)^{2}\right]\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (6.20)
$+C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\left(w_{z}^{h}\right)^{2}+\left(w^{h}\right)^{2}\right)\left(x_{i},y_{j},z,0\right)\right]e^{-2\lambda\left(z+\alpha
t\right)}dzdt.$

Square both sides of both inequalities (6.14) and (6.15). Then multiply the
results by the function $e^{-2\lambda\left(z+\alpha t\right)}$, construct sums
combined with integrals like in the left hand sides of Carleman estimates
(5.5) and (5.9) and then use (6.18)-(6.20). We obtain

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\Delta^{h}\widetilde{w}^{h}-\partial_{z}\tau_{1}^{h}\widetilde{w}_{zt}^{h}\right)^{2}\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\leq
C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\widetilde{w}_{t}^{h}\left(x_{i},y_{j},z,t\right)\right)^{2}e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$+C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\widetilde{w}_{z}^{h}\right)^{2}+\left(\widetilde{w}^{h}\right)^{2}\right]\left(x_{i},y_{j},z,0\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (6.21)
$+C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\left(\widetilde{\tau}^{h}\right)^{2}\right]\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha_{0}t\right)}dzdt.$

And also

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\Delta^{h}\widetilde{\tau}^{h}\right)^{2}\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ $\leq
C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\left(\widetilde{\tau}^{h}\right)^{2}\right]\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (6.22)
$+C_{1}\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\widetilde{w}_{z}^{h}\right)^{2}+\left(\widetilde{w}^{h}\right)^{2}\right]\left(x_{i},y_{j},z,0\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt.$

By (5.4) and the second line of (6.2) Set in Theorem 5.1
$\xi^{h}\left(\mathbf{x}^{h}\right)=\partial_{z}\tau_{1}^{h}\left(\mathbf{x}^{h}\right).$
Then Theorem 5.1 is applicable here. Indeed, it follows from (6.2) and Lemma
6.1 that we can now take in (5.1) $\xi_{0}=1$, $\xi_{1}=n_{0},$ and we can
take $\xi_{2}=M$ in (5.2). Let $\lambda_{0}\geq 1$ be the same as in Theorem
5.1. Thus, apply Carleman estimates (5.5) and (5.9) to the left hand sides of
(6.21) and (6.22) respectively. Then sum up resulting inequalities. In doing
so, we replace in (5.5) $v^{h}\left(\mathbf{x}^{h},t\right)$ with
$\widetilde{w}^{h}\left(\mathbf{x}^{h},t\right)$ and replace in (5.9)
$v^{h}\left(\mathbf{x}^{h},t\right)$ with
$\widetilde{\tau}^{h}\left(\mathbf{x}^{h}\right).$ In addition, use (6.17). We
obtain for all $\lambda\geq\lambda_{0}$

$\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\widetilde{w}_{t}^{h}\left(x_{i},y_{j},z,t\right)\right)^{2}e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$+\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\widetilde{w}^{h}\right)^{2}+\left(\widetilde{w}_{z}^{h}\right)^{2}\right]\left(x_{i},y_{j},z,0\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$+\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\left(\widetilde{\tau}^{h}\right)^{2}\right]\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt.$ $\geq
C_{2}\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\left(\widetilde{w}_{z}^{h}\right)^{2}+\left(\widetilde{w}_{t}^{h}\right)^{2}+\lambda^{2}\left(\widetilde{w}^{h}\right)^{2}\right)\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$+C_{2}\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\lambda^{2}\left(\widetilde{\tau}^{h}\right)^{2}\right)\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$ (6.23)
$+C_{2}\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\widetilde{w}_{z}^{h}\right)^{2}+\lambda^{2}\left(\widetilde{w}^{h}\right)^{2}\right]\left(x_{i},y_{j},z,0\right)e^{-2\lambda
z}dz$
$-C_{2}\lambda^{3}e^{-6\lambda}\left\|\widetilde{w}^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{H^{1,h}\left(\Omega_{h}\right)}^{2}-C_{2}\lambda^{3}\left(\left\|\widetilde{w}^{h}\right\|_{H^{1,h}\left(\Gamma_{h,T_{1}}\right)}^{2}+\left\|\widetilde{w}_{z}^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T1}\right)}^{2}\right)$
$-C_{2}\left\|\widetilde{w}^{h}\right\|_{L_{2}\left(\Theta_{h,T_{1}}\right)}^{2}.$

The multiplier $e^{-6\lambda}$ in the sixth line of (6.23) is due to the fact
that $T_{1}=3/\alpha$ and $e^{-2\lambda\left(z+\alpha T_{1}\right)}\leq
e^{-6\lambda}$ for $z\in\left(0,1\right).$ Choose
$\lambda_{1}=\lambda_{1}\left(M,n_{0},n_{00},h_{0},X,\alpha\right)\geq\lambda_{0}\geq
1$ so large that $C_{2}\lambda_{1}/2>1.$ Then (6.9), (6.16) and (6.23) imply

$C_{2}\lambda^{3}e^{-6\lambda}\left\|\widetilde{w}^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{H^{1,h}\left(\Omega_{h}\right)}^{2}+C_{2}\lambda^{3}\left(\left\|\widetilde{g}_{0}^{h}\right\|_{H^{1,h}\left(\Gamma_{h,T_{1}}\right)}^{2}+\left\|\widetilde{g}_{1}^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T1}\right)}^{2}\right)$
$+C_{2}\left(\left\|\widetilde{w}^{h}\right\|_{L_{2}\left(\Theta_{h,T_{1}}\right)}^{2}+\left\|\widetilde{w}^{h}\right\|_{L_{2}\left(\Theta_{h}\right)}^{2}\right)$
$\geq\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\left(\widetilde{w}_{z}^{h}\right)^{2}+\left(\widetilde{w}_{t}^{h}\right)^{2}+\lambda^{2}\left(\widetilde{w}^{h}\right)^{2}\right)\left(x_{i},y_{j},z,t\right)e^{-2\lambda\left(z+\alpha_{0}t\right)}dzdt$
(6.24)
$+\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{T_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\lambda^{2}\left(\widetilde{\tau}^{h}\right)^{2}\right)\left(x_{i},y_{j},z\right)e^{-2\lambda\left(z+\alpha
t\right)}dzdt$
$+\lambda\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\widetilde{w}_{z}^{h}\right)^{2}+\lambda^{2}\left(\widetilde{w}^{h}\right)^{2}\right]\left(x_{i},y_{j},z,0\right)e^{-2\lambda
z}dz,\text{ }\forall\lambda\geq\lambda_{1}.$

Introduce the number
$t_{1}=1/\alpha\in\left(0,T_{1}\right)=\left(0,3/\alpha\right).$ Then

$e^{-2\lambda\left(z+\alpha t\right)}>e^{-2\lambda\left(1+\alpha
t_{1}\right)}=e^{-4\lambda}\text{ for
}\left(z,t\right)\in\left(0,1\right)\times\left(0,t_{1}\right).$ (6.25)

Divide (6.24) by $\lambda e^{-4\lambda}$ and replace $\lambda^{2}\geq 1$ with
$1$ in the right hand side of the resulting inequality, thus, making it
stronger. Using (6.25), we obtain

$C_{2}\lambda^{2}e^{4\lambda}\left(\left\|\widetilde{g}_{0}^{h}\right\|_{H^{1,h}\left(\Gamma_{h,T_{1}}\right)}^{2}+\left\|\widetilde{g}_{0}^{h}\right\|_{L_{2}^{h}\left(\Gamma_{h,T_{1}}\right)}^{2}\right)$
$+C_{2}e^{4\lambda}\left(\left\|\widetilde{w}^{h}\right\|_{L_{2}\left(\Theta_{h,T_{1}}\right)}^{2}+\left\|\widetilde{w}^{h}\right\|_{L_{2}\left(\Theta_{h}\right)}^{2}\right)+C_{2}\lambda^{2}e^{-2\lambda}\left\|\widetilde{w}^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{H^{1,h}\left(\Omega_{h}\right)}^{2}$
$\geq\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{t_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\left(\widetilde{w}_{z}^{h}\right)^{2}+\left(\widetilde{w}_{t}^{h}\right)^{2}+\left(\widetilde{w}^{h}\right)^{2}\right)\left(x_{i},y_{j},z,t\right)dzdt$
(6.26)
$+\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{t_{1}}\mathop{\displaystyle\int}\limits_{0}^{1}\left(\left(\partial_{z}\widetilde{\tau}^{h}\right)^{2}+\left(\widetilde{\tau}^{h}\right)^{2}\right)\left(x_{i},y_{j},z\right)dzdt$
$+\mathop{\displaystyle\sum}\limits_{i,j=1}^{N}h^{2}\mathop{\displaystyle\int}\limits_{0}^{1}\left[\left(\widetilde{w}_{z}^{h}\right)^{2}+\lambda^{2}\left(\widetilde{w}^{h}\right)^{2}\right]\left(x_{i},y_{j},z,0\right)e^{-2\lambda
z}dz,\text{ }\forall\lambda\geq\lambda_{1}.$

Choose
$\delta_{0}=\delta_{0}\left(M,n_{0},n_{00},h_{0},X,\alpha\right)\in\left(0,1\right)$
such that $\lambda_{1}=\ln\left(\delta_{0}^{-1/3}\right).$ For every
$\delta\in\left(0,\delta_{0}\right)$ we choose
$\lambda=\lambda\left(\delta\right)>\lambda_{1}$ such that

$e^{4\lambda}\delta^{2}=e^{-2\lambda}.$ (6.27)

Hence,
$\lambda=\lambda\left(\delta\right)=\frac{1}{3}\ln\left(\delta^{-1}\right).$
It follows from the third line of (6.2) and the first line of (6.9) that
$\left\|\widetilde{w}^{h}\left(\mathbf{x}^{h},T_{1}\right)\right\|_{H^{1,h}\left(\Omega_{h}\right)}^{2}\leq
C_{2}$. Hence, using (6.10), (6.26) and (6.27), we obtain

$\left\|\widetilde{w}^{h}\right\|_{H^{1,h}\left(Q_{h,1/\alpha}\right)},\left\|\widetilde{\tau}\right\|_{H^{1,h}\left(\Omega_{h}\right)}\leq
C_{2}\delta^{1/3}\ln\left(\delta^{-1}\right).$ (6.28)

Next, by the fourth line of (6.2) and (6.9)

$\left|\widetilde{n}^{h}\right|=\left|\frac{(n_{1}^{h})^{2}-(n_{2}^{h})^{2}}{n_{1}^{h}+n_{2}^{h}}\right|\leq\frac{1}{2}\left|(n_{1}^{h})^{2}-(n_{2}^{h})^{2}\right|=\frac{1}{2}\left||\nabla\tau_{1}^{h}|^{2}-|\nabla\tau_{1}^{h}|^{2}\right|$
$\leq\frac{1}{2}\left|\nabla^{h}\widetilde{\tau}\cdot(\nabla\tau_{1}^{h}+\nabla\tau_{2}^{h})\right|\leq
n_{0}\left|\nabla^{h}\widetilde{\tau}\right|.$

Hence, by (6.28)

$\left\|\widetilde{n}^{h}\right\|_{L_{2}^{h}\left(\Omega_{h}\right)}\leq
C_{2}\delta^{1/3}\ln\left(\delta^{-1}\right).$ (6.29)

Estimates (6.28) and (6.29) imply the target estimates (6.11). $\ \square$

Acknowledgment

The work of V.G. Romanov was supported by a grant from the Siberian Branch of
the Russian Academy of Science, project number FWNF-2022-0009.

## References

  * [1] L. Baudouin, M. de Buhan and S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation, _SIAM J. Nummer. Anal._ , 55, 1578–1613, 2017.
  * [2] L. Baudouin, M. de Buhan, S. Ervedoza, and A. Osses, Carleman-based reconstruction algorithm for the waves, _SIAM Journal on Numerical Analysis,_ 59, 998–1039, 2021.
  * [3] L. Beilina and M.V. Klibanov, _Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems_ , Springer, New York, 2012.
  * [4] M. Bellassoued and M. Yamamoto, _Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems_ , Springer, Japan, 2017\.
  * [5] M. Born and E. Wolf, _Principles of Optics_ , Cambridge University Press, 7th edition, 1999.
  * [6] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, _Soviet Math. Doklady_ , 17, 244–247, 1981.
  * [7] V. A. Khoa, M. V. Klibanov, and L. H. Nguyen, Convexification for a 3D inverse scattering problem with the moving point source _,_ _SIAM J. Imaging Sci_., 13, 871–904, 2020.
  * [8] V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, A. J. Sullivan, Lam Nguyen and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, _Inverse Problems_ , 36, 085007, 2020.
  * [9] M. V. Klibanov and O. V. Ioussoupova, Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, _SIAM J. Math. Anal_., 26, 147–179, 1995.
  * [10] M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem _,_ _SIAM J. Math. Anal.,_ 28, 1371–1388, 1997.
  * [11] M.V. Klibanov and A. Timonov,  _Carleman Estimates for Coefficient Inverse Problems and Numerical Applications_ , VSP, Utrecht, 2004.
  * [12] M.V. Klibanov and V.G. Romanov, Reconstruction procedures for two inverse scattering problem without the phase information, _SIAM J. Appl. Math_., 76, 178-196, 2016.
  * [13] M.V. Klibanov, J. Li and W. Zhang, Convexification for inversion of a time dependent wave front in a heterogeneous medium, _SIAM J. Appl. Math.,_ 79, 1722-1747, 2019.
  * [14] M.V. Klibanov, Travel time tomography with formally determined incomplete data in 3D, _Inverse Problems and Imaging_ , 13,1367-1393, 2019.
  * [15] M.V. Klibanov, A.E. Kolesov and D.-L. Nguyen, Convexification method for an inverse scattering problem and its performance for experimental backscatter data for buried targets, _SIAM J. Imaging Sciences_ , 12, 576-603, 2019.
  * [16] M. V. Klibanov, T. T. Le, Loc H. Nguyen, A. Sullivan and Lam H. Nguyen, Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data, _Inverse Problems and Imaging_ , published online, DOI:10.3934/ipi.2021068, 2021.
  * [17] M. V. Klibanov and J. Li, _Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data_ , De Gruyter, 2021 _._
  * [18] O.A. Ladyzhenskaya, _Boundary Value Problems of Mathematical Physics_ , Springer, 1985.
  * [19] T. T. Le, M.V. Klibanov, Loc H. Nguyen, A. Sullivan and Lam Nguyen, Carleman contraction mapping for a 1D inverse scattering problem with experimental time-dependent data, _Inverse Problems_ , 38, 045002, 2022\.
  * [20] R.G. Novikov, The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, _J. Functional Analysis_ , 103, 409-463, 1992.
  * [21] R.G. Novikov, $\partial-$bar approach to approximate inverse scattering at fixed energy in three dimensions, _International Math. Research Peports_ , 6, 287-349, 2005.
  * [22] Rakesh and M. Salo, The fixed angle scattering problem and wave equation inverse problems with two measurements, _Inverse Problems_ , 36, 035005, 2020.
  * [23] Rakesh and M. Salo, Fixed angle inverse scattering for almost symmetric or controlled perturbations, _SIAM J. Math. Analysis_ , 52, 5467-5499, 2020.
  * [24] V.G. Romanov, _Inverse Problems of Mathematical Physics_ , VNU Press, 1986.
  * [25] V.G. Romanov, _Investigation Methods for Inverse Problems_ , VSP, Utrecht, 2002.
  * [26] V.G. Romanov, Inverse problems for differential equations with memory, _Eur. J. Math. Comput. Appl._ , 2 pp. 51–80, 2014.
  * [27] V.G. Romanov, Phaseless inverse problems that use wave interference, _Siberian Math. J.,_ 59, 494-504, 2018.
  * [28] V.G. Romanov and M.V. Klibanov, Can a single PDE govern well the propagation of the electric wave field in a heterogeneous medium in 3D?_arxiv_ : 2102.02271, _Journal of Inverse and Ill-Posed Problems_ , 2022, DOI: 10.1515/jiip-2021-0085.
  * [29] A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov and A.G. Yagola, _Numerical Methods for the Solution of Ill-Posed Problems_ , Kluwer, London, 1995.
  * [30] M. Yamamoto, Carleman estimates for parabolic equations _._ Topical Review, _Inverse Problems_ , 25, 123013, 2009.