# How does the optimal constant of the Carleman estimate depend on the size of the observation boundary ?

Hamza Ammar, Mahjoub Moncef, Néjib Zemzemi

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Hamza Ammar, Mahjoub Moncef, Néjib Zemzemi. How does the optimal constant of the Carleman estimate depend on the size of the observation boundary?. Inverse Problems, 2025, 10.1088/1361-6420/ae16cd. hal-05332086

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# How does the optimal constant of the Carleman estimate depend on the size of the observation boundary ?

Hamza Ammar<fnref n="1" />, Moncef Mahjoub<fnref n="1" /> ‡ and Néjib Zemzemi<fnref n="2" />

E-mail: hamza.ammar@enit.utm.tn, moncef.mahjoub@enit.utm.tn, nejib.zemzemi@inria.fr

March 2025

**Abstract.** In this paper, we would like to elucidate the question of the Carleman estimate constants dependency on the size of the observation boundary. This allows to quantify the effect of measurements uncertainty with respect to the size of the observation boundary in many inverse and control problems. We take an example of an inverse source problem for a parabolic equation and explicitly calculate Lipschitz stability constants that appears in the $L^2$-estimate of the source function. We deliberately construct a class of space dependent weight functions that depend on the measurement boundary size. Then we identify the optimal weight function that allows to minimize the stability constant. Using our approach, we found that when the measurement boundary covers 80 % or more of the domain boundary, we can explicitly provide the formula of the optimal constant. When the measurement boundary is less than 80 %, we are not able to find the explicit formula of the optimal expression, but we are able to numerically approximate the optimal constant.

This paper presents meticulous calculations that require a rigorous and careful reading to fully comprehend the technical intricacies of the analysis.

**2020 Mathematics Subject Classification:** 35Q93, 35B35, 35R30, 90C31.

**Keywords:** Carleman type estimate, A priori estimate, Explicit estimate, Geometric effect on constants, Weight functions. Submitted to: *Inverse Problems*

‡ Corresponding author

<footnote n="1">¹ University of Tunis El Manar, National Engineers School of Tunis, B.P. 37, 1002 Tunis, Tunisia</footnote>

<footnote n="2">² INRIA Centre de L'université de Bordeaux, Carmen project team. 200 Avenue de la vielle Tour 33405 Talence Cedex, France</footnote>

<footnote n="2">² IHU-LIRYC, avenue du Haut Lévèque 336000 Pessac, France</footnote>
Introduction

Carleman estimates are weighted inequalities introduced by T.Carleman [13] in 1939 to show the uniqueness in Cauchy problems to which Holmgren's theorem are not applicable. This type of weighed energy estimates has become one of the major tools to study strong unique continuation property (SUCP) for many classes of equations, to solve and to prove the stability of inverse and control problems in both deterministic and stochastic partial differential equations. For example, in [1, 2, 3, 4, 5, 6, 7, 9, 10, 15, 22] authors used Carleman estimates to prove Hölder, Lipschitz and logarithmic stability estimates for different types of inverse problems with respect to sub-boundary or sub-domain measurements of the state variable(s). We notice that these stability constants depend explicitly on Carleman estimate ones.

In [18], O. Yu Emanuilov proved null-controllability of linear parabolic equations from an a priori estimate of Carleman type. The control variable is on the right-hand side of the parabolic equation and distributed over an arbitrary subdomain $\omega$ of the domain $\Omega$. He established a Lipschitz estimate of the control $u$ and the state variable $y$ by means of the source function and the initial datum. Also in [19], authors consider a linear heat equation in a bounded domain $\Omega$ of $\mathbb{R}^n$ with Dirichlet boundary conditions and obtained an explicit bound of the cost of approximate controllability, i.e., of the minimal norm of a control $v$ acting through an open subset $\omega$ of $\Omega$ needed to control the system approximately. They have proved that for all $\epsilon > 0$ and $y_1 \in H^2(\Omega) \cap H_0^1(\Omega)$:

$$ C(0, y_1, \epsilon) \le \exp \left( C \left[ \frac{1}{T} + 1 + T \|a\|_{\infty} + \|a\|_{\infty}^2 + \frac{\|a\|_{\infty} \|y_1\|_{L^2} + \|\Delta y_1\|_{L^2}}{\epsilon} \right] \|y_1\|_{L^2} \right), \quad (0.1) $$

where $C(0, y_1, \epsilon)$ is the cost of achieving from zero to the final state $y_1$ at $t = T > 0$ and $C$ is a geometric constant obtained from a Carleman estimate that depends on $\omega$ and $\Omega$. Here the state variable $y$ satisfies the following equation

$$ \partial_t y - \Delta y + ay = v \mathbf{1}_{\omega}. \quad (0.2) $$

From this estimate we can see directly the influence of changing the final time $T$ on the upper-bound of $C(0, y_1, \epsilon)$.

In order to figure out the geometric dependency of all these stability constants and the cost of controllability upper-bounds, we must relate Carleman estimate constants explicitly to both geometric (sub-domain or sub-boundary regions of measurements) and time parameters.

We refer to [8, 11, 12, 24] to see how we manage to obtain explicit bounds of costs of (approximate) controllability for other systems, and especially to see their dependence on the final time $T$.

In [16], authors have analyzed the cost of observability in small times of the one-dimensional heat equation. They consider the following heat equation, set in the
bounded interval $\Omega = ]-L, L[$ and among some time interval $]0, T[$:

$$ egin{cases} 
\partial_t z - \partial_x^2 z = 0, & \text{in } \Omega \times ]0, T[, \\
z(-L, \cdot) = z(L, \cdot) = 0, & \text{in } ]0, T[, \\
z(\cdot, 0) = z_0, & \text{in } \Omega, 
\end{cases} 
\quad (0.3) $$

with initial datum $z_0 \in H_0^1(\Omega)$. They have studied the cost of observability in small times $T$ of the problem (0.3) observed from both sides $x = -L$ and $x = L$, which is the best constant $C_0(L, T)$ that satisfies the following estimate:

$$ \|z(\cdot, T)\|_{L^2(\Omega)} \le C_0(L, T) \left( \|\partial_x z(-L, \cdot)\|_{L^2(]0, T[)} + \|\partial_x z(L, \cdot)\|_{L^2(]0, T[)} \right). \quad (0.4) $$

They found that:

$$ \limsup_{T \to 0} T \log(C_0(L, T)) \le K_0^+ L^2, \quad (0.5) $$

where $K_0 \approx 0.6966$ and the notation "+" in the last estimate means that "any strictly larger number is convenient". But the authors do not claim that this bound is sharp, and this remains, to their knowledge, an open problem. They have also proved that in higher dimensions, if we substitute $\Omega$ by any ball $B_L$ of radius $L$ of $\mathbb{R}^d$ ($d \ge 1$), any solution $z$ of the multi-dimensional heat equation verifies the following estimate:

$$ \left\| z(x, T) \exp \left( \frac{|x|^2}{4T} \right) \right\|_{L^2(\Omega)} \le C_T(L) \exp \left( K \frac{L^2}{T} \right) \| \partial_\nu z \|_{L^2(\partial \Omega \times ]0, T[)} . \quad (0.6) $$

Here, $|\cdot|$ denotes the euclidean norm in $\mathbb{R}^d$ and $C_T$ is a positive constant that depends only on $L$ and $T$ for small times $T$ and goes to zero when $T \to 0$.

A similar result is proved in [23], where they studied the short time observability constant of the heat equation from a ball $\omega$ of radius $r > 0$ of a bounded domain $M$ centred at $x_0 \in M$. The constant is of the form $\exp\left(\frac{\mathcal{R}(\omega)}{T}\right)$ where $\mathcal{R}(\omega)$ verifies the following inequalities:

$$ C_1 |\log(r)|^2 \le \mathcal{R}(\omega) \le C_2 (|\log(r)|^2 + 1). \quad (0.7) $$

Here $C_1$ and $C_2$ are positive constants independent of $r$. We refer to [17, 25] for more examples of observability estimates and the dependence of their bounds on parameters like $T$.

Recently, it is shown in [21] that some degenerate parabolic equations posed on a square domain $\Omega = ]0, 1]^2$ is boundary null controllable at time $T > 0$. Here the control $q \in L^2(0, T; L^2(\partial\Omega))$ is exerted on $\gamma = \{0\} \times \omega$, where $w \subset ]0, 1[$ in a nonempty open set. Furthermore, the control for this degenerate coupled parabolic equation satisfies the control cost $Ce^{C/T}$, where $C$ is a geometric constant independent of $T$.

To the best of our knowledge, this is the only work in the case of sub-boundary observations that studies the effect of changing the size of the observation boundary
on stability, cost of controllability or observability constants. In the literature, authors only focus on the final time dependency.

We take an example of one Carleman estimate that have often been used to prove several results of stability for inverse problems and controllability.

Let $z \in W_2^{2,1}(Q) := \{z(x, y, t); z, \partial_t z, \partial_x z, \partial_y z, \partial_x \partial_y z, \partial_x \partial_t z \in L_2(Q)\}$ be a solution to the following problem:

$$ egin{cases} 
\mathcal{L}z(x, y, t) := \partial_t z - \Delta z = g, & \text{in } Q, \\
\nabla z \cdot \nu + Iz = 0, & \text{on } \Sigma.
\end{cases} 
\tag{0.8} $$

Here $Q := \Omega \times [0, T]$ with $\Omega$ is a bounded open subspace of $\mathbb{R}^2$ with regular border, $\Gamma, T > 0$, $\Sigma := \Gamma \times [0, T]$ and $\Sigma_0 := \Gamma_0 \times [0, T]$, where $\Gamma_0$ is a relative open subspace of $\Gamma$ and

$$ I \in L^{\infty}(\Gamma); \ 0 \le I \text{ a.e on } \Gamma. \tag{0.9} $$

We will study the effect of changing the size of boundary measurements $\Gamma_0$ on constants of the following Carleman estimate which is similar to those given in [20] and [22]. Here, we substitute weight functions $\eta$ and $\bar{\eta}$ by $\eta_\beta$ and $\bar{\eta}_\beta$ as follows:

There exists $\hat{\lambda} > 0$, such that for all $\lambda \ge \hat{\lambda}$, we can choose $s_0(\lambda)$ such that for each $s \ge s_0(\lambda)$, any solution $z \in W_2^{2,1}(Q)$ to the problem (0.8) satisfies the following inequality:

$$ egin{aligned} & \int_Q \{(s\varphi)^{-1} (|\partial_t z|^2 + |\Delta z|^2) + (s\varphi)\lambda^2 |\nabla z|^2 + (s\varphi)^3 \lambda^4 |z|^2\} e^{-2s\eta} dxdt \\ & \le c_{\mathcal{L}}(\lambda) \int_Q |\mathcal{L}z|^2 (e^{-2s\eta_\beta} + e^{-2s\bar{\eta}_\beta}) dxdt \\ & + c_{\text{obs}}(\lambda) \int_{\Sigma_0} (|\partial_t z|^2 + (s\varphi) |\nabla z|^2 + (s\varphi)^3 |z|^2) e^{-2s\eta_\beta} dSdt, \end{aligned} \tag{0.10} $$

such that for all $\beta > 1$:

$$ egin{cases} 
\varphi(x, t) & := e^{\lambda \psi} \theta(t), \\
\eta_\beta(x, t) & := (\beta e^{\lambda \|\psi\|_\infty} - e^{\lambda \psi}) \theta(t), \\
\bar{\eta}_\beta(x, t) & := (\beta e^{\lambda \|\psi\|_\infty} - e^{-\lambda \psi}) \theta(t), \\
\theta(t) & := \frac{1}{t(T-t)}.
\end{cases} 
\tag{0.11} $$

Here, the weight function $\psi \in C^2(\bar{\Omega})$ satisfies the following properties (See [22]):

$$ \psi(x) > 0 \quad \forall x \in \Omega, \quad \psi|_{\partial\Omega \setminus \Gamma_0} = 0, \quad |\nabla\psi(x)| > 0 \quad \forall x \in \bar{\Omega}, \tag{0.12} $$

$$ \text{and} \quad \nabla\psi(x) \cdot \nu(x) < 0, \quad x \in \Gamma'_0 := \partial\Omega \setminus \Gamma_0. $$
To do so, in Section 2 we prove an optimized version of the previous estimate by explicitly calculate all these constants and elucidate their dependencies on problem parameters like T, β, I and ψ where the last one depends implicitly on Ω and Γ₀.

In subsection 1.1, we construct a class of weight functions $(\psi_{\alpha, \theta_M})_{\alpha, \theta_M}$ defined on the unit disc $\Omega$ where $0 < \alpha < \pi$ and $\theta_M \in I_\alpha$ such that $I_\alpha$ depends on $\alpha$. These functions verify all of them properties (0.12) where we choose $\Gamma_0(\alpha)$ any arc of $\partial\Omega$ that measures $2\alpha$.

In Section 3, for each $\alpha$ we determine the optimal $\theta_M$ denoted by $\hat{\theta}(\alpha)$ such that when we put $\psi = \hat{\psi}_\alpha := \psi_{\alpha, \hat{\theta}(\alpha)}$, we are able to minimize the optimal Carleman estimate constant $c_{obs}(\lambda)$ given in (1.2) (see Corollary 1.7 for the details).

Finally, in Section 4, we show how Lipschitz constants, that appear in the identification of the source function, depend on the size of the boundary measurement.

The main contributions of this work are:

(i) We were able to clarify the constants in the Carleman inequality as a function of the size of the geometry where the measurements are available.

(ii) We constructed explicit formulas for the weight functions that satisfy the required properties for the Carleman estimate.

(iii) We determined the exact ratio of the domain's geometry for adjusting the Carleman constants.

(iv) We analyzed the asymptotic behavior of the Carleman constants when the ratio is close to zero.

# 1. Main results

Using the same techniques as in [4, 14, 20, 22] with some modifications to prove the Carleman estimate of Lemma 2.4 in [22], we managed to prove an optimized version of estimate (0.10).

**Theorem 1.1.** For all $\beta > 1$, $\delta > 0$, there exist $\hat{\lambda}(T, I, \psi) > \underline{\lambda}(\psi)$ such that for all $\lambda \ge \hat{\lambda}$ and $s \ge s_\beta(\delta, \lambda) := \frac{e^{\lambda \|\psi\|_\infty} C_0(\beta, \delta, \lambda)}{\underline{\epsilon}(\lambda)}$ any solution $z \in W_2^{2,1}(Q)$ to the problem (0.8) satisfies the following inequality:

$$ \begin{aligned} & \int_Q \{(s\varphi)^{-1} (|\partial_t z|^2 + |\Delta z|^2) + (s\varphi)\lambda^2 |\nabla z|^2 + (s\varphi)^3 \lambda^4 |z|^2\} e^{-2s\eta\beta} dxdt \\ & \le c_{\mathcal{L}}(\delta, \lambda) \int_Q |\mathcal{L}z|^2 (e^{-2s\eta\beta} + e^{-2s\bar{\eta}\beta}) dxdt + c_{\text{obs}}(\lambda) \int_{\Sigma_0} ((s\varphi)^3 |z|^2 + (s\varphi) |\nabla z|^2) e^{-2s\eta\beta} d\Sigma, \end{aligned} \quad (1.1) $$
where

$$
\left\{
\begin{aligned}
c_{\mathcal{L}}(\delta, \lambda) &:= C(\lambda)(1 + \delta), \\
c_{\text{obs}}(\lambda) &:= 4C(\lambda)(1 + \epsilon_2(\lambda))\lambda^3 C_{1,\psi}^3, \\
C(\lambda) &:= \frac{\tilde{c}(\mu)(1 + \epsilon_1(\lambda))}{2\underline{\epsilon}(\lambda)(1 - e^{-\lambda}\|\psi\|_{\infty})}.
\end{aligned}
\right.
\qquad (1.2)
$$

such that

$$
\left\{
\begin{aligned}
\tilde{c}(\mu) &:= \min_{g_{\mu}(X)=f_{\mu}(X), X \in \mathcal{U}} f_{\mu}(x), \\
f_{\mu}(x_1, x_2, x_3) &:= 4(1+x_1)(1+x_2) + \frac{x_3}{\mu^2} + 4 + \frac{1}{\mu^2}, \\
g_{\mu}(x_1, x_2, x_3) &:= (1+x_1)\left(1+\frac{1}{x_2}\right) + \frac{1}{x_1} + \frac{1}{x_3\mu^2} + 1 + \frac{1}{\mu^2} + \frac{1}{\mu^4}.
\end{aligned}
\right.
\qquad (1.3)
$$

$$
\left\{
\begin{aligned}
\epsilon_1(\lambda) &:= 3\mu e^{-\lambda} \|\psi\|_{\infty} \left(3 + \sqrt{5} + 2\sqrt{2}\mu T \lambda^2 e^{-\lambda} \|\psi\|_{\infty}\right), \\
\epsilon_2(\lambda) &:= e^{-\lambda} \|\psi\|_{\infty} \left(1 + \frac{e^{-2\lambda} \|\psi\|_{\infty}}{16}\right), \\
\underline{\epsilon}(\lambda) &:= 1 - \frac{\lambda(\psi)}{\lambda}, \\
C_0(\beta, \delta, \lambda) &:= \frac{T}{2} c_{0,1}(\beta, \lambda) + \left(\frac{T}{2}\right)^2 \left(\frac{1}{\delta} + \lambda + 2\right), \\
c_{0,1}(\beta, \lambda) &:= \frac{4}{\lambda^2 \mu^2} \left(\beta e^{\lambda \|\psi\|_{\infty}} \left(1 + \frac{1}{\mu^2}\right) + 1\right).
\end{aligned}
\right.
\qquad (1.5)
$$

and

$$
\left\{
\begin{aligned}
C_{1,\psi} &:= \max_{x \in \Gamma_0} (|\nabla \psi(x)|), \\
C_{2,\psi} &:= 2 \left( \max \left( \|\partial_x^2 \psi\|_{\infty}, \|\partial_y^2 \psi\|_{\infty} \right) + \|\partial_{x,y} \psi\|_{\infty} \right) + \|\partial_x^2 \psi\|_{\infty} + \|\partial_y^2 \psi\|_{\infty}, \\
\mu = \mu(\psi) &:= \min_{x \in \bar{\Omega}} (|\nabla \psi(x)|) > 0, \\
\underline{\lambda}(\psi) &:= \frac{C_{2,\psi}}{\mu(\psi)^2}.
\end{aligned}
\right.
\qquad (1.6)
$$

Here $\|\cdot\|_{\infty}$ is the infinity norm of a real-valued function over $\bar{\Omega}$.

The proof of this theorem is provided in Section 2.
**Remark 1.2.** As we can see, $\lambda(\psi)$, $s_0(\lambda)$ and $C_{1,\psi}$ depend on our choice of the $\psi$ function and this one itself depends implicitly on $\Gamma_0$. To study the effect of changing the boundary measurement size $\Gamma_0$ on these constants, we will construct a class of weight functions verifying properties (0.12), then we choose the optimal one that minimizes $C_{1,\psi}$.

### 1.1. Construction of a class of weight functions $\psi(=\psi_{\alpha,\theta_M})$

Let $\Omega$ be the unit open disc of the plan $\mathcal{P}$ of center $O$, which is provided with a direct orthonormal reference $\mathcal{R} = (O, \vec{i}, \vec{j})$. We define points the $A$, $B$ and $A'$ by $O\vec{A} = \vec{i}$, $O\vec{B} = \vec{j}$ and $O\vec{A}' = -\vec{i}$.

Let $\Gamma_0(\alpha)$ a part of $\partial\Omega$, defined by the arc $\overarc{CC'}$ that measures $2\alpha$, which is symmetric with respect to the axes $(O, \vec{i})$. Here $0 < \alpha < \pi$. We construct $O'$ on $(O, \vec{i})$ such that $1 = \overline{A'O} < \overline{O'O}$.

For all $\alpha \in ]0, \pi[$, we define $\theta_i(\alpha)$ such that:

(i) If $0 < \alpha < \frac{\pi}{2}$, we construct the point $F$ on axes $(O, \vec{i})$ such that $FOC$ is a right triangle at $C$. Then, we put $\theta_i(\alpha) = \langle \vec{i}, F\vec{C} \rangle \pmod{2\pi} = \frac{\pi}{2} - \alpha \pmod{2\pi}$ and we impose on $O'$ the following condition:

$$ \overline{A'O} < \overline{O'O} < \overline{FO} = \frac{1}{\sin \theta_i}. \qquad (1.7) $$

(ii) If not, we put $\theta_i(\alpha) = 0$.

Finally, $\theta_i(\alpha) = \max(\frac{\pi}{2} - \alpha, 0)$.

We put points $D$ and $D'$ on the circle $C := \partial\Omega$ such that lines $(O'D)$ and $(O'D')$ are tangents to $C$ where the arc $\overarc{DD'}$ measure is positive.

We find that the angle $\langle O\vec{D}, O\vec{D}' \rangle = 2\beta \pmod{2\pi}$ where $\beta < \alpha$. So $\overarc{DD'}$ is situated in $\Gamma_0$.

We provide the plane $\mathcal{P}$ with another direct orthonormal reference $\mathcal{R}' = (O', \vec{i}, \vec{j})$ and for each point $P$ in the right half-plane $\mathcal{P}^+$, we associate its polar coordinates $\varphi(P) = (r, \theta)$ in $\mathcal{R}'$ where $\varphi$ is the diffeomorphism between $\mathcal{P}^+$ and $\mathbb{R}_+^* \times ]-\frac{\pi}{2}, \frac{\pi}{2}]$.

We put $\theta_M = \langle \vec{i}, O'\vec{D} \rangle \pmod{2\pi}$ and $\theta_\alpha(\theta_M) = \langle \vec{i}, O'\vec{C} \rangle \pmod{2\pi}$, we have:

$$ \theta_M \in I_\alpha := ]\theta_i(\alpha), \frac{\pi}{2}[ \quad \text{and} \quad \frac{\pi - \alpha}{2} < \theta_\alpha(\theta_M) < \theta_M. \qquad (1.8) $$

In the figure below, we represented:

(i) Points $A, A', B$, and $O$ on the circle, colored in "blue".

(ii) The geometric angle $\theta := \overarc{NO'A'}$ and the intersection points $N$ and $Q$ between the line $(O'P)$ and the circle, colored in "black".

(iii) The arc $\Gamma_0$, points $C, C', F$, and the geometric angles $\alpha$ and $\theta_i$ on the circle, colored in "red".

(iv) Points $O', D, D'$, and $E$, along with the geometric angles $\theta_M$ and $\theta_\alpha$ on the circle, colored in "green". Here $E$ is the intersection point between the circle $C$ and the line $(O'C)$.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** The unit disc $\bar{\Omega}$ and the constructed points. The observation domain $\Gamma_0$ is the red arc.</figcaption></figure>

Now, we will introduce two fundamental parameters in the construction of the weight functions $\psi_{\alpha, \theta_M}$. Let the set

$$ \mathcal{D}_{\alpha} := \{ (\theta, \theta_M); -\theta_M \le \theta \le \theta_M, \theta_M \in I_{\alpha} = ]\theta_i(\alpha), \frac{\pi}{2}] \}. \quad (1.9) $$

By defining $r_m := O'N$ and $r_M := O'Q$, we have the following lemma:

**Lemma 1.3.** For all $(\theta, \theta_M) \in \mathcal{D}_\alpha$:

$$ \left\{ \begin{array}{l} r_m(\theta, \theta_M) = \frac{\cos \theta - \sqrt{\cos^2 \theta - \cos^2 \theta_M}}{\sin \theta_M}, \\ \\ r_M(\theta, \theta_M) = \frac{\cos \theta + \sqrt{\cos^2 \theta - \cos^2 \theta_M}}{\sin \theta_M}. \end{array} \right. \quad (1.10) $$

*Proof.* Let $(x, y)$ (respectively $(x', y')$) be the coordinates of any point $P \in \Omega$ relative to the direct orthonormal reference $\mathcal{R} = (O, \overrightarrow{OA}, \overrightarrow{OB})$ (respectively to the direct orthonormal reference $\mathcal{R}' = (O', \overrightarrow{OA}, \overrightarrow{OB})$), then we have:

$$ x' = x + \frac{1}{\sin \theta_M}, \quad y' = y. \quad (1.11) $$

Now, if we want to determine the coordinates of $P = N, Q$, it is sufficient to look for any point $P$ such that $OP = 1$ and $\tan(\theta) = \frac{y}{x'}$, which is equivalent to solve:

$$ \left\{ \begin{array}{l} x^2 + y^2 = 1, \\ \tan(\theta) x' = y. \end{array} \right. \quad (1.12) $$
Rewriting in terms of $x'$:

$$ \left(x' - \frac{1}{\sin \theta_M}\right)^2 + \tan^2(\theta) x'^2 = 1. \qquad (1.13) $$

Solving for $x'$, we obtain:

$$ x'_{\pm} = \frac{\cos \theta) \pm \sqrt{\cos^2 \theta - \cos^2 \theta_M}}{\sin \theta_M} \cos \theta. \qquad (1.14) $$

Thus,

$$ O'P = \sqrt{x'_{\pm}^2 + y^2} = \frac{\cos \theta \pm \sqrt{\cos^2 \theta - \cos^2 \theta_M}}{\sin \theta_M}. \qquad (1.15) $$

□

Now, we will use this new parametrization of our study domain. We have:

$$ \left\{ \begin{array}{l} \Omega = \{ P(r, \theta); r_m(\theta, \theta_M) < r < r_M(\theta, \theta_M) \text{ and } -\theta_M < \theta < \theta_M \}, \\ \overline{DD'} = \{ P(r, \theta); r = r_m(\theta, \theta_M) \text{ and } -\theta_M \le \theta \le \theta_M \} \subset \Gamma_0, \\ \partial \Omega \setminus \overline{DD'} = \{ P(r, \theta); r = r_M(\theta, \theta_M) \text{ and } -\theta_M < \theta < \theta_M \}, \\ \Gamma_0 = \overline{CC'}, \\ \partial \Omega \setminus \Gamma_0 = \{ P(r, \theta); r = r_M(\theta, \theta_M) \text{ and } -\theta_\alpha < \theta < \theta_\alpha \}. \end{array} \right. \qquad (1.16) $$

We define the function $\psi_{\alpha, \theta_M} \in C^2(\bar{\Omega})$ by

$$ \forall (x, y) \in \bar{\Omega}; \quad \psi_{\alpha, \theta_M}(x, y) = \Psi_{\alpha, \theta_M} \circ \varphi \left( x + \frac{1}{\sin \theta_M}, y \right), \qquad (1.17) $$

where

$$ \Psi_{\alpha, \theta_M}(r, \theta) = f_{\alpha, \theta_M}(\theta) - r, \qquad (1.18) $$

and $f_{\alpha, \theta_M} \in C^2([-\theta_M, \theta_M])$ is an even function defined on $[0, \theta_M]$ with

$$ f_{\alpha, \theta_M}(\theta) := \begin{cases} r_M(\theta, \theta_M), & \text{if } 0 \le \theta \le \theta_\alpha(\theta_M), \\ \sum_{k=0}^{2} \frac{\partial^k r_M}{\partial \theta^k}(\theta_\alpha, \theta_M) \frac{(\sin \theta - \sin \theta_\alpha)^k}{k! \cos^k \theta_\alpha}, & \text{otherwise.} \end{cases} \qquad (1.19) $$

Here, we recall that $\varphi$ is the diffeomorphism between $\mathcal{P}^+$ and $\mathbb{R}_+^* \times [-\frac{\pi}{2}, \frac{\pi}{2}]$.
**Remark 1.4.** This choice of the weight function $(r, \theta) \mapsto \Psi_{\alpha, \theta_M}(r, \theta)$ is made to ensure that $\psi_{\alpha, \theta_M}|_{\partial\Omega \setminus \Gamma_0} = 0$. Indeed, the boundary portion $\partial\Omega \setminus \Gamma_0$ is a subset of $\partial\Omega \setminus DD'$, which is defined by $\{r = r_M\}$. Thus, the simplest way to define the weight function $\Psi_{\alpha, \theta_M}$ is to set

$$ \Psi_{\alpha, \theta_M} = r_M - r $$

in the subset

$$ \{(r, \theta) | r_m \le r \le r_M, -\theta_\alpha \le \theta \le \theta_\alpha\}. $$

For the remaining part of $\Omega$, we can choose any $C^2$ extension.

Now, we prove that this weight function $\psi_{\alpha, \theta_M}$ verifies all properties in (0.12). Using the chain rule in (1.17), we obtain for all $(x, y) \in \bar{\Omega}$:

$$ |\nabla \psi_{\alpha, \theta_M}(x, y)|^2 = \left( \frac{\Psi_{\alpha, \theta_M}}{\partial r}(r, \theta) \right)^2 + \left( \frac{1}{r} \frac{\Psi_{\alpha, \theta_M}}{\partial \theta}(r, \theta) \right)^2, \quad (1.20) $$

where $(r, \theta) := \varphi \left( x + \frac{1}{\sin \theta_M}, y \right)$.

Hence, by the definition (1.18) of $\Psi_{\alpha, \theta_M}$ we obtain:

$$ |\nabla \psi_{\alpha, \theta_M}(P)|^2 = 1 + \left( \frac{f'_{\alpha, \theta_M}(\theta)}{r} \right)^2, \quad \forall P(r, \theta) \in \bar{\Omega}, \quad (1.21) $$

and since $f'_{\alpha, \theta_M}(0) = 0$, from (1.21) we have:

$$ |\nabla \psi_{\alpha, \theta_M}(P(r, \theta))| \ge 1 = |\nabla \psi_{\alpha, \theta_M}(P(r, 0))|, \quad \forall P(r, \theta) \in \bar{\Omega}. \quad (1.22) $$

We put

$$ S := \mathrm{An}(O', 2\theta_\alpha) \cap \bar{\Omega} = \{ P(r, \theta); r_m(\theta, \theta_M) \le r \le r_M(\theta, \theta_M) \text{ and } -\theta_\alpha \le \theta \le \theta_\alpha \}, \quad (1.23) $$

where $\mathrm{An}(O', 2\theta_\alpha)$ is the angular sector centered at $O'$ of angle $2\theta_\alpha$ around $\overrightarrow{O'\bar{O}}$.

Since all points $P(r, \theta) \in S$ verify $|\theta| \le \theta_\alpha$, from (1.16) and 1.18 we have:

$$ \Psi_{\alpha, \theta_M}(r, \theta) = r_M(\theta, \theta_M) - r \ge 0, \quad \forall P(r, \theta) \in S. \quad (1.24) $$

Furthermore, using the expression of the $C^2$ extension of $r_M(., \theta_M)$ in (1.19), we conclude that:

$$ \Psi_{\alpha, \theta_M}(r, \theta) \ge 0, \quad \forall P(r, \theta) \in \bar{\Omega}. \quad (1.25) $$

In fact, to prove the previous inequality, from (1.24) and that $\psi$ is even w.r.t its second variable, it is sufficient to prove that $\psi \ge 0$ on the sub-domain $S'_+$ of $\bar{\Omega}$ where

$$ S'_+ := (\bar{\Omega} \setminus S) \cap (\mathbb{R} \times \mathbb{R}_+) = \{ P(r, \theta); r_m(\theta, \theta_M) \le r \le r_M(\theta, \theta_M) \text{ and } \theta_\alpha \le \theta \le \theta_M \}. \quad (1.26) $$

For all $P(r, \theta) \in S'_+$, we have

$$ \Psi_{\alpha, \theta_M}(r, \theta) = f_{\alpha, \theta_M}(\theta) - r \ge f_{\alpha, \theta_M}(\theta) - r_M(\theta, \theta_M) := \Psi_{\alpha, \theta_M}(r_M(\theta, \theta_M), \theta) \quad (1.27) $$
Hence, if $\psi_{|\overline{CE}} \ge 0$, then $\psi_{|S'_+} \ge 0$, which reduces the study domain to the arc $\overline{CE}$.
Using the change of variables $x_\alpha = \sin(\theta_\alpha)$, $x = \sin(\theta)$ and $x_M = \sin(\theta_M)$ in (1.19), for all $\theta \in [\theta_\alpha, \theta_M]$ we obtain:

$$f_{\alpha, \theta_M}(\theta) = F_{\alpha, x_M}(x) := R_M(x_\alpha) + R'_M(x_\alpha)(x-x_\alpha) + \frac{R''_M(x_\alpha)}{2}(x-x_\alpha)^2 - \frac{R'_M(x_\alpha)}{(1-x_\alpha^2)^{\frac{3}{2}}}(x-x_\alpha), \quad (1.28)$$

$$\text{where } r_M(\theta, \theta_M) = R_M(x) := \frac{\sqrt{x_M^2 - x^2} + \sqrt{1 - x^2}}{x_M}.$$

Now, if we put $\Phi_{\alpha,x_M}(x) := F_{\alpha,x_M}(x) - R_M(x)$, we find that

$$\Psi_{\alpha,x_M}(r_M(\theta, \theta_M), \theta) = \Phi_{\alpha,x_M}(x), \quad (1.29)$$

which implies that

$$\Phi_{\alpha,x_M} : \begin{array}{rcl} [x_\alpha, x_M] & \to & \mathbb{R} \\ x & \mapsto & \Phi_{\alpha,x_M}(x) \end{array} \quad (1.30)$$

is a parametrization of $\psi$ on $\overline{CE}$.

For all $x \in [x_\alpha, x_M]$ we have:

$$\begin{align} \Phi'_{\alpha,x_M}(x) &:= R'_M(x_\alpha) + R''_M(x_\alpha)(x-x_\alpha) - R'_M(x_\alpha) - \frac{R'_M(x_\alpha)}{(1-x_\alpha^2)^{\frac{3}{2}}} \nonumber \\ &= -\left(\int_{x_\alpha}^x \frac{(x-t)^2}{2} R_M^{(3)}(t) \, dt + \frac{R'_M(x_\alpha)}{(1-x_\alpha^2)^{\frac{3}{2}}}\right) \tag{1.31} \\ &> 0, \nonumber \end{align}$$

because

$$R'_M(x_\alpha) = - \frac{\frac{x_\alpha}{\sqrt{x_M^2 - x_\alpha^2}} + \frac{x_\alpha}{\sqrt{1-x_\alpha^2}}}{x_M} < 0,$$

$$R_M^{(3)}(x) = - \frac{\frac{3x^3}{(x_M^2 - x^2)^{5/2}} + \frac{3x^3}{(1-x^2)^{5/2}} + \frac{3x}{(x_M^2 - x^2)^{3/2}} + \frac{3x}{(1-x^2)^{3/2}}}{x_M} < 0.$$

Therefore, $\Phi_{\alpha,x_M}$ increases strictly on $[x_\alpha, x_M]$. In particular, we obtain for all $x \in [x_\alpha, x_M]$:

$$\Phi_{\alpha,x_M}(x) \ge \Phi_{\alpha,x_M}(x_\alpha) = 0, \quad (1.32)$$

Hence, $\psi$ is positive on $\overline{CE}$ which completes the proof of (1.25).

From (1.24), we have:

$$\forall \theta \in [-\theta_{\alpha}, \theta_{\alpha}], \quad \Psi_{\alpha, \theta_M}(r_M(\theta, \theta_M), \theta) = r_M(\theta, \theta_M) - r_M(\theta, \theta_M) = 0, \quad (1.33)$$

which implies from the expression of $\partial\Omega\setminus\Gamma_0$ in (1.16) that

$$\psi_{\alpha, \theta_M} |_{\partial \Omega \setminus \Gamma_0} = 0. \quad (1.34)$$

Hence, from (1.24) and (1.34 we conclude that:

$$\nabla \psi_{\alpha, \theta_M} \cdot \nu < 0, \quad \text{on } \Gamma'_0 := \partial \Omega \setminus \Gamma_0 \quad (1.35)$$
**Remark 1.5.** If we want $\psi_{\alpha, \theta_M}$ to be of regularity $C^n(\bar{\Omega})$, we add derivatives of superior order to the sum in (1.19) of $f_{\alpha, \theta_M}$ to be like: $\sum_{k=0}^n \frac{\partial^k r_M}{\partial \theta^k}(\theta_\alpha, \theta_M) \frac{(\sin \theta - \sin \theta_\alpha)^k}{k! \cos^k \theta_\alpha}$.

We put $\mathcal{A}$ the domain of admissible geometric parameters for this construction of $\psi_{\alpha, \theta_M}$ defined as:

$$ \mathcal{A} := \{ (\alpha, \theta_M); \alpha \in ]0, \pi[ \text{ and } \theta_M \in I_\alpha \}, \quad (1.36) $$

where $I_\alpha$ is defined in (1.8).

For all $(\alpha, \theta_M) \in \mathcal{A}$, we put:

$$ C_1(\alpha, \theta_M) := C_{1, \psi_{\alpha, \theta_M}} = \max_{P(r, \theta) \in \Gamma_0^+} |\nabla \psi_{\alpha, \theta_M}(P)|, $$

where $C_{1, \psi_{\alpha, \theta_M}}$ is $C_{1, \psi}$ defined in (1.6), for $\psi = \psi_{\alpha, \theta_M}$ and $\Gamma_0^+$ is the part of $\Gamma_0$ contained in the positive half-plan (above the x-axis).

Our goal is to find for each $\alpha$ the smallest constant among all $C_1(\alpha, \theta_M)$ where $\theta_M \in I_\alpha$ and we denote it $\hat{C}_1(\alpha)$. So:

$$ \hat{C}_1(\alpha) := \inf_{\theta_M \in I_\alpha} C_1(\alpha, \theta_M). \quad (1.37) $$

From now on, we denote by $C_\alpha$ the class of weight functions $(\psi_{\alpha, \theta_M})_{\theta_i(\alpha) < \theta_M < \frac{\pi}{2}}$.

Following the construction of the weight functions class $C_\alpha$, we announce the theorem of the principle result of this work.

**Theorem 1.6.** For all $\alpha \in ]0, \pi[$, there exists a unique $\hat{\theta} \in \bar{I}_\alpha$ such that:

$$ \hat{C}_1(\alpha) = C_{1, \hat{\psi}_\alpha}, \quad (1.38) $$

where $\hat{\psi}_\alpha$ is the point-wise limit of the class of functions $C_\alpha$ when $\theta_M$ goes to $\hat{\theta}$.

Moreover, if we put $\hat{C}_2(\alpha) := C_{2, \hat{\psi}_\alpha}$, $\mu(\alpha) := \mu(\hat{\psi}_\alpha)$ and $\lambda(\alpha) := \lambda(\hat{\psi}_\alpha)$ (see definitions (1.6)) we find that:

(i) For every $\alpha \in ]0, \pi[$, $\hat{\psi}_\alpha$ verifies properties (0.12) where $\|\hat{\psi}_\alpha\|_\infty = 2$ and $\mu(\alpha) = 1$.

(ii) $\hat{C}_1 : \alpha \mapsto \hat{C}_1(\alpha)$ and $\hat{C}_2 : \alpha \mapsto \hat{C}_2(\alpha) = \lambda(\alpha)$ are both decreasing and continuous functions on $]0, \pi[$.

(iii)

$$ \left\{ \begin{array}{l} \hat{C}_1(\alpha) \underset{\alpha \to 0^+}{\sim} \frac{2}{\alpha^2}, \\ \\ \hat{C}_2(\alpha) \underset{\alpha \to 0^+}{\sim} \frac{1}{9} \left(\frac{2}{\alpha}\right)^6. \end{array} \right. \quad (1.39) $$

(iv) There exists $\gamma_0 \approx 2.512 \in ]\frac{\pi}{2}, \pi[$ such that for all $\alpha \in [\gamma_0, \pi[$ we have:

$$ \left\{ \begin{array}{l} \hat{C}_1(\alpha) = \frac{(\sin \alpha - 1) \sqrt{3 \sin^2 \alpha + 4 \sin \alpha + 2}}{\cos^3 \alpha}, \\ \\ \hat{C}_2(\alpha) = \frac{-3}{\cos^3 \alpha}, \end{array} \right. \quad (1.40) $$
and $\forall (x, y) \in \bar{\Omega}$,

$$ \hat{\psi}_{\alpha}(x, y) = \begin{cases} \sqrt{1-y^2} - x, & \text{if } |y| \le \sin \alpha, \\ -\frac{-y^2 + 2 |y| \sin^3 \alpha + 2x \cos^3 \alpha + 3 \cos^2 \alpha - 1}{2 \cos^3 \alpha}, & \text{else.} \end{cases} \quad (1.41) $$

The proof of Theorem 1.6 is presented in Section 3.

**Corollary 1.7.** For all $\alpha \in ]0, \pi]$, optimal formulas for $\hat{c}_{\mathcal{L}}$ and $\hat{c}_{\text{obs}}$ are given by

$$ \begin{cases} \hat{c}_{\mathcal{L}}(\delta, \alpha) := C(\lambda_{\text{obs}})(1 + \delta), \\ \hat{c}_{\text{obs}}(\alpha) := 4C(\lambda_{\text{obs}}) \left(1 + e^{-2\lambda_{\text{obs}}} \left(1 + \frac{e^{-4\lambda_{\text{obs}}}}{16}\right)\right) \lambda_{\text{obs}}^3 \hat{C}_1(\alpha)^3. \end{cases} \quad (1.42) $$

where $C(\lambda)$ for all $\lambda \ge \underline{\lambda} = \hat{C}_2(\alpha)$ is given by:

$$ C(\lambda) = \begin{cases} \frac{\tilde{c}(1)\lambda(1+3e^{-2\lambda}(3+\sqrt{5}+2\sqrt{2}T\lambda^2e^{-2\lambda}))}{2(\lambda-\tilde{C}_2(\alpha))(1-e^{-2\lambda})}, & \text{if } \lambda > \underline{\lambda} = \hat{C}_2(\alpha) \\ +\infty, & \text{else} \end{cases} \quad (1.43) $$

and

$$ \left\{ \begin{aligned} \lambda_{\text{obs}}(\alpha, I, T) &:= \underset{\underline{\lambda} \le \lambda}{\operatorname{argmin}} \left( C(\lambda) \left( 1 + e^{-2\lambda} \left( 1 + \frac{e^{-4\lambda}}{16} \right) \right) \lambda^3 \right), \\ \hat{\lambda}(\alpha, I, T) &:= \max \left( \frac{1}{2} \log \left( \frac{M(I,T)}{2^4} \right), \hat{C}_2(\alpha) \right), \\ M(I, T) &:= \max \left( \left(1 + \left(\frac{T}{2}\right)^2\right)^2 \tilde{c}(1)^2, \left(\frac{\|I\|_{\infty} T}{4}\right) \right), \\ \tilde{c}(1) &= \frac{\sqrt{109}}{2} + \frac{15}{2} \approx 12.720. \end{aligned} \right. \quad (1.44) $$

*Proof.* In Theorem 1.1, we wrote Carleman constants $c_{\mathcal{L}}(\delta, \lambda)$ and $c_{\text{obs}}(\lambda)$ in terms of parameters $\delta, \lambda, \mu, T, \|\psi\|_{\infty}$ and $C_{1,\psi}$ as shown in system (1.2):

$$ \left\{ \begin{aligned} c_{\mathcal{L}}(\delta, \lambda) &:= C(\lambda)(1 + \delta), \\ c_{\text{obs}}(\lambda) &:= 4C(\lambda)(1 + \epsilon_2(\lambda))\lambda^3 C_{1,\psi}^3, \\ C(\lambda) &:= \frac{\tilde{c}(\mu)(1 + \epsilon_1(\lambda))}{2\underline{\epsilon}(\lambda)(1 - e^{-\lambda}\|\psi\|_{\infty})}. \end{aligned} \right. $$

where

* $\delta$ is any non negative number which influences the initial value $s_{\beta}$ of parameter $s$.

*

$$ C_{1,\psi} := \max_{x \in \Gamma_0} (|\nabla \psi(x)|). $$
• The initial value $\hat{\lambda}$ of the parameter $\lambda$ depend on $C_{2,\psi}$, $\mu$, $T$ and $I$ since

$$ 
\left\{
\begin{aligned}
\lambda &\ge \hat{\lambda} > \underline{\lambda}(\psi) := \frac{C_{2,\psi}}{\mu(\psi)^2} \\
C_{2,\psi} &:= 2 \left( \max \left( \|\partial_x^2 \psi\|_{\infty}, \|\partial_y^2 \psi\|_{\infty} \right) + \|\partial_{x,y} \psi\|_{\infty} \right) + \|\partial_x^2 \psi\|_{\infty} + \|\partial_y^2 \psi\|_{\infty}, \\
\mu &= \mu(\psi) := \min_{x \in \bar{\Omega}} (|\nabla \psi(x)|) > 0.
\end{aligned}
\right.
$$

The value of $\hat{\lambda}$ is related to $C_{2,\psi}$ and $\tilde{\lambda}$ which represents the solution of a minimization problem as shown in (2.20). Since any minimizer $\tilde{\lambda}$ of the previously mentioned problem (2.20) does not have in general an explicit expression, we gave an estimate of its value in (2.22) using parameters $\|\psi\|_{\infty}$, $I$, $\mu$, $T$.

$$ 
\left\{
\begin{aligned}
\epsilon_1(\lambda) &:= 3\mu e^{-\lambda \|\psi\|_{\infty}} \left(3 + \sqrt{5} + 2\sqrt{2}\mu T \lambda^2 e^{-\lambda \|\psi\|_{\infty}}\right), \\
\epsilon_2(\lambda) &:= e^{-\lambda \|\psi\|_{\infty}} \left(1 + \frac{e^{-2\lambda \|\psi\|_{\infty}}}{16}\right), \\
\epsilon(\lambda) &:= 1 - \frac{\underline{\lambda}(\psi)}{\lambda}.
\end{aligned}
\right.
$$

• $\tilde{c}(\mu)$ is the solution of the minimization problem (1.3).

The only terms in Carleman constants that depend on the size of the boundary measurements are $C_{1,\psi}$ and $C_{2,\psi}$, because they are both related to the weight function $\psi$ as shown in the first two equations of (1.6), and the function $\psi$ itself is related to $\Gamma_0$ by its properties in (0.12).

In Theorem 1.6. (iv), for $\gamma_0 \le \alpha \le \pi$ we succeeded to identify the optimal couple of constants $(\hat{C}_1(\alpha), \hat{C}_2(\alpha))$ among all couples $(C_{1,\psi}, C_{2,\psi})$ where we choose the optimal $\psi$ function denoted by $\hat{\psi}_\alpha$ from the class of weight functions $C_\alpha = (\psi_{\alpha,\theta_M})_{\theta_i(\alpha)<\theta_M<\frac{\pi}{2}}$ such that each function verifies properties (0.12) which are related to the boundary measurement size parameterized by the angle $\alpha$ of the arc $\Gamma_0$.

We showed in Theorem 1.6 (i) that $\|\hat{\psi}_\alpha\|_{\infty} = 2$ and $\mu(\hat{\psi}_\alpha) = 1$ for all $\alpha$ between 0 and $\pi$. We showed that when there exists 80% or more of the boundary measurements, precisely when $\alpha \in [\gamma_0, \pi]$ ($\frac{\gamma_0}{\pi} \approx 0.799772234617420$) that

$$ 
\left\{
\begin{aligned}
\hat{C}_1(\alpha) &= \frac{(\sin \alpha - 1) \sqrt{3 \sin^2 \alpha + 4 \sin \alpha + 2}}{\cos^3 \alpha}, \\
\hat{C}_2(\alpha) &= \frac{-3}{\cos^3 \alpha},
\end{aligned}
\right.
$$

Since $\mu = 1$, we explicitly solve the minimization problem (1.3) and we show that

$$ \tilde{c}(1) = \frac{\sqrt{109}}{2} + \frac{15}{2} \approx 12.720. \qquad (1.45) $$
Now, we are able to write the optimal value of each Carleman constant. We begin by giving the initial value $\hat{\lambda}$ of $\lambda$. So we put:

$$ \hat{\lambda}(\alpha, I, T) = \max \left( \lambda_0(I, T), \hat{C}_2(\alpha) \right) $$

where

$$ \left\{ \begin{array}{l} \lambda_0(I, T) = \left( \frac{1}{2} \log \left( \frac{M(I, T)}{2^4} \right) \right), \\ \\ M(I, T) = \max \left( \left( 1 + \left( \frac{T}{2} \right)^2 \right)^2 \tilde{c}(1)^2, \left( \frac{\|I\|_{\infty} T}{4} \right) \right). \end{array} \right. $$

and the parameter $I$ is used in the boundary condition of (0.8).

The optimal value of $\lambda$ depend only on $c_{obs}$ since $\lambda \mapsto c_{\mathcal{L}}(\lambda)$ is a decreasing function. This value is given by:

$$ \lambda_{obs}(\alpha, I, T) := \underset{\hat{\lambda} \le \lambda}{\operatorname{argmin}} \left( C(\lambda) \left( 1 + e^{-2\lambda} \left( 1 + \frac{e^{-4\lambda}}{16} \right) \right) \lambda^3 \right). \quad (1.46) $$

where $C(\lambda)$ for all $\lambda > \underline{\lambda} = \hat{C}_2(\alpha)$ is given by:

$$ C(\lambda) = \frac{\tilde{c}(1)\lambda(1 + 3e^{-2\lambda}(3 + \sqrt{5} + 2\sqrt{2}T\lambda^2e^{-2\lambda}))}{2(\lambda - \hat{C}_2(\alpha))(1 - e^{-2\lambda})}, \quad (1.47) $$

and $C(\lambda) = +\infty$.

Finally, for all $\alpha \in ]0, \pi]$ optimal Carleman constants are given by:

$$ \left\{ \begin{array}{l} \hat{c}_{\mathcal{L}}(\delta, \alpha) := C(\lambda_{obs})(1 + \delta), \\ \\ \hat{c}_{obs}(\alpha) := 4C(\lambda_{obs}) \left( 1 + e^{-2\lambda_{obs}} \left( 1 + \frac{e^{-4\lambda_{obs}}}{16} \right) \right) \lambda_{obs}^3 \hat{C}_1(\alpha)^3. \end{array} \right. \quad (1.48) $$

□

### Remarks 1.8.

(i) For the value $\gamma_0 \approx 2.512$, the interval $]\gamma_0, \pi[$ corresponds to the ratio of 80% of the boundary where the measurements are available. In this case, explicit formula for the Carleman constants can be provided since we have explicit formulas of $\hat{C}_1$ and $\hat{C}_2$ with respect to $\alpha$ (see (1.40)).

(ii) For all values of $\alpha$ between 0 and $\gamma_0$, which cover less than 80% of the boundary, although we do not have explicit formulas for $\hat{C}_1(\alpha)$ and $\hat{C}_2(\alpha)$, we can still simulate these functions using minimization algorithms. For a clearer understanding of the origin of $\gamma_0$, we refer to subsection 3.1.

(iii) As an application of Corollary (1.8), we calculated the exact values of the optimal Carleman constants in the following particular cases:
(a) For $T = 1$, $I \le \left(\frac{5\tilde{c}(1)}{2}\right)^2$ and the observation is on the whole boundary $\Gamma$ (i.e $\alpha = \pi$):
We find that $\lambda_{\text{obs}} \approx 4.015652$, $\hat{C}_{\mathcal{L}} \approx 25.283274 * (1 + \delta)$, $\hat{C}_{\text{obs}} \approx 18528.816397$. This particular case includes both Neumann and Robin boundary conditions.

(b) For $I \ge \left(\frac{5\tilde{c}(1)}{2}\right)^2$, $T \le T_0(\alpha, I) :=$ and the observation covers 80% of the boundary $\Gamma$ (i.e $\alpha = \frac{4\pi}{5} > \gamma_0$):
We find that $\lambda_{\text{obs}} \approx 7.554221$, $\hat{C}_{\mathcal{L}} \approx 25.439956 * (1 + \delta)$, $\hat{C}_{\text{obs}} \approx 258816.691897$.

In the following corollary, we give the asymptotic behavior of the Carleman constants $c_{\mathcal{L}}(\delta, \lambda)$ and $c_{\text{obs}}$ when $\alpha$ is sufficiently close to 0.

**Corollary 1.9.** Under the same hypothesis of Theorem 1.6, constants in Theorem 1.1 satisfy for $\alpha$ sufficiently close to 0:

$$ \exists \lambda > \lambda(\alpha) := \hat{C}_2(\alpha), \quad \exists \delta > 0; \quad c_{\mathcal{L}}(\delta, \lambda) = \left(\frac{\tilde{c}(1)}{2}\right)^{+}, \qquad (1.49) $$

$$ \min_{\lambda > \lambda(\alpha)} c_{\text{obs}}(\lambda) = c_{\text{obs}}\left(\frac{4}{3}\lambda\right) \underset{\alpha \to 0^+}{\sim} \frac{2^{30}\tilde{c}(1)}{3^9\alpha^{24}}, \qquad (1.50) $$

We mean by $x^+$ any number strictly larger than $x$.

The proof of Corollary 1.9 is a direct deduction from Theorem 1.6.

**Remark 1.10. Generalization of the domain $\Omega$**

For simplicity reasons, we fix our spatial domain to the unit disc knowing that all proven properties in our work have similar versions for every domain diffeomorphic to the unit disc. Since the change of the domain can be represented by a diffeomorphism $\varphi : \Omega' \to \Omega$ of class $C^2$, we can replace all weight functions $\psi$ in this work by:

$$ \tilde{\psi} := \psi \circ \varphi, \qquad (1.51) $$

where $\Gamma'_0 := \varphi^{-1}(\Gamma_0)$.

Since any function $\tilde{\psi}$ that verifies properties in (0.12) with respect to $\Omega'$ is not necessarily written in the form $\psi \circ \varphi$ where $\psi$ verifies (0.12) with respect to $\Omega$, we will provide upper bounds of optimal Carleman constants.

In fact, the equivalent version of $C_{1,\psi}$ and $C_{2,\psi}$ (defined in (1.6)) with respect to the new domain $\Omega'$ are given by:

$$ \left\{ \begin{array}{l} C_{1,\tilde{\psi}} := \max_{(x',y') \in \Gamma'_0} (|\nabla \tilde{\psi}(x', y')|), \\ C_{2,\tilde{\psi}} := 2 \left( \max \left( \|\partial_x^2 \tilde{\psi}\|_{\infty}, \|\partial_y^2 \tilde{\psi}\|_{\infty} \right) + \|\partial_{x,y} \psi\|_{\infty} \right) + \|\partial_x^2 \tilde{\psi}\|_{\infty} + \|\partial_y^2 \tilde{\psi}\|_{\infty}. \end{array} \right. \qquad (1.52) $$
Using the chain rule we find that:

$$ \left| \nabla \tilde{\psi}(x', y') \right| \le C_{\Omega'} |\nabla \psi(x, y)|, \\ \left| \frac{\partial^2 \tilde{\psi}}{\partial x'^2} \right| + \left| \frac{\partial^2 \tilde{\psi}}{\partial y'^2} \right| + \left| \frac{\partial^2 \tilde{\psi}}{\partial x' \partial y'} \right| \le C_{\Omega'} \left( \left| \nabla \psi(x, y) \right| + \left| \frac{\partial^2 \psi}{\partial x^2} \right| + \left| \frac{\partial^2 \psi}{\partial y^2} \right| + \left| \frac{\partial^2 \psi}{\partial x \partial y} \right| \right) \quad (1.53) $$

where $(x, y) := \varphi(x', y') \in \Omega$ and by $C_{\Omega'}$ we denote a generic constant that is related to $\varphi$. This gives:

$$ \left\{ \begin{array}{l} C_{1, \tilde{\psi}} \le C_{\Omega'} C_{1, \psi}, \\ C_{2, \tilde{\psi}} \le C_{\Omega'} (C_{1, \psi} + C_{2, \psi}), \end{array} \right. \qquad (1.54) $$

Now, if we replace $\psi$ by the optimal weight function $\tilde{\psi}_{\alpha}$ given in Theorem 1.6 and we put $\tilde{\psi}_{\alpha} := \tilde{\psi}_{\alpha} \circ \varphi$ we find that:

$$ \left\{ \begin{array}{l} \tilde{C}_1(\alpha) := C_{1, \tilde{\psi}_{\alpha}} \le C_{\Omega'} C_{1, \psi_{\alpha}} = C_{\Omega'} \hat{C}_1(\alpha), \\ \tilde{C}_2(\alpha) := C_{2, \tilde{\psi}_{\alpha}} \le C_{\Omega'} (C_{1, \psi_{\alpha}} + C_{2, \psi_{\alpha}}) = C_{\Omega'} (\hat{C}_1(\alpha) + \hat{C}_2(\alpha)), \end{array} \right. \qquad (1.55) $$

This gives an upper bound of both constants $\tilde{C}_1(\alpha)$ and $\tilde{C}_2(\alpha)$ with respect to our new domain $\Omega'$. Since both new Carleman constants $\tilde{c}_{\Sigma}$ and $\tilde{c}_{\text{obs}}$ are expressed via these constants using formulas in (1.42)-(1.44). Using the same type of formulas as in Corollary 1.7, we obtain upper bounds of both new optimal constants $\tilde{C}_1(\alpha)$ and $\tilde{C}_2(\alpha)$ for the new domain $\Omega'$ in term of $\alpha$ when $\gamma_0 \le \alpha \le \pi$.

## 1.2. Applying our Carleman estimate on an inverse problem

Now, we give an explicit estimate of Lipschitz constants that appears when establishing the Lipschtiz stability result in the following inverse source problem.
We consider

$$ \left\{ \begin{array}{ll} \partial_t y - \Delta y = g(x, t), & \text{in } Q, \\ \nabla y \cdot \nu + I(x) y = 0, & \text{on } \Sigma, \\ y(t_0, \cdot) = y_0, & \text{in } \Omega. \end{array} \right. \qquad (1.56) $$

Here $Q, \Omega, T, \Sigma, \Sigma_0, I$ are well defined in the introduction. Moreover, the function $g(x, t)$ satisfies

$$ |\partial_t g(x, t)| \le C_g \left| g \left( x, \frac{T}{2} \right) \right| \quad \text{for almost every } (x, t) \in \bar{Q}. \qquad (1.57) $$

Our inverse problem consists of determining $g(x, t)$ from overdetermining data $(\partial_t y)|_{\Gamma_0 \times ]0, T[}$, $(\nabla \partial_t u)|_{\Gamma_0 \times ]0, T[}$ which has been proved by O. Y Imanuvilov and M. Yamamoto in [22]. Here, we did simplify the elliptic operator in order to have a simpler form of constants appearing in the following stability result.
**Theorem 1.11.** Using the same notations as in Theorem 1.1, for $g(x,t) \in W_2^{0,1}(Q) := \{y(x,t); y, \partial_t y \in L^2(Q)\}$ satisfy (1.57), if $y \in W_2^{2,1}(Q)$ satisfies (1.56) then:

$$ \|g\|_{L^2(Q)} \le c_0(\lambda_0) \|\Delta y_0\|_{L^2(\Omega)} + c_{\Sigma_0}(\lambda_0) \left( \|(\partial_t y)|_{\Sigma_0}\|_{L^2(\Sigma_0)} + \|(\nabla \partial_t y)|_{\Sigma_0}\|_{L^2(\Sigma_0)} \right), \quad (1.58) $$

where $\lambda_0$ is chosen in $]\lambda, +\infty[$, $]\tilde{\lambda}, +\infty[$, $[1, +\infty[$ to minimize $c_{\Sigma_0}$ defined by:

$$ 
\left\{
\begin{aligned}
c_{\Sigma_0}(\lambda) &:= \frac{(2(1+c_\mu e^{-\lambda\|\psi\|_\infty}) e^{\lambda\|\psi\|_\infty} C_{1,\psi})^3}{\min(1, \frac{T}{2})^4 (1-e^{-\lambda\|\psi\|_\infty}) (\lambda-\tilde{\lambda})} (\lambda s_0(\lambda))^2 c_0(\lambda), \\
c_0(\lambda) &:= \frac{2}{1-e^{-\lambda\|\psi\|_\infty}} \exp\left(\frac{4e^{\lambda\|\psi\|_\infty}(e^{\lambda\|\psi\|_\infty}-1)}{\underline{\epsilon}(\lambda)} \left[ \frac{3(1+\mu^2)}{(2\lambda\|\psi\|_\infty)^2} + (3+\lambda) + \frac{c_{0,1}(1,\lambda)}{T} \right] \right), \\
\tilde{\lambda} &:= \frac{2\log M}{\|\psi\|_\infty}, \\
M &:= \max\left(\frac{\sqrt{\mu \|I\|_\infty}}{2\|\psi\|_\infty^2} T^{1/2}, \frac{3\mu^2(1+3\|\psi\|_\infty)}{2\|\psi\|_\infty^3} T, \frac{8(1+\mu^2)\|\psi\|_\infty C_g}{\mu^2(1-e^{-\|\psi\|_\infty})} \sqrt{\frac{2\pi}{3}} T^{3/2}\right), \\
c_\mu &:= \max(3, \mu^2).
\end{aligned}
\right.
\quad (1.59) $$

Moreover, in the particular case where $\Omega$ is the unit disc and $\Gamma_0 = \Gamma_0(\alpha)$ as in Theorem 1.6, we find that:

$$ c_0(\lambda_0), c_{\Sigma_0}(\lambda_0) \underset{\lambda \to 0^+}{=} o\left(\exp\left(\frac{\exp\left(\frac{(2^6)^+}{\alpha^6}\right)}{T}\right)\right); \quad \forall T > 0, \quad (1.60) $$

$$ c_0(\lambda_0), c_{\Sigma_0}(\lambda_0) \underset{T \to 0^+}{=} o\left(\exp\left(\frac{\mathcal{R}(\alpha)}{T}\right)\right); \quad \forall \alpha \in ]0, \pi[, \quad (1.62) $$

where $\mathcal{R}(\alpha) := \frac{2^4(1+2e^{2\lambda_0})e^{4\lambda_0}}{\lambda_0(\lambda_0-\tilde{\lambda})}$ is a geometric constant that depends only on $\alpha$. Here $\lambda_0(\alpha) \in ]\tilde{\lambda}(\alpha), \tilde{\lambda}(\alpha) + \frac{1}{2}[$ is the unique solution to the following equation:

$$ (4\lambda^2 - (4\tilde{\lambda} + 2)\lambda + \tilde{\lambda})(3 + e^{-2\lambda}) = -(2\lambda - \tilde{\lambda}). \quad (1.63) $$

The proof of Theorem 1.11 can be found in Section 4.

**Remark 1.12.** Since we can change the scale of $t$, we can prove a similar result by replacing $\frac{T}{2}$ in (1.57) by any $t_0 \in ]0, T[$. This technique has been used in these two references [10, 22].
## 2. Proof of Theorem 1.1

We define for all $s > 0$: $\omega := e^{-s\eta_\beta}z$, and $\bar{\omega} = e^{-s\bar{\eta}_\beta}z$.

Using the same procedure as in [4] to prove Carleman estimate in Theorem 4.3. We introduce operators $L_1, L_2$, as follows

$$ egin{cases} L_1 y = -\Delta y - \lambda^2 s^2 \varphi^2 |\nabla \psi|^2 y + s \partial_t \eta_\beta y, \\ L_2 y = \partial_t y + 2 s \lambda \varphi \nabla \psi \cdot \nabla y, \end{cases} \qquad (2.1) $$

so

$$ L_1\omega + L_2\omega = f_s(\omega), \qquad (2.2) $$

where

$$ f_s(\omega) = e^{-s\eta_\beta}g - \lambda s\varphi\Delta\psi\omega + \lambda^2 s\varphi|\nabla\psi|^2\omega. \qquad (2.3) $$

If we apply the Peter-Paul inequality: $|ab| \le (\frac{a^2\delta}{2} + \frac{b^2}{2\delta})$, $\forall a, b \in \mathbb{R}$, $\delta > 0$ when establishing the following estimate, we obtain for all $\lambda > \bar{\lambda}(\psi)$, $\delta > 0$ and $s \ge s_\beta(\delta, \lambda)$:

$$ \begin{aligned} & \|L_1(\omega)\|_{L^2(Q)}^2 + \|L_2(\omega)\|_{L^2(Q)}^2 + 2C_Q(\lambda, s) \int_Q \lambda^4 s^3 \varphi^3 |\nabla \psi|^4 \omega^2 + \lambda^2 s \varphi |\nabla \psi|^2 |\nabla \omega|^2 dx dt \\ & \le (1 + \delta) \int_Q g^2 (e^{-2s\eta_\beta} + e^{-2s\bar{\eta}_\beta}) dx dt \\ & + \int_{\Sigma_0} \left( 4\lambda^3 C_1(\psi)^3 (1+\delta_1(\lambda, s))(s\varphi)^3 |z|^2 + 2\lambda C_1(\psi)(1+\delta_2(\lambda, s))(s\varphi) |\nabla z|^2 \right) e^{-2s\eta_\beta} dSdt, \end{aligned} \qquad (2.4) $$

where

$$ egin{cases} \bar{\lambda}(\psi) := \frac{C_{2,\psi}}{\mu(\psi)^2}, \\ C_{1,\psi} := \max_{x \in \Gamma_0} |\nabla \psi(x)| \\ C_{2,\psi} := 2 \left( \max \left( \|\partial_x^2 \psi\|_{\infty}, \|\partial_y^2 \psi\|_{\infty} \right) + \|\partial_{x,y} \psi\|_{\infty} \right) + \|\partial_x^2 \psi\|_{\infty} + \|\partial_y^2 \psi\|_{\infty}, \\ \mu(\psi) := \min_{x \in \bar{\Omega}} |\nabla \psi(x)| > 0. \end{cases} \qquad (2.5) $$
and

$$
\left\{
\begin{aligned}
s_{\beta}(\delta, \lambda) &:= \frac{e^{\lambda \|\psi\|_{\infty}} C_0(\beta, \delta, \lambda)}{\epsilon(\lambda)}, \\
C_0(\beta, \delta, \lambda) &:= 2T \left( \frac{\beta e^{\lambda \|\psi\|_{\infty}}}{\lambda^2} \left( \frac{1}{\mu^2} + \frac{1}{\mu^4} \right) + \frac{T}{8} \left( \frac{1}{\delta} + \lambda + 2 \right) + \frac{1}{\lambda^2 \mu^2} \right), \\
C_Q(\lambda, s) &:= \epsilon(\lambda) \left( 1 - e^{-\lambda \|\psi\|_{\infty}} \frac{s_{\beta}}{s} \right), \\
\epsilon(\lambda) &:= 1 - \frac{\lambda}{\lambda}, \\
\delta_1(\lambda, s) &:= \frac{\beta T e^{\lambda \|\psi\|_{\infty}}}{\mu^2 \lambda^2 s} + \frac{(\frac{T}{2})^2}{s} \left[ 1 + \frac{T}{4\mu^2 \lambda^2 s} \right], \\
\delta_2(\lambda, s) &:= \frac{T^2 \|I\|_{\infty}}{8\mu\lambda s}.
\end{aligned}
\right.
\quad (2.6)
$$

So, if we put $\forall \lambda \ge 1$, $\forall \delta > 0$:

$$
\left\{
\begin{aligned}
\epsilon_2(\beta, \delta, \lambda) &:= \delta_1(\lambda, s_\beta(\delta, \lambda)), \\
\lambda_1 &:= \min \left\{ \lambda \ge 1; \frac{1 + \delta_2(\lambda, s_\beta(\delta, \lambda))}{2\lambda^2 C_1(\psi)^2 (1 + \tilde{\delta}_1(\delta, \lambda))} \le 1 \right\},
\end{aligned}
\right.
\quad (2.7)
$$

we find that for all $\lambda > \bar{\lambda}$ such that $\lambda \ge \lambda_1$, for all $\beta > 1$, $\delta > 0$ and $s \ge s_\beta(\delta, \lambda)$:

$$
\begin{equation}
\begin{aligned}
& \|L_1(\omega)\|_{L^2(Q)}^2 + \|L_2(\omega)\|_{L^2(Q)}^2 + 2C_Q(\lambda, s) \int_Q \lambda^4 s^3 \varphi^3 |\nabla\psi|^4 \omega^2 + \lambda^2 s \varphi |\nabla\psi|^2 |\nabla\omega|^2 dx dt \le \\
& (1+\delta) \int_Q g^2 (e^{-2s\eta} + e^{-2s\bar{\eta}\beta}) dx dt + 4(1+\epsilon_2)\lambda^3 C_1(\psi)^3 \int_{\Sigma_0} \left( (s\varphi)^3 |z|^2 + (s\varphi) |\nabla z|^2 \right) e^{-2s\eta} dSdt.
\end{aligned}
\tag{2.8}
\end{equation}
$$

Now, if we use again Peter-Paul's inequality in both definitions (2.1) of operators $L_1$ and $L_2$, we obtain for all $X = (x_1, x_2, x_3) \in \mathcal{U} := ]0, 1[ \times (\mathbb{R}_+^*)^2$:

$$
\begin{align*}
& \int_Q \left\{ (s\varphi)^{-1} (|\partial_t z|^2 + |\Delta z|^2) + (s\varphi)\lambda^2 |\nabla z|^2 + (s\varphi)^3 \lambda^4 |z|^2 \right\} e^{-2s\eta\beta} dx dt \\
& \le c_1 \|L_1(\omega)\|_{L^2(Q)}^2 + c_2 \|L_2(\omega)\|_{L^2(Q)}^2 + (f(X) + \epsilon_1) \int_Q \lambda^2 s \varphi |\nabla \psi|^2 |\nabla \omega|^2 dx dt \\
& \quad + (g(X) + \epsilon_0) \int_Q \lambda^4 s^3 \varphi^3 |\nabla \psi|^4 \omega^2 dx dt, \tag{2.9}
\end{align*}
$$
where $f,g: \mathcal{U} \to \mathbb{R}_+$ are two maps of class $\mathcal{C}^1$ defined by:

$$ egin{cases} f_{\mu}(x_1, x_2, x_3) := 4(1+x_1)(1+x_2) + \frac{x_3}{\mu^2} + 4 + \frac{1}{\mu^2}, \\ g_{\mu}(x_1, x_2, x_3) := (1+x_1)\left(1+\frac{1}{x_2}\right) + \frac{1}{x_1} + \frac{1}{x_3\mu^2} + 1 + \frac{1}{\mu^2} + \frac{1}{\mu^4}. \end{cases} \tag{2.10} $$

$$ egin{cases} c_1 = c_1(\delta_4, s) &:= \frac{T^2\left(1+\frac{1}{x_1}\right)\left(1+\frac{1}{\delta_4}\right)}{2s}, \\ c_2 = c_2(\delta_6, \delta_7, s) &:= \frac{T^2\left(1+\frac{1}{\delta_6}\right)(1+\delta_7)}{2s}, \\ \epsilon_1 = \epsilon_1(\delta_5, \delta_6, \delta_7) &:= 4\left(\delta_5(1+x_1)(1+x_2) + \delta_6 + \delta_7 + \delta_6\delta_7\right), \\ \epsilon_0 = \epsilon_0(\delta_4, \dots, \delta_7, \lambda, s) &:= \delta_5(1+x_1)\left(1+\frac{1}{x_2}\right) + \delta_4\left(1+\frac{1}{x_1}\right) + \frac{c_3(\delta_4, \dots, \delta_7, \lambda)}{s^2} \end{cases} \tag{2.11} $$

and

$$ c_3(\delta_4, \dots, \delta_7, \lambda) := \frac{T^2 \left( \beta e^{\lambda \|\psi\|_{\infty}} - 1 \right)^2}{\lambda^4} \left( \frac{2 \left( 1 + \frac{1}{x_1} \right) \left( 1 + \frac{1}{\delta_4} \right)}{\mu^2} + \frac{1}{\mu^4} \right) + \frac{\left( \left( \frac{T}{2} \right)^2 \left( 1 + \left( \frac{T}{2} \right)^2 \right) \right)^2 (1 + x_1) \left( 1 + \frac{1}{\delta_5} \right)}{\mu^4} + \frac{T^4 \left( 1 + \frac{1}{\delta_6} \right) (1 + \delta_7)}{2} \tag{2.12} $$

Here $\delta_4, \dots, \delta_7 > 0$ will be fixed later and we put $p = (\delta_4, \dots, \delta_7)$.

Now, if we solve this optimization problem

$$ (\mathcal{P}_{\mu}) : \min_{g(X) \le f(X), X \in \mathcal{U}} f(X), $$

then by K.K.T theorem, this constraint has to be active, so we find that problem $(\mathcal{P})$ has a solution denoted by $\tilde{c}(\mu)$ verifying:

$$ \tilde{c}(\mu) = \min_{g_{\mu}(X) = f_{\mu}(X), X \in \mathcal{U}} f_{\mu}(X). \tag{2.13} $$

Using the Lagrange multipliers, we find also that this problem has a unique minimizer that we denote by $\tilde{X}(\mu) = (\tilde{x}_1, \tilde{x}_2, \tilde{x}_3)$ and

$$ \tilde{c}(\mu) = \min_{f_{\mu}(X) \le g_{\mu}(X), X \in \mathcal{U}} g_{\mu}(X). \tag{2.14} $$
From now on, by $X$ we mean the minimizer $\tilde{X}$.

Let $C'_Q := \tilde{c}(\mu) + \max(\epsilon_0, \epsilon_1)$ and $\tau(p, \lambda, s) := \frac{C'_Q}{2C_Q}$, then by (2.8) we have:

$$ \begin{aligned} & \frac{1}{\tau} \int_Q \{(s\varphi)^{-1} (|\partial_t z|^2 + |\Delta z|^2) + (s\varphi)\lambda^2 |\nabla z|^2 + (s\varphi)^3 \lambda^4 |z|^2\} e^{-2s\eta} dxdt \\ & \le \frac{c_1}{\tau} \|L_1(\omega)\|_{L^2(Q)}^2 + \frac{c_2}{\tau} \|L_2(\omega)\|_{L^2(Q)}^2 + 2C_Q(\lambda, s) \int_Q \lambda^4 s^3 \varphi^3 |\nabla \psi|^4 \omega^2 + \lambda^2 s \varphi |\nabla \psi|^2 |\nabla \omega|^2 dxdt. \end{aligned} \quad (2.15) $$

By numerical consideration, we can see that $x_2: \mu \mapsto x_2(\mu)$ is a decreasing function such that $x_2(\mu) \to x_{2,\infty} > \frac{1}{4}$ where $X_\infty := (x_{1,\infty}, x_{2,\infty}) = (\frac{\sqrt{5}-1}{4}, \sqrt{5}-2)$ is solution to the problem $(\mathcal{P}_\infty)$:

$$ f_\infty(x_1, x_2) := 4(1 + x_1)(1 + x_2) + 4, $$

$$ g_\infty(x_1, x_2) := (1 + x_1) \left(1 + \frac{1}{x_2}\right) + \frac{1}{x_1} + 1, $$

hence $4(1 + x_2) > \left(1 + \frac{1}{x_2}\right)$ which implies that for $\lambda$ large enough and by taking $\delta_6$ and $\delta_7$ sufficiently large with respect to $\delta_4$ we would have $\epsilon_1 \ge \epsilon_0$, which justifies the final condition in (2.17).

Hence, we would have for all $\lambda > \underline{\lambda}$ and $s \ge s_\beta(\delta, \lambda)$:

$$ \frac{C'_Q}{2C_Q} = \tau(p, \lambda, s) \le \tau_0(p', \lambda) := \frac{\tilde{c}(\mu) + \epsilon_1(p')}{2\underline{\epsilon}(\lambda)(1 - e^{-\lambda}\|\psi\|_\infty)}, \quad (2.16) $$

where $p' = (\delta_5, \delta_6, \delta_7)$.

Using both inequalities (2.8) and (2.15), we find that for all $\delta > 0$, parameters $p = (\delta_4, \dots, \delta_7) \in (\mathbb{R}_+^*)^4$ and $\lambda > \underline{\lambda}$ that verifies:

$$ \left\{ \begin{array}{ll} \displaystyle \frac{c_1(\delta_4, s_\beta(\delta, \lambda))}{\tau_0(p', \lambda)} & \le 1, \\ \\ \displaystyle \frac{c_2(\delta_6, \delta_7, s_0(\delta, \lambda))}{\tau_0(p', \lambda)} & \le 1, \\ \\ \displaystyle \epsilon_0(p, \lambda, s_0(\lambda)) & \le \epsilon_1(p'), \\ \\ \lambda_1 & \le \lambda. \end{array} \right. \quad (2.17) $$

For all $s \ge s_\beta(\delta, \lambda)$ we obtain:

$$ \begin{aligned} & \int_Q \{(s\varphi)^{-1} (|\partial_t z|^2 + |\Delta z|^2) + (s\varphi)\lambda^2 |\nabla z|^2 + (s\varphi)^3 \lambda^4 |z|^2\} e^{-2s\eta\beta} dxdt \le \\ & c_{\underline{\epsilon}} \int_Q g^2 (e^{-2s\eta\beta} + e^{-2s\bar{\eta}\beta}) dxdt + c_{\text{obs}} \int_{\Sigma_0} \left( (s\varphi)^3 |z|^2 + (s\varphi) |\nabla z|^2 \right) e^{-2s\eta\beta} dSdt, \end{aligned} \quad (2.18) $$
where

$$ egin{cases} c_{\mathcal{L}}(p', \delta, \lambda) & := \tau_0(p', \lambda)(1 + \delta), \\ c_{\text{obs}}(p', \delta, \lambda) & := 4\tau_0(p', \lambda) (1 + \epsilon_2(\beta, \delta, \lambda)) \lambda^3 C_1(\psi)^3. \end{cases} \tag{2.19} $$

The optimal choice of parameters $p'$ and the initial value $\tilde{\lambda}$ of $\lambda$ is any minimizer $(\tilde{p}', \tilde{\lambda}) \in ]0, +\infty]^4$ of the following minimization problem (P):

$$ \min_{(p', \lambda) \in \tilde{\mathcal{A}}} \frac{(\tilde{c}(\mu) + \epsilon_1(p')) (1 + \epsilon_2(\beta, \delta, \lambda)) \lambda^3}{\underline{\epsilon}(\lambda) (1 - e^{-\lambda \|\psi\|_{\infty}})}, \tag{2.20} $$

where $\tilde{\mathcal{A}}$ is the set of all quadruples $(p', \lambda) = (\delta_5, \delta_6, \delta_7, \lambda)$ such that there exists $\delta_4 > 0$ where parameters $(p, \lambda)$ verify conditions in (2.17).

We put $\tilde{\epsilon}_1 := \epsilon_1(\tilde{p}')$, $\tilde{\epsilon}_2(\beta, \delta) := \epsilon_2(\beta, \delta, \tilde{\lambda})$ and if $\tilde{\lambda} \le \underline{\lambda}(\psi)$ we can take $\tilde{\lambda}$ any number greater than $\underline{\lambda}(\psi)$, otherwise we put $\tilde{\lambda} = \tilde{\lambda}$.

Because of the complexity to solve explicitly the constrained minimization problem (P) for each $\beta > 1$, $\delta > 0$ and the non use of an exact formula for parameters $\tilde{p}'$ and $\tilde{\lambda}$, we would rather give an estimate of $\tilde{\epsilon}_1$, $\tilde{\epsilon}_2$ and $\tilde{\lambda}$. We find that for all $\beta > 1$, $\delta > 0$ and $\lambda > \underline{\lambda}$ such that $\lambda \ge \tilde{\lambda}$:

$$ egin{cases} \tilde{\epsilon}_1 \le 3\mu \tilde{c}(\mu) e^{-\lambda \|\psi\|_{\infty}} (3 + \sqrt{5} + 2\sqrt{2}\mu T \lambda^2 e^{-\lambda \|\psi\|_{\infty}}), \\ \tilde{\epsilon}_2 \le e^{-\lambda \|\psi\|_{\infty}} \left(1 + \frac{e^{-2\lambda \|\psi\|_{\infty}}}{16}\right), \end{cases} \tag{2.21} $$

where

$$ egin{cases} \tilde{\lambda} \le \max \left( \frac{1}{\|\psi\|_{\infty}} \log \left( \frac{M(I, \mu, T)}{\|\psi\|_{\infty}^4} \right), \log \left( 2\mu(3 + \sqrt{5}) \right), \log \left( \frac{1}{2(2 - \sqrt{2})\mu} \right) \right), \\ M(I, \mu, T) = \max \left( \left( 1 + \left( \frac{T}{2} \right)^2 \right)^2 \mu^4 \tilde{c}(\mu)^2, \left( \frac{\|I\|_{\infty} \mu T}{4} \right) \right). \end{cases} \tag{2.22} $$

### 3. Proof of Theorem 1.6

In this section, we present our proof in three parts. First, we establish all necessary results except for the asymptotic behavior of $\hat{C}_1$ and the behavior of $\hat{C}_2$. Next, we analyze the asymptotic behavior of $\hat{C}_1$, and finally, we examine the behavior of $\hat{C}_2$.
### 3.1. Proof of major results

As $\Gamma_0^+ = \Gamma_{0,1}^+ \cup \Gamma_{0,2}^+ \cup \Gamma_{0,3}^+$, where:

$$ egin{cases} 
\Gamma_{0,1}^+ := \overline{EA'} = \{P(r_m(\theta, \theta_M), \theta); 0 \le \theta \le \theta_\alpha\}, \\
\Gamma_{0,2}^+ := \overline{DE} = \{P(r_m(\theta, \theta_M), \theta); \theta_\alpha(\theta_M) \le \theta \le \theta_M\}, \\
\Gamma_{0,3}^+ := \overline{CD} = \{P(r_M(\theta, \theta_M), \theta); \theta_\alpha(\theta_M) \le \theta \le \theta_M\},
\end{cases} 
\quad (3.1) $$

we obtain for all $P(r, \theta) \in \Gamma_0^+$,

$$ |\nabla \psi_{\alpha, \theta_M}(P)|^2 = \begin{cases} 1 + \left( \frac{f'_{\alpha, \theta_M}(\theta)}{r_m(\theta, \theta_M)} \right)^2, & \text{if } P \in \Gamma_{0,1}^+ \cup \Gamma_{0,2}^+, \\ 1 + \left( \frac{f'_{\alpha, \theta_M}(\theta)}{r_M(\theta, \theta_M)} \right)^2, & \text{else.} \end{cases} \quad (3.2) $$

As $r_m(\theta, \theta_M) \le r_M(\theta, \theta_M)$, $\forall \theta_\alpha \le \theta \le \theta_M$, we find that the maximum of $|\nabla \psi_{\alpha, \theta_M}|$ is reached on $\Gamma_{0,1}^+$ or $\Gamma_{0,2}^+$.

For this reason, we put:

$$ C_{1,i}(\alpha, \theta_M) := \max_{P(r,\theta) \in \Gamma_{0,i}^+} |\nabla \psi_{\alpha, \theta_M}(P)|, \quad i \in \{1,2\}. \quad (3.3) $$

By defining the function $g_{\alpha, \theta_M}$ on $[0, \theta_M]$ by:

$$ g_{\alpha, \theta_M} : \theta \mapsto \sqrt{1 + \left( \frac{f'_{\alpha, \theta_M}(\theta)}{r_m(\theta, \theta_M)} \right)^2}, \quad (3.4) $$

using (3.2), we find that for all $(\alpha, \theta_M) \in \mathcal{A}$:

$$ \forall \theta \in [0, \theta_M], \quad |\nabla \psi_{\alpha, \theta_M}(P(r_m(\theta, \theta_M), \theta))| = g_{\alpha, \theta_M}(\theta). \quad (3.5) $$

It follows from the expression (3.4) of $g_{\alpha, \theta_M}$ that its maximum is also the same of this function:

$$ h_{\alpha, \theta_M}(\theta) : \theta \mapsto -\frac{f'_{\alpha, \theta_M}(\theta)}{r_m(\theta, \theta_M)}, \quad (3.6) $$

on $[0, \theta_M]$, because for all $\alpha \in ]0, \pi[$ and $(\theta, \theta_M) \in \mathcal{D}_\alpha$, $f'_{\alpha, \theta_M}(\theta) \le 0$, so $0 \le h_{\alpha, \theta_M}$.

By putting for all $(\alpha, \theta_M) \in \mathcal{A}$:

$$ egin{cases} h_M(\alpha, \theta_M) := \max_{\theta \in [0, \theta_M]} h_{\alpha, \theta_M}(\theta), \\ h_{M,1}(\alpha, \theta_M) := \max_{\theta \in [0, \theta_\alpha(\theta_M)]} h_{\alpha, \theta_M}(\theta), \\ h_{M,2}(\alpha, \theta_M) := \max_{\theta \in [\theta_\alpha(\theta_M), \theta_M]} h_{\alpha, \theta_M}(\theta). \end{cases} \quad (3.7) $$
we have:

$$ egin{cases} C_1 = \sqrt{1 + h_{\mathcal{M}}^2}, \\ C_{1,i} = \sqrt{1 + h_{\mathcal{M},i}^2}, \quad i \in \{1, 2\}. \end{cases} \qquad (3.8) $$

Now, in the following lemma we want to give an explicit formulas and some properties of $h_{\mathcal{M}}$ and $h_{\mathcal{M},i}$, $i = 1, 2$. The proof of this lemma is given in Appendix A.

**Lemma 3.1.** There exists three real-valued functions $\theta_{M,1} \in C^1([\theta_c, \frac{\pi}{2}], [0, \frac{\pi}{2}])$, $\theta_{M,2} \in C^1(\mathcal{A}, [0, \frac{\pi}{2}])$ and $\theta_{int} \in C([0, \pi], [0, \frac{\pi}{2}])$ such that:

(i) $\theta_{int}(0) = \frac{\pi}{2}$ and $\theta_{int}|_{[\frac{\pi}{2}, \pi]} = id|_{[\frac{\pi}{2}, \pi]}$.

(ii) $\theta_{int}$ is strictly decreasing (resp. increasing) on $[0, \alpha_0]$ (resp. on $[\alpha_0, \pi]$).

(iii) For all $(\alpha, \theta_M) \in \mathcal{A}$, if we put:

$$ egin{cases} h_1(\theta_M) & := -\frac{\frac{\partial r_M}{\partial \theta}(\theta_{M,1}(\theta_M), \theta_M)}{r_M(\theta_{M,1}(\theta_M), \theta_M)}, \\ h_2(\alpha, \theta_M) & := h_{\alpha, \theta_M}(\theta_{M,2}(\alpha, \theta_M)), \end{cases} \qquad (3.9) $$

we have:

$$ h_{\mathcal{M}}(\alpha, \theta_M) = \begin{cases} h_{\mathcal{M},2}(\alpha, \theta_M) = h_2(\alpha, \theta_M), & \text{if } \theta_M \le \theta_{int}(\alpha), \\ h_{\mathcal{M},1}(\alpha, \theta_M) = h_1(\theta_M), & \text{if } \theta_{int}(\alpha) \le \theta_M, \end{cases} \qquad (3.10) $$

and

$$ h_2(\alpha, \theta_M) \underset{\theta_M \to \theta_i(\alpha)}{\to} \begin{cases} +\infty, & \text{if } \alpha \le \frac{\pi}{2}, \\ \frac{\sin^3 \alpha - 1}{\cos^3 \alpha}, & \text{else,} \end{cases} \qquad (3.11) $$

where

$$ egin{cases} x_c & := \sqrt{\frac{7\sqrt{7}-17}{2}}, \\ \theta_c & := \arcsin(x_c), \\ X_{M,c} & := \sin^2(\theta_{M,1}(\theta_c)), \\ \alpha_0 & := \arccos\left(\frac{X_{M,c} - \sqrt{(X_{M,c}-1)(X_{M,c}-x_c^2)}}{x_c}\right) \approx 1.19. \end{cases} \qquad (3.12) $$

(See (1.36) for the definition of $\mathcal{A}$).
We make the following change of variables:

$$ 
\left\{
\begin{aligned}
x_M &:= \sin(\theta_M), \\
x_{M,1}(x_M) &:= \sin(\theta_{M,1}(\theta_M)), \\
x_{M,2}(\alpha, x_M) &:= \sin(\theta_{M,2}(\alpha, \theta_M)), \\
x_{\text{int}}(\alpha) &:= \sin(\theta_{\text{int}}(\alpha)), \\
H_1(x_M) &:= h_1(\theta_M), \\
H_2(\alpha, x_M) &:= h_2(\alpha, \theta_M).
\end{aligned}
\right.
\qquad (3.13) $$

First, we prove that:

$$ H_1 \text{ is strictly increasing on } I_c := [x_c, 1]. \qquad (3.14) $$

In fact, for all $x_M \in I_c$, $x_{M,1}(x_M)$ is a critical point of $F_1(\cdot, x_M)$ in $[\frac{1}{\sqrt{2}}, x_M[$ (see (A.1)), we find that:

$$ 
\begin{aligned}
H'_1(x_M) &= \frac{\partial F_1}{\partial x_2}(x_{M,1}(x_M), x_M) \\
&= \frac{x_{M,1} x_M \left( 2\sqrt{1-x_{M,1}^2} \sqrt{x_M^2 - x_{M,1}^2} - \sqrt{1-x_M^2} \right) \left( \sqrt{1-x_{M,1}^2} + \sqrt{x_M^2 - x_{M,1}^2} \right)^2}{(x_M^2 - x_{M,1}^2)^{3/2} (1-x_M^2)^2} \\
&> \frac{x_{M,1} x_M \sqrt{1-x_{M,1}^2} \left( 2\sqrt{x_M^2 - x_{M,1}^2} - \sqrt{1-x_M^2} \right) \left( \sqrt{1-x_{M,1}^2} + \sqrt{x_M^2 - x_{M,1}^2} \right)^2}{(x_M^2 - x_{M,1}^2)^{3/2} (1-x_M^2)^2}.
\end{aligned}
\qquad (3.15) $$

As this map $u \mapsto 2\sqrt{u^2 - x_{M,1}(u)^2} - \sqrt{1-u^2}$ is strictly increasing on $I_c$, we find that (see case (ii) in Corollary Appendix A.2):

$$ 
\begin{aligned}
& 2\sqrt{x_M^2 - x_{M,1}^2} - \sqrt{1-x_M^2} \ge 2\sqrt{x_c^2 - x_{M,c}^2} - \sqrt{1-x_c^2} \\
&= \frac{(\sqrt{2}\sqrt{59\sqrt{7}-151}) - (\sqrt{3}\sqrt{19-7\sqrt{7}})}{\sqrt{6}} \approx 0.814 > 0. \quad (3.16)
\end{aligned}
$$

Using proven limits in (3.11), for all $\alpha > \frac{\pi}{2}$ we can extend the definition of $H_2(\alpha, \cdot)$ (respectively of $h_2(\alpha, \cdot)$) for $x_M = x_i(\alpha)$ (respectively for $\theta_M = \frac{\pi}{2} - \alpha$). By putting the real interval $\hat{I}_\alpha$ such that

$$ 
\hat{I}_\alpha = \begin{cases} 
[x_i(\alpha), x_{\text{int}}(\alpha)], & \text{if } \alpha < \frac{\pi}{2}, \\
[0, 1[, & \text{if } \alpha = \frac{\pi}{2}, \\
[0, 1[, & \text{else,}
\end{cases}
$$
by (1.37), (3.8), (3.10) and 3.14, we have for all $\alpha \in ]0, \pi[$:

$$ \hat{C}_1(\alpha) = \sqrt{1 + \inf_{x_M \in \hat{I}_\alpha} H_2(\alpha, x_M)^2}. \quad (3.17) $$

We assume that function $H_2$ satisfies the following properties:

$$ \forall \alpha \in ]0, \pi[, \quad H_2(\alpha, \cdot) \text{ is convex on } \hat{I}_\alpha, \quad (3.18) $$

and

$$ \forall x_M \in ]0, 1[, \quad \alpha \mapsto \frac{\partial K_2}{\partial x_M}(\alpha, x_M) \quad \text{is strictly increasing on } ]\arccos(x_M), \frac{\pi}{2}]. \quad (3.19) $$

These properties can be verified with multiple numerical simulations, so we can avoid all the complexity of these calculations.

Using properties (3.11) and (3.18), we obtain:

$$ \forall \alpha \in ]0, \pi[, \quad \exists! \hat{x}(\alpha) \in \hat{I}_{\alpha}; \quad \hat{C}_1(\alpha) = 1 + H_2(\alpha, \hat{x}(\alpha))^2. \quad (3.20) $$

Here $\hat{x}(\alpha)$ is the unique minimizer of $H_2(\alpha, \cdot)$ on $\hat{I}_\alpha$.

We define function slope on $]0, \frac{\pi}{2}[$ by

$$ \text{slope}(\alpha) = \frac{\partial K_2}{\partial x_M}(\alpha, x_{\text{int}}(\alpha)), \forall \alpha \in ]0, \frac{\pi}{2}]. $$

By properties (ii) from Lemma 3.1, (3.18) and (3.19), we conclude that $\alpha \mapsto \text{Slope}(\alpha)$ is strictly increasing on $[\alpha_0, \frac{\pi}{2}]$ where $\alpha_0$ is defined in Lemma 3.1 using equation (3.12). To study this function on $]0, \alpha_0]$, we re-scale it using the real bijection $W$, where

$$ W(x) = \begin{cases} x, & \text{if } -1 \le x \le 1, \\ \text{sign}(x) (\log_{10}(|x|) + 1), & \text{else.} \end{cases} $$

Performing a numerical simulation of the composed function $W \circ \text{Slope}$ on $[0.01, \alpha_0]$ (Figure 2), we conclude that $\text{Slope}$ is strictly increasing on $]0, \frac{\pi}{2}]$ and has a unique zero, which we denote by $\beta_0$.

Using Newton's method to solve the following system:

$$ \left\{ \begin{aligned} H_1(\alpha, x_M) &= H_2(\alpha, x_M), \\ \frac{\partial H_2}{\partial x_M}(\alpha, x_M) &= 0, \end{aligned} \right. \quad (3.21) $$

we find the unique solution $(\alpha, x_M)$ of the system above is equal to $(\beta_0, x_{\text{int}}(\beta_0)) \approx (0.619, 0.918)$ which is an approximation of order $10^{-3}$.

For all $\alpha \in ]0, \pi[$, we find that:

$$ \left\{ \begin{array}{ll} \hat{x}(\alpha) = x_{\text{int}}(\alpha), & \text{if } \alpha \le \beta_0, \\ \hat{x}(\alpha) = \underset{x_M \in \hat{I}_\alpha}{\operatorname{argmin}} H_2(\alpha, x_M) < x_{\text{int}}(\beta_0), & \text{else.} \end{array} \right. \quad (3.22) $$
<figure><img src="image_3.png" /><figcaption>Figure 2. The graph of $W \circ \text{Slope}$ for $\alpha \in [0.01, \alpha_0]$.</figcaption></figure>

Using (ii) of Lemma 3.1, (3.18) and (3.19) we find that:

$$ \hat{x} : \alpha \mapsto \hat{x}(\alpha) \text{ is decreasing on } ]0, \pi[. \qquad (3.23) $$

From now on, we define $\hat{\theta}(\alpha) := \arcsin(\hat{x}(\alpha))$. By solving numerically equations (3.22), we were able to perform a simulation of $\hat{\theta}$ (See Figure 3).

<figure><img src="image_4.png" /><figcaption>Figure 3. Graph of $\hat{\theta}$ on $]0, \pi[$.</figcaption></figure>

By the mean of (3.10) and (3.22), we have for all $\alpha \in ]0, \beta_0]$:

$$ \hat{C}_1(\alpha) = \sqrt{1 + H_1(\alpha, x_{\text{int}}(\alpha))^2}. \qquad (3.24) $$
We deduce from (3.14) and (ii) of Lemma 3.1, that $C_1(\alpha)$ is strictly decreasing on $]0, \beta_0]$. In addition, using (3.22), for all $\alpha \in ]\beta_0, \pi[$:

$$ \hat{C}_1(\alpha) = \sqrt{1 + H_2(\alpha, \hat{x}(\alpha))^2}. \qquad (3.25) $$

If we define

$$ \hat{H}(\alpha) := \begin{cases} H_1(\alpha, \hat{x}(\alpha)), & \text{if } \alpha \le \beta_0, \\ H_2(\alpha, \hat{x}(\alpha)), & \text{else,} \end{cases} \qquad (3.26) $$

we obtain

$$ \hat{C}_1 = \sqrt{1 + \hat{H}^2}. \qquad (3.27) $$

Taking into consideration in performed calculations below that $x_M = \hat{x}(\alpha)$ is a critical point of $H_2(\alpha, \cdot)$, we find that for all $\alpha \in ]\beta_0, \pi[$:

$$ \begin{aligned} \hat{H}'(\alpha) = \frac{\partial H_2}{\partial \alpha}(\alpha, \hat{x}(\alpha)) = & \frac{\partial F_2}{\partial x_0}(x_\alpha(x_M), x_{M,2}(\alpha, x_M), x_M) \frac{\partial x_\alpha}{\partial \alpha}(x_M) = \\ & - \frac{x_\alpha (\sqrt{x_M^2 - x_{M,2}^2} + \sqrt{1 - x_{M,2}^2}) (x_{M,2} - x_\alpha) (2x_M^2(1-x_\alpha^2) + x_M^2 - x_\alpha^4)}{(1-x_\alpha^2)^{3/2} (x_M^2 - x_\alpha^2)^{5/2}} \times \\ & \frac{x_M (x_M - \cos \alpha) (1 - x_M \cos \alpha)}{(x_M^2 - 2 \cos \alpha x_M + 1)^{3/2}} < 0, \quad (3.28) \end{aligned} $$

which implies that $\hat{C}_1$ is strictly decreasing on $]0, \pi[$.

For all $\alpha \in ]0, \pi[$, we make the change of variables $\hat{\theta}(\alpha) = \arcsin(\hat{x})$.

For all $\alpha \in ]\pi/2, \pi[$, using (A.45) in $\frac{\partial H_2}{\partial x_M}$ formula, we find that:

$$ \frac{\partial H_2}{\partial x_M}(\alpha, x_M) \rightarrow D(\alpha), \qquad (3.29) $$

where

$$ D(\alpha) := \frac{-\sin^3 \alpha - 4 \sin^2 \alpha + \sin \alpha + 1}{(1 - \sin \alpha) (1 + \sin \alpha)^2}. \qquad (3.30) $$

We notice that function $D: \alpha \mapsto D(\alpha)$ is strictly increasing, moreover it has a unique zero $\gamma_0 = \pi - \arcsin(x_0) \approx 2.512$ on $]\pi/2, \pi[$. Here $x_0 = 2 \cos \left( \frac{\arccos\left(\frac{-137\sqrt{19}}{722}\right)}{3} \right) \sqrt{\frac{19}{9}} - \frac{4}{3}$ is the only root of $P = x^3 + 4x^2 - x - 1$ between 0 and 1. Using (3.23), we conclude that $\forall \alpha \in [\gamma_0, \pi[$:

$$ \left\{ \begin{array}{l} \hat{x}(\alpha) = 0, \\ \\ \hat{C}_1(\alpha) = \sqrt{1 + H_2(\alpha, 0)^2} = \frac{(\sin \alpha - 1) \sqrt{3 \sin^2 \alpha + 4 \sin \alpha + 2}}{\cos^3 \alpha}, \end{array} \right. \qquad (3.31) $$

and because $H_2(\alpha, \cdot)$ is strictly increasing, we find that function $\hat{C}_1: \alpha \mapsto \hat{C}_1(\alpha)$ is strictly decreasing on $]0, \pi[$ and it goes to $\sqrt{2}$ as $\alpha$ goes to $\pi$.
From now on, for all $\alpha \in ]0, \pi[$ we denote by $\hat{\psi}_{\alpha}$ the point-wise limit of $(\psi_{\alpha, \theta_M})_{\theta_i(\alpha) < \theta_M < \frac{\pi}{2}}$ as $\theta_M \to \hat{\theta}(\alpha)^+$. If we put $\theta_M = \hat{\theta}(\alpha)$, we find that:

$$ \hat{\psi}_{\alpha} = \begin{cases} \psi_{\alpha, \theta_M} = f_{\alpha, \theta_M}(\theta) - r, & \text{if } \alpha < \gamma_0, \\ \tilde{\psi}_{\alpha}, & \text{else.} \end{cases} \qquad (3.32) $$

where $\forall (x, y) \in \bar{\Omega}$,

$$ \hat{\psi}_{\alpha}(x, y) = \begin{cases} \sqrt{1-y^2} - x, & \text{if } |y| \le \sin \alpha, \\ -\frac{-y^2 + 2 |y| \sin^3 \alpha + 2 x \cos^3 \alpha + 3 \cos^2 \alpha - 1}{2 \cos^3 \alpha}, & \text{else.} \end{cases} \qquad (3.33) $$

Finally, we define $\hat{C}_2 := C_{2, \hat{\psi}_{\alpha}}$, which completes the proof of the major result of our Theorem.

### 3.2. asymptotic behavior of $\hat{C}_1$

In this subsection, we will find an asymptotic expansion of $\alpha \mapsto \hat{C}_1(\alpha)$ near zero. We recall that from (3.10) and (3.24) that for all $0 < \alpha < \beta_0$:

$$ \hat{C}_1(\alpha) = \sqrt{1 + H_2(\alpha, x_{\text{int}}(\alpha))^2} = \sqrt{1 + H_1(x_{\text{int}}(\alpha))^2}, \qquad (3.34) $$

so, to find a power series of $\hat{C}_1$ near zero, it is sufficient to study the equation $H_1(x_{\text{int}}(\alpha)) = H_2(\alpha, x_{\text{int}}(\alpha))$ near zero. For this reason, we put:

$$ x_{\text{int}}(\alpha) = \cos \alpha + c_i(\alpha) \alpha^3 < x_f(\alpha), \qquad (3.35) $$

where $\alpha \mapsto c_i(\alpha)$ is necessarily positive on $]0, \beta_0[$ because $x_i(\alpha) = \cos(\alpha) < \hat{x}(\alpha)$. We will prove that $c_i$ is also bounded on $]0, \beta_0[$ by the mean of power expansions near zero. Using (A.12) and (A.20), we find that:

$$ \begin{cases} x_{M,1}(x_{\text{int}}(\alpha)) = \frac{\sqrt{2} (12 c_i^2 + 7) \alpha^6}{24} - \frac{\sqrt{2} c_i \alpha^5}{2} + \frac{\sqrt{2} \alpha^4}{8} + \frac{\sqrt{2}}{2} + o(\alpha^6), \\ x_{M,2}(x_{\text{int}}(\alpha)) = x_M - \left(\frac{9 c_i^4}{8}\right) \alpha^6 - \left(\frac{15 c_i^5}{4}\right) \alpha^7 - \left(\frac{c_i^4 (149 c_i^2 - 9)}{16}\right) \alpha^8 + o(\alpha^8). \end{cases} \qquad (3.36) $$

It follows that

$$ H_1(x_{\text{int}}(\alpha)) = \frac{2}{\alpha^2} + \frac{4 c_i}{\alpha} + \left(8 c_i^2 + \frac{2}{3}\right) + \left(16 c_i^3 + \frac{2 c_i}{3}\right) \alpha + \left(32 c_i^4 + 2 c_i^2 + \frac{19}{30}\right) \alpha^2 \\ \left(64 c_i^5 + \frac{16 c_i^3}{3} - \frac{107 c_i}{90}\right) \alpha^3 + \left(128 c_i^6 + \frac{40 c_i^4}{3} + \frac{4 c_i^2}{15} + \frac{323}{378}\right) \alpha^4 + o(\alpha^4), \qquad (3.37) $$
and

$$ H_2(\alpha, x_{int}(\alpha)) = \frac{3}{2c_i\alpha^2} + \frac{1}{\alpha} + \frac{19c_i^2 - 2}{8c} + \left(\frac{23c_i^2}{4} - \frac{1}{3}\right)\alpha + \frac{1120c_i^4 + 15c_i^2 + 1}{80c_i}\alpha^2 \\ \left(\frac{269c_i^4}{8} + \frac{5c_i^2}{12} - \frac{1}{45}\right)\alpha^3 + \frac{1059345c_i^6 + 22120c_i^4 + 2282c_i^2 - 4}{13440c_i}\alpha^4 + o(\alpha^4). \quad (3.38) $$

Using (3.37)-(3.38), and that $H_1(x_{int}(\alpha)) = H_2(\alpha, x_{int}(\alpha))$, we find that:

$$ c_i(\alpha) = \frac{3}{4} \left( -\frac{31190743\alpha^6}{6193152} + \frac{50529\alpha^5}{20480} - \frac{10179\alpha^4}{10240} + \frac{5\alpha^3}{192} + \frac{41\alpha^2}{64} - \alpha + 1 \right) + o(\alpha^6), $$

so

$$ x_{int}(\alpha) = \frac{209\alpha^6}{11520} + \frac{123\alpha^5}{256} - \frac{17\alpha^4}{24} + \frac{3\alpha^3}{4} - \frac{\alpha^2}{2} + 1 + o(\alpha^6). \quad (3.39) $$

We conclude that:

$$ \hat{C}_1(\alpha) = \hat{C}_{1,0}(\alpha) + o(\alpha^4), \quad (3.40) $$

where

$$ \hat{C}_{1,0}(\alpha) := \frac{2}{\alpha^2} + \frac{3}{\alpha} + \frac{13}{6} + \frac{11\alpha}{64} + \frac{829\alpha^2}{480} + \frac{47603\alpha^3}{30720} + \frac{5689727\alpha^4}{1548288}. \quad (3.41) $$

Here, we plot the figure of $\hat{C}_1$ to illustrate its significance (Figure 4). In fact, this

<figure><img src="image_5.png" /><figcaption>Figure 4. Graph of $\hat{C}_1$ on $[10^{-3}, \pi]$</figcaption></figure>

representation of $\hat{C}_1$ is crucial, as $\hat{C}_1$ is the most important function in this work. Theorem 1.6 is based on this function, which represents the infimum over $\theta_M$ of the maximum gradient of the functions $\psi_{\alpha,\theta_M}$. Moreover, even $\hat{C}_2$ is constructed from $\hat{C}_1$. Specifically, $\hat{C}_1$ is defined as

$$ \hat{C}_1 := \inf(C_{1,\psi}), \quad (3.42) $$
where $\psi = \psi_{\alpha, \theta_M}$ (see Equation (1.36) and the preceding one). Furthermore, the term $C_{1,\psi}$ is cubic in $c_{obs}$ from the optimized Carleman inequality (1.1) in the first theorem. On the other hand, following Equations (3.40) and (3.41), it is clear that $\hat{C}_{1,0}$ represents the asymptotic behavior of $\hat{C}_1$. As for $\gamma_0$, it explicitly appears in the formula for $\hat{C}_1$ within the system of equations (3.31), which defines both $\hat{x}$ and $\hat{C}_1$ over the interval $]\gamma_0, \pi[$. This interval represents approximately 20% of the range of variation of $\alpha$, as indicated in the conclusion. Finally, $\sqrt{2}$ corresponds to the minimum value of $\hat{C}_1$, which can be easily obtained by substituting $\alpha = \pi$ into the expression for $\hat{C}_1$ in Equation (3.31). This highlights the importance of plotting this figure.

### 3.3. Behavior of $\hat{C}_2$

We recall that

$$ \hat{C}_2(\alpha) := 2 \left( \max \left( \|\partial_x^2 \hat{\psi}_\alpha\|_\infty, \|\partial_y^2 \hat{\psi}_\alpha\|_\infty \right) + \|\partial_{x,y} \hat{\psi}_\alpha\|_\infty \right) + \|\partial_x^2 \hat{\psi}_\alpha\|_\infty + \|\partial_y^2 \hat{\psi}_\alpha\|_\infty. $$

For all $\alpha \in ]0, \pi[$, we find that if $\alpha < \gamma_0$:

$$ \left\{ \begin{aligned} \partial_x^2 \hat{\psi}_\alpha(P) &= -\frac{\sin^2 \theta}{r} + \frac{2 \sin \theta \cos \theta}{r^2} f'_{\alpha, \theta_M}(\theta) + \frac{\sin^2 \theta}{r^2} f''_{\alpha, \theta_M}(\theta) \le 0, \\ \partial_y^2 \hat{\psi}_\alpha(P) &= -\frac{\cos^2 \theta}{r} - \frac{2 \sin \theta \cos \theta}{r^2} f'_{\alpha, \theta_M}(\theta) + \frac{\cos^2 \theta}{r^2} f''_{\alpha, \theta_M}(\theta) < 0, \\ \partial_x \partial_y \hat{\psi}_\alpha(P) &= \frac{\sin \theta \cos \theta}{r} + \frac{2 \sin^2 \theta - 1}{r^2} f'_{\alpha, \theta_M}(\theta) - \frac{\sin \theta \cos \theta}{r^2} f''_{\alpha, \theta_M}(\theta) \ge 0, \end{aligned} \right. \quad (3.43) $$

else

$$ \left\{ \begin{aligned} \partial_x^2 \hat{\psi}_\alpha(P) &= \partial_x \partial_y \hat{\psi}_\alpha(P) = 0, \\ \partial_y^2 \hat{\psi}_\alpha(P) &= \begin{cases} \frac{1}{\sqrt{1-y^2}}, & \text{if } |y| \le \sin \alpha, \\ \frac{1}{\cos^3 \alpha}, & \text{else,} \end{cases} \end{aligned} \right. \quad (3.44) $$

where $P(r, \theta) = (x, y) \in \bar{\Omega}$ and $(r, \theta)$ are polar coordinates of $P$ with respect to $\mathcal{R}'$.

We find also that all second order derivatives of $\hat{\psi}_\alpha$ reach their maximum only if $r = r_{min}(., \alpha)$, that is on $\Gamma_{0,1}^+ \cup \Gamma_{0,2}^+$. For all $0 < \alpha < \gamma_0$ and for all $\theta \in [0, \theta_M(\alpha)]$ we put:

$$ \left\{ \begin{aligned} \hat{\psi}_{1,\alpha}(\theta) &:= -\partial_x^2 \hat{\psi}_\alpha(P(r_{min}(\theta, \alpha), \theta)), \\ \hat{\psi}_{2,\alpha}(\theta) &:= -\partial_y^2 \hat{\psi}_\alpha(P(r_{min}(\theta, \alpha), \theta)), \\ \hat{\psi}_{3,\alpha}(\theta) &:= \partial_x \partial_y \hat{\psi}_\alpha(P(r_{min}(\theta, \alpha), \theta)). \end{aligned} \right. \quad (3.45) $$

We prove that there exists a triplet $(\alpha_1, \alpha_2, \alpha_3) \in ]0, \frac{\pi}{2}]^3$ such that for all $0 < \alpha < \gamma_0$:
(i) For $i \in \{1, 3\}$

$$ \|\psi_{i,\alpha}\|_{\infty} = \begin{cases} \psi_{i,\alpha}(\theta_{\alpha}), & \text{if } \alpha \le \alpha_i, \\ \psi_{i,\alpha}(\theta_{i,\alpha}), & \text{else,} \end{cases} \qquad (3.46) $$

where $\alpha_i$ is the unique root of function $\alpha \mapsto \psi_{i,\alpha'}(\theta_\alpha)$ (right derivative) and $\theta_{i,\alpha}$ is the maximizer of $\psi_{i,\alpha}$ on $]\theta_\alpha, \theta_M(\alpha)]$.

(ii)

$$ \|\psi_{2,\alpha}\|_{\infty} = \begin{cases} \psi_{2,\alpha}(0) = \frac{-\sin^2 \theta_M + 2 \sin \theta_M + 1}{(\sin \theta_M - 1)^2}, & \text{if } \alpha \le \alpha_2, \\ \psi_{2,\alpha}(\theta_\alpha), & \text{else,} \end{cases} \qquad (3.47) $$

where $\alpha_2$ the unique root of equation $\psi_{2,\alpha}(0) = \psi_{2,\alpha}(\theta_\alpha)$.

Here $(\alpha_1, \alpha_2, \alpha_3) \approx (1.419, 1.357, 1.808)$.

Then we find that:

$$ \max \left( \|\partial_x^2 \hat{\psi}_\alpha\|_\infty, \|\partial_y^2 \hat{\psi}_\alpha\|_\infty \right) = \begin{cases} \psi_{1,\alpha}(\theta_\alpha), & \text{if } \alpha \le \beta_1, \\ \psi_{2,\alpha}(\theta_\alpha), & \text{if } \beta_1 \le \alpha < \gamma_0, \\ -\frac{1}{\cos^3 \alpha}, & \text{else.} \end{cases} \qquad (3.48) $$

Here $\beta_1 \approx 1.1381$ is the unique solution of equation $\psi_{1,\alpha}(\theta_\alpha) = \psi_{2,\alpha}(\theta_\alpha)$. It follows from expressions in (3.46), (3.47) and (3.48) that $\hat{C}_2$ is strictly decreasing (see Figure 5). Especially, using (3.39) we obtain:

$$ \hat{C}_2(\alpha) \sim \hat{C}_{2,0}(\alpha) := \frac{2^6}{9\alpha^6}, \qquad (3.49) $$

and for all $\alpha \in [\gamma_0, \pi[$:

$$ \hat{C}_2(\alpha) = -\frac{3}{\cos^3 \alpha} \qquad (3.50) $$

Here, we provide a brief description of Figure 5 to make it more readable. Just like the variable $\hat{C}_1$, the function $\hat{C}_2$ also plays a crucial role, as it is constructed from the function $\hat{C}_1$. In fact, By its definition in Theorem 1.6, we have:

$$ \hat{C}_2(\alpha) := C_{2,\hat{\psi}_\alpha}, \qquad (3.51) $$

where $\hat{\psi}_\alpha$ is the optimal weight function, and $C_{2,\hat{\psi}_\alpha}$ corresponds to $C_{2,\psi}$ (defined by the second equation in system (??)) for $\psi = \hat{\psi}_\alpha$. This quantity is linked to the minimal value of $\lambda$, denoted by $\underline{\lambda}$, from the first theorem 1.1 (which ensures the validity of the Carleman estimate) through the relation:

$$ \underline{\lambda}(\psi) := \frac{C_{2,\psi}}{\mu(\psi)^2} \quad (\text{see the last equation in (??)}). \qquad (3.52) $$
<figure><img src="image_6.png" /><figcaption>Figure 5. Graph of $\hat{C}_2$ on $[10^{-3}, \pi[$.</figcaption></figure>

In our case, for the class of weight functions $C_\alpha := (\psi_{\alpha,\theta_M})_{\theta_M}$, we have $\mu(\psi) := \mu(\alpha) = 1$ for all $\alpha$ (see property (ii) of Theorem 1.6). Consequently:

$$ \underline{\lambda}(\psi) = C_{2,\psi} = \hat{C}_2. \qquad (3.53) $$

Thus, the Carleman constants depend on $\hat{C}_2(\alpha)$, as it represents the minimal value of $\lambda$. Moreover, when $\alpha$ approaches zero, we explicitly observe the dependence of the Carleman constants on $\alpha$, as demonstrated by the first and the second equations in Corollary 1.9. Also, we plotted the function $\alpha \mapsto \frac{2^6}{9\alpha^6}$ (formula (3.50)), which we denoted by $\hat{C}_{2,0}$. This highlights the importance of plotting its graph.

## 4. Proof of Theorem 1.11

In order to estimate constants in Theorem 1.11 as optimal as possible, we state this particular case of Carleman estimate (1.1)).

**Lemma 4.1.** *Using the same notations in Proposition 1.1, we obtain for all $\lambda > \underline{\lambda}(\psi)$ such that $\lambda \ge \tilde{\lambda}'$:*

$$ \int_Q \left( \frac{|\hat{\partial}_t z|^2}{(s_0(\lambda)\varphi)^{-1}} + 4\lambda^4((s_0(\lambda)\varphi)^3|z|^2) \right) e^{-2s_0(\lambda)\eta_{\beta(\lambda)}} dx dt \le c'_{\mathcal{L}} \int_Q |\mathcal{L}z(x,t)|^2 e^{-2s_0\eta_0(x)} dx dt \\ + c'_{\text{obs}} \int_{\Sigma_0} (|z(x,t)|^2 + |\nabla z(x,t)|^2) e^{-2s_0\eta_0(x)} d\Sigma, \quad (4.1) $$
where

$$
\left\{
\begin{aligned}
s_0(\lambda) &:= s_{\beta(\lambda)}(1, \lambda), \\
\beta(\lambda) &:= 1 + \frac{3\mu^4 T}{16 \|\psi\|_{\infty}^2} e^{-\lambda \|\psi\|_{\infty}}, \\
c'_{\mathcal{L}}(\lambda) &:= \frac{8}{(1 - e^{-\lambda \|\psi\|_{\infty}})} \left( \frac{1}{\underline{\epsilon}(\lambda)} + 3 e^{-\lambda \|\psi\|_{\infty}} \right), \\
c'_{\text{obs}}(\lambda) &:= c_T c'_{\mathcal{L}}(\lambda) (1 + \epsilon_2(\lambda)) (\lambda s_0(\lambda) e^{\lambda \|\psi\|_{\infty}})^3 C_{1,\psi}^3, \\
c_T &:= \left(\frac{2}{T}\right)^2 \max\left(\frac{2}{T}, 1\right)^4, \\
\eta_0(\lambda, x) &:= \eta_{\beta(\lambda)}(x, \frac{T}{2}), \quad \forall x \in \Omega, \\
\tilde{\lambda}' &:= \frac{2}{\|\psi\|_{\infty}} \log (M'(I, \psi, T)), \\
M'(I, \psi, T) &:= \max \left( \frac{3\mu^2(1 + 3\|\psi\|_{\infty})}{2\|\psi\|_{\infty}^3} T, \frac{\sqrt{\|I\|_{\infty}\mu}}{2\|\psi\|_{\infty}^2} T^{\frac{1}{2}} \right).
\end{aligned}
\right.
\quad (4.2)
$$

The proof of this lemma is similar to the one of Theorem 1.1.

As the stability part of Theorem 1.11 is already proved in [22], we will only explain a few details in order to keep tracking constants. By (1.56), the function $z(x, t) := \partial_t y(x, t)$ satisfies the following system of equations:

$$
\left\{
\begin{array}{ll}
\partial_t z - \Delta z = \partial_t g(x, t), & \text{in } Q, \\
\\
\nabla z \cdot \nu + I(x) z = 0, & \text{on } \Sigma, \\
\\
z(t_0, \cdot) = \Delta y_0 + g(\cdot, t_0), & \text{in } \Omega.
\end{array}
\right.
\qquad (4.3)
$$

where we put $t_0 := \frac{T}{2}$.

Using Carleman estimate (4.1) in Lemma 4.1 for $z := \partial_t y$ and inequality (1.57), we obtain for all $\lambda > \bar{\lambda}$ such that $\lambda \ge \tilde{\lambda}'$:

$$
\begin{align}
I_1 &:= \int_Q \left\{ (s_0(\lambda)\varphi)^{-1} |\partial_t z|^2 + 4\lambda^4 (s_0(\lambda)\varphi)^3 |z|^2 \right\} e^{-2s_0(\lambda)\eta_{\beta(\lambda)}} dx dt \nonumber \\
&\le C_g c'_{\mathcal{L}} \int_Q |g(x, t_0)|^2 e^{-2s_0\eta_0(x)} dx dt + c'_{\text{obs}} \int_{\Sigma_0} (|z(x, t)|^2 + |\nabla z(x, t)|^2) e^{-2s_0\eta_0(x)} d\Sigma. \tag{4.4}
\end{align}
$$
On the other hand, using $z(x, 0)e^{-2s_0\eta_\beta(x,0)} = 0$, we find that:

$$ 
\begin{aligned}
\int_{\Omega} s_0 z(x, t_0)^2 e^{-2s_0\eta_0(x)} dx &= \int_0^{t_0} \left( \partial_t \int_{\Omega} s_0 z(x, t)^2 e^{-2s_0\eta_\beta(x,t)} dx \right) dt \\
&\le \int_Q 2s_0^2 |\partial_t\eta| z^2 e^{-2s_0\eta_\beta} dxdt + \int_Q 2 \left( \frac{\partial_t z e^{-s_0\eta_\beta}}{\sqrt{s_0\varphi}} \right) \left( s_0^{\frac{3}{2}} \varphi^{\frac{1}{2}} z e^{-s_0\eta_\beta} \right) dxdt \le C_{1,T} I_1
\end{aligned} 
\quad (4.5) $$

where

$$ C_{1,T}(\lambda) := \left(\frac{T}{2\lambda}\right)^2 (1 + \mu^2 e^{-\lambda\|\psi\|_{\infty}}), $$

Using the last equation in (4.3), we find that:

$$ \int_{\Omega} g(x, t_0)^2 e^{-2s_0 \eta_0(x)} dx \le 2 \int_{\Omega} (z(x, t_0)^2 + |\Delta y_0(x)|^2) e^{-2s_0 \eta_0(x)} dx, \quad (4.6) $$

Also we find that for all $x \in \Omega$,

$$ \int_0^T e^{-2s_0 \eta(x,t)} dt \le \sqrt{\frac{2\pi}{(\beta(\lambda) - 1)s_0(\lambda)}} t_0^3 e^{-\frac{\lambda}{2}\|\psi\|_{\infty}} e^{-2s_0(\lambda)\eta_0(\lambda,x)}. \quad (4.7) $$

Using (4.7) in (4.4), from (4.5) and (4.6), we deduce that for all $\lambda > \bar{\lambda}$ such that $\lambda \ge \tilde{\lambda}$:

$$ 
\begin{aligned}
\int_{\Omega} g(x, t_0)^2 e^{-2s_0 \eta_0} dx \le C'_{g}(\lambda) \int_{\Omega} g(x, t_0)^2 e^{-2s_0 \eta_0} dx \\
+ C_{\Sigma_0}(\lambda) \int_{\Sigma_0} (|z|^2 + |\nabla z|^2) e^{-2s_0 \eta_0} dSdt + 2 \int_{\Omega} |\Delta y_0|^2 e^{-2s_0 \eta_0} dx, \quad (4.8)
\end{aligned} $$

where

$$ 
\left\{ 
\begin{aligned}
C'_{g}(\lambda) &:= \frac{2 t_0^3 C_{1,T}(\lambda) C_g c'_{\mathcal{L}}(\lambda)}{s_0(\lambda)^{\frac{3}{2}}} \sqrt{\frac{2\pi}{\beta(\lambda) - 1}} e^{-\frac{\lambda}{2}\|\psi\|_{\infty}}, \\
C_{\Sigma_0}(\lambda) &:= \frac{2}{s_0(\lambda)} C_{1,T}(\lambda) c'_{\text{obs}}(\lambda).
\end{aligned} 
\right. 
\quad (4.9) $$

Finally, if we put $\tilde{\lambda} = \max \left( 1, \tilde{\lambda}', \frac{2}{\|\psi\|_{\infty}} \log \left( \frac{8(1 + \mu^2) \|\psi\|_{\infty} C_g}{\mu^2 (1 - e^{-\|\psi\|_{\infty}})} \sqrt{\frac{2\pi}{3} T^{\frac{3}{2}}} \right) \right)$, we find that for all $\lambda > \bar{\lambda}(\psi)$ such that $\lambda \ge \tilde{\lambda}$:

$$ 0 < 1 - e^{-\lambda \|\psi\|_{\infty}} \le 1 - C'_{g}(\lambda), \quad (4.10) $$

and from (4.7) and the following inequality

$$ \frac{(\beta(\lambda) - 1) e^{\lambda \|\psi\|_{\infty}}}{t_0^2} \le \eta_0 \le \frac{\beta(\lambda) e^{\lambda \|\psi\|_{\infty}} - 1}{t_0^2}, \quad (4.11) $$

we conclude that:

$$ \|g\|_{L^2(Q)} \le c_0(\lambda) \|\Delta y_0\|_{L^2(\Omega)} + c_{\Sigma_0}(\lambda) \left( \|(\partial_t y)|_{\Sigma_0}\|_{L^2(\Sigma_0)} + \|(\nabla \partial_t y)|_{\Sigma_0}\|_{L^2(\Sigma_0)} \right), \quad (4.12) $$
where

$$
\left\{
\begin{aligned}
c_0(\lambda) &:= \frac{2}{1 - e^{-\lambda \|\psi\|_\infty}} \exp \left( \frac{2s_0(\lambda)}{t_0^2} (e^{\lambda \|\psi\|_\infty} - 1) \right), \\
c_{\Sigma_0}(\lambda) &:= \frac{C_{\Sigma_0}(\lambda)}{1 - e^{-\lambda \|\psi\|_\infty}} \exp \left( \frac{2s_0(\lambda)}{t_0^2} (e^{\lambda \|\psi\|_\infty} - 1) \right).
\end{aligned}
\right.
\qquad (4.13)
$$

Here the optimal choice of $\lambda \ge \tilde{\lambda}$ would be the one that minimizes $\lambda \mapsto c_{\Sigma_0}(\lambda)$ which has necessarily a unique minimizer $\lambda_0 \in ]\tilde{\lambda}, +\infty[ \cap ]\tilde{\lambda}, +\infty[$.

In the particular case where we choose the triplet $(\Omega, (\Gamma_0)_{0<\alpha<\pi}, (\hat{\psi}_\alpha)_{0<\alpha<\pi})$ as in Theorem 1.6, from (1.39) and corollary 1.9 we find that for all $T > 0$:

$$
\forall \lambda > \underline{\lambda}(\alpha), c_0(\lambda), c_{\Sigma_0}(\lambda) \underset{\alpha \to 0^+}{=} o\left(\exp\left(\frac{2^4 \beta}{T} \times \frac{e^{6\lambda}}{\lambda(\lambda - \underline{\lambda})}\right)\right). \quad (4.14)
$$

Here, the simplest choice of $\lambda$ is the one that minimizes the function $\lambda \mapsto \frac{e^{6\lambda}}{\lambda(\lambda - \underline{\lambda})}$ over $]\hat{\lambda}, +\infty[$. We find that this minimizer is

$$
\lambda_0(\alpha) = \frac{3\underline{\lambda}}{4} + \frac{\sqrt{9\underline{\lambda}^2 + 2\underline{\lambda} + 1}}{4} + \frac{1}{4},
$$

hence

$$
\frac{2^4 \beta(\lambda_0)}{T} \times \frac{e^{6\lambda_0}}{\lambda_0 (\lambda_0 - \underline{\lambda})} \underset{\alpha \to 0^+}{\sim} \frac{2^6 e^{\frac{3}{4}} e^{9\underline{\lambda}}}{3T \underline{\lambda}^2} \quad (4.15)
$$

As $\underline{\lambda} = \hat{C}_2(\lambda)$ by definition (\ref{eq:optimalConstant}) of $\underline{\lambda}$, from (3.49), (4.14) and (4.15) we conclude that:

$$
c_0(\lambda_0), c_{\Sigma_0}(\lambda_0) \underset{\alpha \to 0^+}{=} o\left( \exp\left( \frac{\exp\left(\frac{(2^6)^+}{\alpha^6}\right)}{T} \right) \right). \quad (4.16)
$$

For all $\alpha \in ]0, \pi[$ and $T > 0$ small enough we have:

$$
\forall \lambda > \underline{\lambda}(\alpha), \frac{2s_0(\lambda)}{t_0^2} (e^{2\lambda} - 1) \underset{T \to 0^+}{=} \frac{2^4 (1 + 2e^{2\lambda}) e^{2\lambda}}{\lambda(\lambda - \underline{\lambda})T} (e^{2\lambda} - 1) + O(1). \quad (4.17)
$$

which implies that:

$$
c_0(\lambda_0), c_{\Sigma_0}(\lambda_0) \underset{T \to 0^+}{=} o\left(\exp\left(\frac{\mathcal{R}(\alpha)}{T}\right)\right), \quad (4.18)
$$

where $\mathcal{R}(\alpha) = \min_{\lambda > \underline{\lambda}(\alpha)} f(\lambda)$ and $f : ]\underline{\lambda}(\alpha), +\infty[$ is a real valued function defined by:

$$
f(\lambda) = \frac{2^4 (1 + 2 e^{2\lambda}) e^{4\lambda}}{\lambda(\lambda - \underline{\lambda})}. \qquad (4.19)
$$

We find that $\mathcal{R}(\alpha) = f(\lambda_0)$ such that $\lambda_0$ is the unique minimizer of $f$ verifying:

$$
(i) \quad \lambda_0(\alpha) \in ]\underline{\lambda}(\alpha), \underline{\lambda}(\alpha) + \frac{1}{2}[,
$$

(ii) $\lambda_0$ is the unique solution to the following equation:

$$
(4\lambda^2 - (4\underline{\lambda} + 2)\lambda + \underline{\lambda})(3 + e^{-2\lambda}) = -(2\lambda - \underline{\lambda}), \quad (4.20)
$$

which completes the proof of Theorem 1.11.
5. Discussion and Conclusions

In this work, we where able to write Carleman estimate (1.1) constants $c_{\mathcal{L}}$ and $c_{\text{obs}}$ in
terms of the weight function $\psi$ and $T$ in an optimal way, so they would be minimal as
possible. The sense of optimality here is not absolute, it is related to the established
procedure by O. Yu Imanuvilov and A. V. Fursikov [18, 20] to prove similar Carleman
estimates. As any $\psi$ verifying properties (0.12) depends on $\Gamma_0$ and $\Omega$, and because
it is not unique, we have to choose the one that minimizes $c_{\text{obs}}$ where it depends
on $C_{1,\psi} := \max_{x \in \Gamma_0} |\nabla\psi(x)|$. The problem of finding an absolute Carleman constant is
challenging and very complex. To simplify the problem, we studied the particular
case where $\Omega$ is the unit disc in $\mathbb{R}^2$ and $\Gamma_0 = \Gamma_0(\alpha)$ is an arc of the circle $\partial\Omega$ that
measures $0 < 2\alpha < 2\pi$. For each $\alpha \in ]0, \pi[$, we constructed a class of weight functions
$C_\alpha = (\psi_{\alpha,\theta_M})_{\theta_i(\alpha)<\theta_M<\frac{\pi}{2}}$ such that each one verifies properties (0.12) for the couple
$(\Omega, \Gamma_0(\alpha))$. Here $\theta_i(\alpha) = \max(0, \frac{\pi}{2} - \alpha)$.

After that, we chose the optimal $\psi \in C_\alpha$ that minimizes $C_{1,\psi}$ and we denoted it $\hat{\psi}_\alpha$. We
found that $\hat{\psi}_\alpha$ is the point-wise limit of the class of functions $C_\alpha$ when $\theta_M$ goes to some
parameter $\hat{\theta}(\alpha)$ between $\theta_i(\alpha)$ and $\frac{\pi}{2}$. Moreover, for all $\alpha \in [\gamma_0, \pi[$ where $\gamma_0 \approx 2.512$ we
proved that $\hat{\theta}(\alpha) = 0$, hence we have an explicit expression of $\hat{\psi}_\alpha$ (see eq.(3.33)). Even
if $\alpha < \gamma_0$, we are still able to numerically compute $\hat{\theta}(\alpha)$ by solving equations (3.22)
(See Figure 3). Also we were able to predict the asymptotic behavior of of all previous
constants when $\alpha$ goes to 0 (see Theorem 1.6, Corollary 1.9 and Theorem 1.11).

The most valuable results of this work, is that in 20% of all cases (when $\gamma_0 \le \alpha < \pi$),
we were able to write all constants of Carleman estimate (1.1) and stability ones of the
inverse source problem (1.56) in terms of $\alpha$ and $T$ only. In the remaining cases which
represent 80% of all angles $\alpha \in ]0, \pi[$, we were able to numerically compute $\hat{C}_1(\alpha)$ and
$\hat{C}_2(\alpha)$ (See figures 4-5), which leads to the computation of all Carleman constants and
stability ones of the inverse source problem. Also we were able to study the asymptotic
behavior of these constants when the size of the observation boundary is very small ($\alpha$
goes to 0) or when $T$ goes to 0. We conclude that inverse problem stability constants
are of order $\exp\left(\frac{\mathcal{R}(\alpha)}{T}\right)$ where $\mathcal{R}(\alpha)$ is of order $\frac{1}{\alpha^6}$ when $\alpha \to 0$, and we can also
explicitly calculate this geometric constant following the detailed procedure in Theorem
1.11 when $T \to 0$.

We notice the last two results ($\alpha, T \to 0$) and established costs of controllability and
observability in many works [16, 19, 21, 23] are similar in their form of dependency on
$T$. Some similarities come from the fact that when establishing these results they have
used Carleman estimates. So we can apply our result in previous ones to see the effect
of changing the size of boundary measurements on the cost of control for some parabolic
systems.

In the future, we hope to improve our result by taking into consideration a wider class
of weight functions that is dense in the space of admissible weight functions, to try to
identify the absolute optimal Carleman constants.
Acknowledgements

This work has been supported by SPIMCY cooperative research program, funded by INRIA international laboratory LIRIMA. The LAMSIN researcher's work is supported on a regular basis by the Tunisian Ministry of Higher Education, Scientific Research and Technology. This work was also supported by the Agence Nationale de la Recherche (Grant IHU LIRYC ANR-10-IAHU-04).

Appendix A. Proof of Lemma 3.1

We begin this proof by determining $h_{M,1}$.

First we put this change of variables $(x, x_M) := (\sin(\theta), \sin(\theta_M))$, then we have for all $\alpha \in ]0, \pi[$:

$$h_{\alpha, \theta_M}(\theta) = H_{1,x_M}(x) := F_1(x, x_M); \quad \forall (\theta, \theta_M) \in \mathcal{D}_{\alpha,1}, \qquad (A.1)$$

where

$$\left\{ \begin{array}{l} F_1(x_1, x_2) := \frac{x_1 \left( \sqrt{x_2^2 - x_1^2} + \sqrt{1-x_1^2} \right)}{\sqrt{1-x_1^2} \sqrt{x_2^2 - x_1^2 + x_1^2 - x_2^2}}, \quad \forall 0 \le x_1 < x_2, \\ \mathcal{D}_{\alpha,1} := \left\{ (\theta, \theta_M) \in \mathcal{D}_{\alpha}; \ 0 \le \theta \le \theta_{\alpha}(\theta_M) \right\} \end{array} \right.$$

(See equation (1.9) for the definition of $\mathcal{D}_{\alpha}$).

We put

$$\left\{ \begin{array}{l} x_{\alpha}(x_M) := \sin(\theta_{\alpha}(\theta_M)) = \frac{x_M \sin \alpha}{\sqrt{x_M^2 - 2 \cos \alpha x_M + 1}}, \\ x_i(\alpha) := \sin(\theta_i(\alpha)) = \cos(\alpha). \end{array} \right. \qquad (A.2)$$

For all $0 \le x < x_M < 1$, we have:

$$\frac{\partial F_1}{\partial x_1}(x, x_M) = - \frac{(\sqrt{x_M^2 - x^2} + \sqrt{1-x^2}) (2x^4 - x^2 x_M^2 - x_M^2 + 2x^2 \sqrt{1-x^2} \sqrt{x_M^2 - x^2} + x_M^2 \sqrt{1-x^2} \sqrt{x_M^2 - x^2})}{\sqrt{1-x^2} \sqrt{x_M^2 - x^2} (\sqrt{1-x^2} \sqrt{x_M^2 - x^2} + x^2 - x_M^2)^2}, \qquad (A.3)$$

which implies that

$$\mathrm{sign}(\frac{\partial F_1}{\partial x_1}(x, x_M)) = \mathrm{sign} \left( x^2 x_M^2 - 2x^4 + x_M^2 - \sqrt{1-x^2} \sqrt{x_M^2-x^2} (2x^2+x_M^2) \right), \qquad (A.4)$$

especially,
$$ \frac{\partial F_1}{\partial x_1}(x, x_M) = 0 \qquad (A.5) $$

$$ \iff (E_1) : x^2 x_M^2 - 2 x^4 + x_M^2 = \sqrt{1-x^2} \sqrt{x_M^2 - x^2} (2x^2 + x_M^2), \qquad (A.6) $$

$$ \overset{X=x^2}{\implies} (E'_1) : X^3 - X^2 x_M^2 - \frac{X x_M^4}{4} + \frac{x_M^4}{4} = 0, \qquad (A.7) $$

$$ \overset{Z=X-x_M^2/3}{\iff} (E''_1) : Z^3 + p_M Z + q_M = 0, \qquad (A.8) $$

$$ \text{where } p_M = -\frac{7x_M^4}{12} \text{ and } q_M = \frac{x_M^4}{4} - \frac{17x_M^6}{108}. $$

If we put $\Delta_1 = -(4p_M^3 + 27q_M^2)$, we find that

$$ \Delta_1 = \frac{1}{8} x_M^8 \left( x_M^2 + \frac{7\sqrt{7} + 17}{2} \right) (x_M + x_c)(x_M - x_c), \qquad (A.9) $$

$$ \text{where } x_c = \sqrt{\frac{7\sqrt{7}-17}{2}} \approx 0.8719. $$

Let

$$ I_c := [x_c, 1]. \qquad (A.10) $$

Using Cardano's method, if $x_M < x_c$, we find that $\Delta_1 < 0$, so the unique real solution $Z_0$ of (A.8) is strictly negative which is automatically rejected, hence function $F_1(., x_M)$ is strictly increasing on $[0, x_\alpha]$. Otherwise, if $x_M \in I_c$, we find that equation (A.7) have one strictly negative solution and two other ones between 0 and 1 such that:

$$ \left\{ \begin{array}{l} X_{m,1}(x_M) := Z_0(x_M) + \frac{x_M^2}{3}, \\ X_{M,1}(x_M) := Z_2(x_M) + \frac{x_M^2}{3}, \end{array} \right. \qquad (A.11) $$

where

$$ Z_k(x_M) = 2 \cos \left( \frac{\arccos \left( \frac{3\sqrt{3}q_M \sqrt{-\frac{1}{p_M}}}{2p_M} \right)}{3} + \frac{2\pi k}{3} \right) \sqrt{-\frac{p_M}{3}}, \quad k \in \{0, 2\}. \qquad (A.12) $$

**Remark Appendix A.1.** The purpose of Figure Appendix A is to represent $X_{M,1}$ and $X_{m,1}$, which are the squares of the solutions $x_{M,1}$ and $x_{m,1}$ of equation (A.5), in order to better understand the behavior of the functions $\theta_{M,1} = \arcsin(x_{M,1})$ and $\theta_{m,1} = \arcsin(x_{m,1})$ (see (A.15)), which appear in the expression of $h_{M,1}$ (A.16).

Now, we will discuss the number of solutions of Equation (A.8) as well as the monotonicity of the function $F_1(., x_M)$ depending on the relative position between $x_M$ and $x_c$. We have the following corollary:
<figure><img src="image_7.png" /><figcaption>Figure A1. Graphs of $X_{M,1}$ and $X_{m,1}$.</figcaption></figure>

**Corollary Appendix A.2.** For all $x_M \in ]0, 1[$ we have:

(i) if $x_M < x_c$, $F_1(., x_M)$ is strictly increasing on $[0, x_M]$.

(ii) if $x_M = x_c$,

$$X_{M,c} := X_{M,1}(x_c) = X_{m,1}(x_c) = -\frac{3q_M}{2p_M}, \qquad (A.13)$$

is the double solution of equation (A.8) and $\sqrt{X_{M,c}}$ is an inflection point of $F_1(., x_c)$, moreover it is strictly increasing.

(iii) else,

$$0 < X_{M,1}(x_M) < X_{m,1}(x_M) < 1. \qquad (A.14)$$

Function $F_1(., x_M)$ has only one local maximum for $x = x_{M,1} := \sqrt{X_{M,1}}$ and two other local minimums respectively for $x = 0$ and $x = x_{m,1} := \sqrt{X_{m,1}}$.

Let

$$\left\{
\begin{aligned}
\theta_c &:= \arcsin(x_c) \approx 1.058975 \\
\theta_{m,1}(\theta_M) &: \theta_M \mapsto \arcsin(x_{m,1}(\sin(\theta_M))) \\
\theta_{M,1}(\theta_M) &: \theta_M \mapsto \arcsin(x_{M,1}(\sin(\theta_M))) \\
h_0(\alpha, \theta_M) &:= h_{\alpha,\theta_M}(\theta_\alpha(\theta_M)), \\
h_1(\theta_M) &:= -\frac{\frac{\partial r_M}{\partial \theta}(\theta_{M,1}(\theta_M), \theta_M)}{r_M(\theta_{M,1}(\theta_M), \theta_M)} \\
h_2(\alpha, \theta_M) &:= h_{\alpha,\theta_M}(\theta_{M,2}(\alpha, \theta_M)).
\end{aligned}
\right.
\qquad (A.15)$$
We find that for all $(\alpha, \theta_M) \in \mathcal{A}$:

$$h_{\mathcal{M},1}(\alpha, \theta_M) = \begin{cases} h_0(\alpha, \theta_M), & \text{if } \theta_M \le \theta_c \text{ or } \theta_\alpha(\theta_M) \le \theta_{M,1}(\theta_M) \\ h_1(\theta_M), & \text{if } \theta_{M,1}(\theta_M) \le \theta_\alpha(\theta_M) \le \theta_{m,1}(\theta_M) \\ \max(h_0(\alpha, \theta_M), h_1(\theta_M)), & \text{else.} \end{cases} \quad (A.16)$$

Now, we will determine $h_{\mathcal{M},2}$.

It follows from the definition (3.1) of $\Gamma_{0,2}^+$, if we use the same change of variables as we did previously to obtain (A.1), for all $\alpha \in ]0, \pi[$ we have:

$$h_{\alpha, \theta_M}(\theta) = H_{2, \alpha, x_M}(x) := F_2(x_\alpha(x_M), x, x_M), \quad \forall (\theta, \theta_M) \in \mathcal{D}_{\alpha, 2}, \quad (A.17)$$

where

$$\left\{
\begin{aligned}
F_2(x_0, x_1, x_2) &:= \frac{(x_1 p_1(x_0, x_2) - p_2(x_0, x_2)) \left(\sqrt{1-x_1^2} + \sqrt{x_2^2-x_1^2}\right)}{\sqrt{1-x_0^2} (x_2^2-x_0^2)^{3/2} (1-x_2^2)}, \\
&\forall 0 < x_0 \le x_1 \le x_2 < 1; x_0 \ne x_2 \\
\mathcal{D}_{\alpha, 2} &:= \{(\theta, \theta_M) \in \mathcal{D}_\alpha; \theta_\alpha(\theta_M) \le \theta \le \theta_M\},
\end{aligned}
\right.
\quad (A.18)$$

and

$$\left\{
\begin{aligned}
p_1(x_0, x_2) &:= p_{1,\alpha}(x_M) := x_2^2 + x_0^4 - 2x_2^2 x_0^2 + \sqrt{1-x_0^2} (x_2^2 - x_0^2)^{3/2} > x_2^2(1-x_2^2), \\
p_2(x_0, x_2) &:= p_{2,\alpha}(x_M) := x_0^3 (1-x_2^2) > 0.
\end{aligned}
\right.
\quad (A.19)$$

For all $(\alpha, x_M) \in \hat{\mathcal{A}} := \{(\alpha, \sin(\theta_M)); (\alpha, \theta_M) \in \mathcal{A}\}$ and $0 \le x \le x_M$ we have:

$$H'_{2,\alpha,x_M}(x) = \frac{\sqrt{x_M^2 - x^2} \sqrt{1-x^2} p_{1,\alpha}(x_M) - x (x p_{1,\alpha}(x_M) - p_{2,\alpha}(x_M))}{\sqrt{1-x^2} \sqrt{1-x_\alpha^2} \sqrt{x_M^2 - x^2} (x_M^2 - x_\alpha^2)^{3/2} \left(\sqrt{1-x^2} - \sqrt{x_M^2 - x^2}\right)},$$

hence

$$H'_{2,\alpha,x_M}(x) = 0 \iff (E_2) : \quad \varphi_{\alpha,x_M}(x) = 0, \quad (A.20)$$

where

$$\varphi_{\alpha,x_M}(x) := \sqrt{x_M^2 - x^2} \sqrt{1-x^2} p_{1,\alpha}(x_M) - x (x p_{1,\alpha}(x_M) - p_{2,\alpha}(x_M)). \quad (A.21)$$

Using (A.19), we find that for all $(\alpha, x_M) \in \hat{\mathcal{A}}$:

$$\left\{
\begin{aligned}
\varphi_{\alpha,x_M}(0) &= x_M p_{1,\alpha}(x_M) > 0, \\
\varphi_{\alpha,x_M}(x_M) &= -x_M (x_M p_{1,\alpha}(x_M) - p_{2,\alpha}(x_M)) < -x_M(1-x_M^2)(x_M^3 - x_\alpha(x_M)^3) < 0.
\end{aligned}
\right.
\quad (A.22)$$
So $H_{2,\alpha,x_M}$ has at least one critical point in $]0, x_M[$. For all $(\alpha, x_M) \in \hat{\mathcal{A}}$ and $x \in ]0, x_M[$:

$$H'_{2,\alpha,x_M}(x) = 0 \iff \varphi_{\alpha,x_M}(x) = 0 \quad (A.23)$$

$$\iff x \left( x p_{1,\alpha}(x_M) - p_{2,\alpha}(x_M) \right) = \sqrt{x_M^2 - x^2} \sqrt{1-x^2} p_{1,\alpha}(x_M) \quad (A.24)$$

$$\implies -2p_{1,\alpha}p_{2,\alpha}x^3 + x^2 \left( (1+x_M^2)p_{1,\alpha}^2 + p_{2,\alpha}^2 \right) - x_M^2 p_{1,\alpha}^2 = 0 \quad (A.25)$$

$$\implies (E'_2) : x^3 - x^2 \left( \frac{(1+x_M^2)p_{1,\alpha}^2 + p_{2,\alpha}^2}{2p_{1,\alpha}p_{2,\alpha}} \right) + x_M^2 \frac{p_{1,\alpha}}{2p_{2,\alpha}} = 0 \quad (A.26)$$

$$\stackrel{x=z+s_{\alpha,x_M}/3}{\iff} z^3 + p_{\alpha,x_M}z + q_{\alpha,x_M} = 0, \quad (A.27)$$

where

$$\left\{
\begin{aligned}
s_{\alpha,x_M} &:= \frac{(1+x_M^2)p_{1,\alpha}^2 + p_{2,\alpha}^2}{2p_{1,\alpha}p_{2,\alpha}} \\
p_{\alpha,x_M} &:= -\frac{\left( (x_M^2+1) p_{1,\alpha}^2 + p_{2,\alpha}^2 \right)^2}{12 p_{1,\alpha}^2 p_{2,\alpha}^2} \\
q_{\alpha,x_M} &:= -2r_p^3 + \frac{x_M^2 p_{1,\alpha}}{2p_{2,\alpha}} \\
r_p &:= \sqrt{-\frac{p_{\alpha,x_M}}{3}} = \frac{s_{\alpha,x_M}}{3} = \frac{(x_M^2+1) p_{1,\alpha}^2 + p_{2,\alpha}^2}{6 p_{1,\alpha} p_{2,\alpha}}.
\end{aligned}
\right.
\quad (A.28)$$

If we put $\Delta_2 := -(4p_{\alpha,x_M}^3 + 27q_{\alpha,x_M}^2)$, we find that $\Delta_2$ is strictly positive and equation (A.26) has 3 real solutions where the only positive one which we denote by $x_{M,2}(\alpha, x_M)$ is:

$$x_{M,2}(\alpha, x_M) = r_p \left( 1 - 2 \cos \left( \frac{\pi + \arccos(S(p_{\alpha,x_M}, q_{\alpha,x_M}))}{3} \right) \right), \quad (A.29)$$

where

$$S(p, q) = -\frac{q_{\alpha,x_M}}{2r_p^3}.$$

It follows that for all $(\alpha, x_M) \in \hat{\mathcal{A}}$, $x_{M,2}(\alpha, x_M)$ is the unique critical point of $H_{2,\alpha,x_M}$ on $]0, x_M[$. Moreover,

$$\forall (\alpha, x_M) \in \hat{\mathcal{A}}, \quad \mathrm{sign}(H'_{2,\alpha,x_M}) = \mathrm{sign}(\varphi_{\alpha,x_M}). \quad (A.30)$$

Using (A.22), For all $(\alpha, x_M) \in \hat{\mathcal{A}}$ we find that:

$H_{2,\alpha,x_M}$ is concave and it has a unique global maximum at on $]0, x_M[$ at $x = x_{M,2}(\alpha, x_M)$. \hfill (A.31)

If we define:

$$\left\{
\begin{array}{l}
\theta_{M,2}(\alpha, \theta_M) := \arcsin(x_{M,2}(\alpha, x_M)) \in ]0, \theta_M[; \forall (\alpha, \theta_M) \in \mathcal{A}, \\
h_2(\alpha, \theta_M) := h_{\alpha,\theta_M}(\theta_{M,2}(\alpha, \theta_M)),
\end{array}
\right.
\quad (A.32)$$
using (A.31), we conclude that for all $(\alpha, \theta_M) \in \mathcal{A}$:

$$h_{\mathcal{M},2}(\alpha, \theta_M) = \begin{cases} h_0(\alpha, \theta_M), & \text{if, } \theta_{\mathcal{M},2}(\alpha, \theta_M) \le \theta_\alpha(\theta_M) \\ h_2(\alpha, \theta_M), & \text{else} \end{cases} \qquad (A.33)$$

To complete this proof, we will use the following lemma.

**Lemma Appendix A.3.** There exists a continuous function $\theta_f: [0, \pi] \to [\theta_c, \frac{\pi}{2}]$ and $\alpha_0 \in ]0, \frac{\pi}{2}]$ such that $\theta_f(0) = \frac{\pi}{2}$, $\theta_f|_{[\frac{\pi}{2}, \pi]} = \text{id}|_{[\frac{\pi}{2}, \pi]}$ and for all $(\alpha, \theta_M) \in \mathcal{A}$ we have

$$\left\{ \begin{array}{ll} \theta_{\alpha}(\theta_M) \le \theta_{\mathcal{M},2}(\alpha, \theta_M), & \text{if } \theta_M \le \theta_f(\alpha), \\ \theta_{\mathcal{M},2}(\alpha, \theta_M) < \theta_{\alpha}(\theta_M), & \text{else.} \end{array} \right. \qquad (A.34)$$

Especially, for all $\theta_c \le \theta_M \le \frac{\pi}{2}$, we have:

(i) If $\alpha < \alpha_0$:

$$\left\{ \begin{array}{ll} \theta_{m,1}(\theta_M) < \theta_\alpha(\theta_M), & \text{if } \theta_M < \theta_f(\alpha), \\ \theta_{\mathcal{M},1}(\theta_M) < \theta_\alpha(\theta_M) \le \theta_{m,1}(\theta_M), & \text{else.} \end{array} \right. \qquad (A.35)$$

(ii) If $\alpha = \alpha_0$, then $\theta_f(\alpha_0) = \theta_c$ and

$$\left\{ \begin{array}{ll} \theta_{\mathcal{M},1}(\theta_M) = \theta_\alpha(\theta_M) = \theta_{m,1}(\theta_M), & \text{if } \theta_M = \theta_c, \\ \theta_{\mathcal{M},1}(\theta_M) < \theta_\alpha(\theta_M) < \theta_{m,1}(\theta_M), & \text{else.} \end{array} \right. \qquad (A.36)$$

(iii) Else $(\alpha \in ]\alpha_0, \pi[)$

$$\left\{ \begin{array}{ll} \theta_{\alpha}(\theta_M) < \theta_{\mathcal{M},1}(\theta_M), & \text{if } \theta_M < \theta_f(\alpha), \\ \theta_{\mathcal{M},1}(\theta_M) \le \theta_\alpha(\theta_M) < \theta_{m,1}(\theta_M), & \text{else.} \end{array} \right. \qquad (A.37)$$

The proof of this lemma is given in Appendix B.

Using this lemma in (A.16) and (A.33), we obtain for all $(\alpha, \theta_M) \in \mathcal{A}$:

$$h_{\mathcal{M}}(\alpha, \theta_M) = \begin{cases} h_2(\alpha, \theta_M), & \text{if } (\theta_M \le \theta_c), \\ \max(h_1(\theta_M), h_2(\alpha, \theta_M)), & \text{if } \theta_c \le \theta_M \le \theta_f(\alpha), \\ h_1(\theta_M), & \text{if } \theta_f(\alpha) \le \theta_M. \end{cases} \qquad (A.38)$$

Especially, by Corollary Appendix A.2 we find that for all $\alpha_0 \le \alpha$:

$$h_{\mathcal{M}}(\alpha, \theta_M) = h_2(\alpha, \theta_M) \text{ if } \theta_M \le \theta_f(\alpha), \qquad (A.39)$$
<figure><img src="image_8.png" /><figcaption>Figure A2. Graphs of $\frac{K'_{\alpha}}{|K'_{\alpha}(x_{i,c})|}$ for different $\alpha \in ]0, \alpha_0[$.</figcaption></figure>

so, for all $\alpha \in [\alpha_0, \frac{\pi}{2}]$ the unique solution to equation

$$ (E_{\alpha}): \quad h_{1}(\theta_{M}) = h_{2}(\alpha, \theta_{M}) \qquad (A.40) $$

is $\theta_M = \theta_f(\alpha)$. From now on, for all $\alpha_0 \le \alpha$ we denote $\theta_f(\alpha)$ by $\theta_{int}(\alpha)$. For all $\alpha < \alpha_0$, we study (A.40) on $J_{\alpha} \subset I_{\alpha}$ where

$$ J_{\alpha} = \begin{cases} [x_i(\alpha), x_f(\alpha)], & \text{si } \alpha \le \alpha_c, \\ [x_c, x_f(\alpha)], & \text{else.} \end{cases} \qquad (A.41) $$

We put $K_{\alpha}: J_{\alpha} \to \mathbb{R}; x_M \mapsto H_1(x_M) - H_2(\alpha, x_M)$ such that

$$ H_1(x_M) := h_1(\theta_M) \quad \text{and} \quad H_2(\alpha, x_M) := h_2(\alpha, \theta_M). $$

We have

$$ \forall \alpha \in ]0, \alpha_0[, \quad K'_{\alpha} > 0. \qquad (A.42) $$

We would rather generate many numerical simulations of $K'_{\alpha}$ for different values of $\alpha \in ]0, \alpha_0[$ on $J_{\alpha}$, than giving the explicit proof of property (A.42). So we made these simulations on one real interval $[x_{i,c}(\alpha), x_f(\alpha)]$, where

$$ x_{i,c} = \begin{cases} x_i + 0.1(x_f - x_i), & \text{if } \alpha \le \alpha_c, \\ x_c, & \text{else,} \end{cases} \qquad (A.43) $$

and we arranged them in Figure A2.

Using definitions (A.2), for all $\alpha \in ]0, \pi[$ we find that:

$$ \left\{ \begin{array}{ll} \underset{x_M \to x_i(\alpha)}{x_{\alpha}(x_M)} \sim x_M, & \text{if } \alpha \le \frac{\pi}{2}, \\ \underset{x_M \to x_i(\alpha)}{x_{\alpha}(x_M)} \sim x_M \sin \alpha, & \text{else,} \end{array} \right. \qquad (A.44) $$
and for all $\alpha \in ]\frac{\pi}{2}, \pi[$, using (A.20), we obtain:

$$x_{M,2}(\alpha, x_M) = \lim_{x_M \to 0^+} x_M - \frac{(\sin^3 \alpha - 1)^2}{2} x_M^3 + o(x_M^3). \quad (A.45)$$

Then, by definition (A.18) of $H_{2,\alpha,x_M}(x)$ and (A.44) we find that:

$$H_2(\alpha, x_M) \rightarrow \begin{cases} +\infty, & \text{if } \alpha \le \frac{\pi}{2}, \\ \frac{\sin^3 \alpha - 1}{\cos^3 \alpha}, & \text{else.} \end{cases} \quad (A.46)$$

By Corollary Appendix A.2 and the first inequality in (A.35) we find that for all $\alpha \in ]\alpha_c, \alpha_0[$, $h_{\alpha,\theta_c}$ is strictly increasing on $[0, \theta_\alpha(\theta_c)]$, moreover $\theta_\alpha(\theta_c) < \theta_{M,2}(\alpha, \theta_c)$. Taking (A.33) into consideration, we find that:

$$H_1(x_c) < H_2(\alpha, x_c). \quad (A.47)$$

Also, we find that

$$\forall \alpha \in ]0, \alpha_0[, \quad x_M \in I_c, \quad x_\alpha(x_M) = x_{m,1}(x_M) = x_{M,2}(\alpha, x_M) \iff x_M = x_f(\alpha). \quad (A.48)$$

So, taking the case number (iii) of Corollary Appendix A.2 into consideration, we have:

$$H_1(x_f(\alpha)) > H_0(\alpha, x_f(\alpha)) = H_2(\alpha, x_f(\alpha)). \quad (A.49)$$

Using (A.42), (A.46), (A.47) and (A.49) we conclude that:

$$\forall \alpha \in ]0, \alpha_0[, \quad \exists! x_{int}(\alpha) \in J_{\alpha}; \quad K_{\alpha}(x_{int}(\alpha)) = 0. \quad (A.50)$$

<figure><img src="image_9.png" /><figcaption>Figure A3. $x_{int}$ graph on $]0, \alpha_0[$ with $10^{-7}$ precision.</figcaption></figure>
Hence, we are able to define $\theta_{int}: \alpha \mapsto \theta_{int}(\alpha)$ on $]0, \pi[$ by:

$$ \forall \alpha \in ]0, \alpha_0[, \quad \theta_{int}(\alpha) := \arcsin(x_{int}(\alpha)). \qquad (A.51) $$

Using a numerical simulation to find $\theta_{int}$, by Newton method we obtain Figure A3. We find that $\theta_{int}$ is strictly decreasing on $]0, \alpha_0[$.

Hence (A.38) becomes:

$$ h_M(\alpha, \theta_M) = \begin{cases} h_{M,2}(\alpha, \theta_M) = h_2(\alpha, \theta_M), & \text{if } \theta_M \le \theta_{int}(\alpha), \\ h_{M,1}(\alpha, \theta_M) = h_1(\theta_M), & \text{if } \theta_{int}(\alpha) \le \theta_M. \end{cases} \qquad (A.52) $$

□

## Appendix B. Proof of Lemma Appendix A.3

We define function $X_0$ on $\hat{D} := ]0, \pi[ \times ]0, 1]$ by:

$$ X_0 : (\alpha, x_M) \mapsto \left( \left( \frac{\frac{1}{x_M} - \cos \alpha}{\sin \alpha} \right)^2 + 1 \right)^{-1}, \qquad (B.1) $$

which is equal to $x_\alpha^2$ on $\hat{A} := \{(\alpha, \sin \theta_M); (\alpha, \theta_M) \in A\}$. We refer to (1.36) for further details on $A$.

Here, we will study the relative position between these three functions $X_0$, $X_{m,1}$ and $X_{M,1}$ on

$$ \tilde{A}_c := \{(\alpha, x_M) \in ]0, \pi[ \times I_c; x_i(\alpha) \le x_M\}, \qquad (B.2) $$

where $I_c := [x_c, 1]$ is defined in (A.10). If we put $\alpha_c = \frac{\pi}{2} - \theta_c \approx 0.511821$, we find that:

$$ \tilde{A}_c = \{(\alpha, x_M) \in \hat{D} = ]0, \pi[ \times ]0, 1]\}; \quad x_M \in I_{\alpha,c} \}, \qquad (B.3) $$

where

$$ I_{\alpha,c} := \begin{cases} ]x_{\alpha,c}, 1] := ]x_i(\alpha), 1] & \text{if } \alpha \le \alpha_c, \\ [x_{\alpha,c}, 1] := I_c & \text{else.} \end{cases} \qquad (B.4) $$

First, we begin by determining points of intersection between each of previous function graphs. Let $\alpha \in ]0, \frac{\pi}{2}]$, for all $x_M$ in $I_{\alpha,c}$, $x_{m,1}(x_M)$ and $x_{M,1}(x_M)$ are the only solutions between 0 and 1 to equation (A.6). To determine admissible points of intersection $(\alpha, x_M) \in \tilde{A}_c$ between these graphs, it is sufficient to solve (A.6) by replacing $x$ with $x_\alpha(x_M)$, which is equivalent to this one in $x_M \in I_{\alpha,c}$:

$$ (E'_\alpha) : \cos \alpha x_M^3 + (1 - 4 \cos^2 \alpha) x_M^2 + (2 \cos^3 \alpha + \cos \alpha) x_M - 1 = 0. \qquad (B.5) $$

By Cardano's method, we find that this equation have a unique solution $x_f(\alpha) \in I_{\alpha,c}$, moreover we notice that $x_f : ]0, \frac{\pi}{2}]$; $\alpha \mapsto x_f(\alpha)$ is a continuous function having it's
<figure><img src="image_10.png" /><figcaption>Figure B1. Graph of $x_f$.</figcaption></figure>

values in $I_c$ and $x_f(0^+) = x_f(\frac{\pi}{2}) = 1$. Hence it has a unique global minimum $x_M^* \in [x_c, 1[$ reached at $\alpha^* \in ]0, \frac{\pi}{2}[$ (See Figure B1). We find that the following equation

$$ (E_c) : X_0(x_c, \cdot) = X_{M,1}(x_c) \qquad (B.6) $$

in $\alpha$ is equivalent to a 2$^{nd}$ degree one in $\cos(\alpha)$ that has two solutions

$$ \cos(\alpha_{\pm}) = \frac{X_{M,c} \pm \sqrt{(X_{M,c} - 1)(X_{M,c} - x_c^2)}}{x_c}, $$

where $X_{M,c} := X_{M,1}(x_c)$, hence

$$ \left\{ \begin{array}{ll} X_{M,1}(x_c) < X_0(x_c, \alpha), & \text{if } \alpha_+ < \alpha < \alpha_-, \\ X_0(x_c, \alpha) \le X_{M,1}(x_c), & \text{else.} \end{array} \right. \qquad (B.7) $$

Moreover, the only admissible solution of (B.6) is $\alpha_0 := \alpha_- \approx 1.191429$ where we choose $x_M = x_c$, because $\alpha_+ \approx 0.2013027 < \alpha_c \Rightarrow \theta_i(\alpha_+) = \frac{\pi}{2} - \alpha_+ > \theta_c$, hence $\theta_c \notin I_{\alpha_+}$ and $(\alpha_+, \theta_c) \notin \mathcal{A}$ which implies that, for all $\alpha_c < \alpha \le \alpha_0$, $x_M \in I_c$,

$$ \left\{ \begin{array}{ll} X_{M,1}(x_M) = X_0(\alpha, x_M), & \text{if } (\alpha, x_M) = (\alpha_0, x_c), \\ X_{M,1}(x_M) < X_0(\alpha, x_M), & \text{else.} \end{array} \right. \qquad (B.8) $$

Especially $\alpha = \alpha_0$ is the only angle such that $x_c$ is the solution to (B.6) (See (B.5)), so we deduce that

$$ (\alpha^*, x_M^*) = (\alpha_0, x_c) \text{ and } x_f(\alpha_0) = x_c. \qquad (B.9) $$
We also prove that:

$$ \forall 0 < \alpha \le \alpha_c, \quad x_M \in I_\alpha, \quad X_{M,1}(x_M) < X_0(\alpha, x_M). \qquad (B.10) $$

In fact, $X_{0,i}: \alpha \mapsto X_0(x_i, \alpha) = \cos(\alpha)^2$ is strictly decreasing, $X_{M,i}: \alpha \mapsto X_{M,i}(x_i(\alpha))$ is strictly increasing on $]0, \alpha_c]$ and

$$ X_{0,i}(\alpha_c) = \cos^2 \alpha_c \approx 0.7601 > 0.588 \approx X_{M,c} = X_{M,i}(\alpha_c), \qquad (B.11) $$

so $X_0(x_M, \alpha) > X_0(x_i, \alpha) := X_{0,i}(\alpha) > X_{0,i}(\alpha_c) > X_{M,i}(\alpha_c) > X_{M,i}(\alpha) > X_{M,1}(x_M)$ for all $x_M \in I_\alpha$.

From (B.8)-(B.10) we have for all $0 < \alpha \le \alpha_0$:

$$ \forall x_M \in I_{\alpha,c}; \quad X_{M,1}(x_M) \le X_0(\alpha, x_M), \qquad (B.12) $$

where we have equality if and only if $\alpha = \alpha_0$ and $x_M = x_f(\alpha_0) = x_c$. If $\alpha_c \le \alpha < \alpha_0$, then $x_{\alpha,c} = x_c$ (i.e $I_{\alpha,c} = I_c$) and

$$ X_{M,c} := X_{m,1}(x_c) = X_{M,1}(x_c) < X_0(\alpha, x_c). \qquad (B.13) $$

Otherwise, when $0 < \alpha < \alpha_c$, we have $x_{\alpha,c} = x_i(\alpha)$ and

$$ X_{m,1}(x_i(\alpha)) = \frac{\cos^2 \alpha}{3} + \frac{\sqrt{7} \cos^2 \alpha \cos\left(\frac{\operatorname{acos}(y_\alpha)}{3}\right)}{3}. \qquad (B.14) $$

Here $y_a = -\frac{\sqrt{7} \left(\frac{27}{\cos^2 a} - 17\right)}{49} < 0$ for all $a \in [0, \alpha_c]$. We find that

$$ y_{\alpha c} < y_{\alpha} < y_0 < 0, \qquad (B.15) $$

where $y_{\alpha c} = -\frac{\sqrt{7} \left(\frac{27}{x_c^2} - 17\right)}{49} = -1$.

Therefore,

$$ \sqrt{7} \cos \left( \frac{\operatorname{acos}(y_\alpha)}{3} \right) < \sqrt{7} \cos \left( \frac{\operatorname{acos}(y_0)}{3} \right), \qquad (B.16) $$

and

$$ X_{m,1}(x_i(0)) = 1 = \frac{1}{3} + \frac{\sqrt{7} \cos \left( \frac{\operatorname{acos}(y_0)}{3} \right)}{3}, \qquad (B.17) $$

then

$$ \sqrt{7} \cos \left( \frac{\operatorname{acos}(y_0)}{3} \right) = 2. \qquad (B.18) $$

From (B.14) and (B.18), we conclude that

$$ X_{m,1}(x_i(\alpha)) < \frac{\cos^2 \alpha}{3} + \frac{2 \cos^2 \alpha}{3} = \cos^2 \alpha = X_0(\alpha, x_i(\alpha)). \qquad (B.19) $$
Using (B.13) and (B.19) we find that

$$X_{m,1}(x_{\alpha,c}) < X_0(\alpha, x_{\alpha,c}). \qquad (B.20)$$

Using (B.12), (B.20) and

$$X_{m,1}(1) = 1 > \frac{1 + \cos \alpha}{2} = X_0(\alpha, 1), \qquad (B.21)$$

we have, for all $0 < \alpha < \alpha_0$, $x_{\alpha,c} < x_M < 1$,

$$\left\{ \begin{array}{ll} X_{m,1}(x_M) < X_0(\alpha, x_M), & \text{if } x_M < x_f(\alpha), \\ X_{M,1}(x_M) < X_0(\alpha, x_M) < X_{m,1}(x_M), & \text{if } x_f(\alpha) < x_M. \end{array} \right. \qquad (B.22)$$

For all $\alpha_0 < \alpha < \frac{\pi}{2}$, using (B.7), we find that:

$$\left\{ \begin{array}{l} X_0(x_c, \alpha) < X_{M,1}(x_c), \\ X_{M,1}(1) = 0.5 < \frac{1 + \cos \alpha}{2} = X_0(1, \alpha). \end{array} \right. \qquad (B.23)$$

By the mean value theorem, we find that $x_M = x_f(\alpha) \in \hat{I}_c$ is the unique solution to $X_0(\alpha, x_M) = X_{M,1}(x_M)$. Hence, from (A.14) we find that $X_0(\alpha, x_f(\alpha)) < X_{m,1}(x_f(\alpha))$ which implies that for all $x_M \in I_{\alpha,c}$, $X_0(\alpha, x_M) < X_{m,1}(x_M)$.

From previous results, we conclude that, for all $\alpha_0 < \alpha < \frac{\pi}{2}$, $x_M \in I_c$,

$$\left\{ \begin{array}{ll} X_0(\alpha, x_M) < X_{M,1}(x_M), & \text{if } x_M < x_f(\alpha), \\ X_{M,1}(x_M) < X_0(\alpha, x_M) < X_{m,1}(x_M), & \text{if } x_f(\alpha) < x_M. \end{array} \right. \qquad (B.24)$$

For $\alpha = \frac{\pi}{2}$ we find that $x_f(\frac{\pi}{2}) = 1$. So for all $x_M \in I_c$ we would have $X_0(\alpha, x_M) \le X_{M,1}(x_M)$ with equality only if $x_M = 1$.

Finally, for all $\frac{\pi}{2} < \alpha < \pi$, because $X_0(\alpha, \cdot)$ (resp. $X_{M,1}$) is strictly increasing (resp. decreasing) on $I_c$ and $X_0(\alpha, 1) = \frac{1+\cos \alpha}{2} < \frac{1}{2} = X_{M,1}(1)$, we find that:

$$\forall \frac{\pi}{2} < \alpha < \pi, \quad x_M \in I_c, \quad X_0(\alpha, x_M) < X_{M,1}(x_M). \qquad (B.25)$$

From now on, we put $\theta_f(\alpha) = \arcsin(x_f(\alpha))$ and extend the function $\theta_f$ to $] \frac{\pi}{2} , \pi[$ by the identity function. From the definition of $h_{\alpha,\theta_M}$ (3.6), (B.22), (B.24), (B.25), and optimums of $F_1(\cdot, x_M)$ for each $x_M = \sin \theta_M \in I_c$ given in Corollary Appendix A.2, for all $(\alpha, \theta_M) \in \mathcal{A}$ we have:

$$\left\{ \begin{array}{ll} h'_{\alpha,\theta_M}(\theta_\alpha(\theta_M)) > 0, & \text{if } \theta_M < \theta_f(\alpha), \\ h'_{\alpha,\theta_M}(\theta_\alpha(\theta_M)) = 0, & \text{if } \theta_M = \theta_f(\alpha), \\ h'_{\alpha,\theta_M}(\theta_\alpha(\theta_M)) < 0, & \text{else.} \end{array} \right. \qquad (B.26)$$
Hence, by (A.17) and the concavity (A.31) of $H_{2,\alpha,x_M}$, we find that:

$$ egin{cases} \theta_{\alpha}(\theta_M) \le \theta_{M,2}(\alpha, \theta_M), & \text{if } \theta_M \le \theta_f(\alpha), \\ \theta_{M,2}(\alpha, \theta_M) < \theta_{\alpha}(\theta_M), & \text{else,} \end{cases} \qquad (B.27) $$

which completes the proof. □

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