Comment on 'Revisiting the probe and enclosure methods'

Let $v(x, \tau, t): = e^{-\tau(t-x\cdot\rho)+i \sqrt{\tau^2+k^2} x \cdot \rho^{\perp}}$, where ρ is a unit vector, with $\rho\cdot \rho^{\perp} = 0$, and τ and t are two positive parameters. The enclosure method is a reconstruction scheme proposed, two decades ago, by Ikehata to reconstruct (geometrical features of) interfaces from remote measurements using v as a test function (and eventually other test functions). In the case where the mathematical model has a lower order term (i.e. of Helmholtz type), some extra conditions were needed to justify this method. In [1], an approach was proposed to remove these extra conditions. It is based on appropriate a priori estimates that control the lower order terms to deal with the lack of positivity. In a recent manuscript, see [2], section 4, the author made some comments saying that the justification of the estimates in lemma 3.5 of [1], see the estimates (1) and (2) below, might not be complete. The object of this note is to clarify this issue. To go straight to the point, we consider only the critical value of the parameter t: $t = h_D(\rho)$ where $h_D(\rho): = \sup_{x\in D}x\cdot\rho$ and use the notation $v(x, \tau): = v(x, \tau, h_D(\rho))$ (as in lemma 3.5 of [1]).

Let D be a bounded and Lipschitz regular domain in $\mathbb{R}^3$. Following the notations in [1], section 3.2, we introduce the sets $D_{j,\delta} \subset D$, $D_\delta \subset D$ as follows. For any $\alpha \in \partial D \cap \{x \cdot \rho = h_D(\rho)\} = : K$, define $B(\alpha, \delta) : = \{x \in \mathbb{R}^3 ; \vert x -\alpha\vert \lt \delta\} (\delta \gt 0)$. Then, $K \subset \cup_{\alpha \in K} B(\alpha, \delta)$. Since K is compact, there exists $\{\alpha_1,\ldots ,\alpha_N \} \subset K$ such that $K \subset B(\alpha_1,\delta) \cup \cdot \cdot \cdot\cup B(\alpha_N,\delta)$. Then, we define $D_{j,\delta} : = D \cap B(\alpha_j, \delta)$ and $D_\delta : = \cup^N_{j = 1} D_{j,\delta}$. We have $\int_{D \setminus D_\delta} \vert v(x, \tau)\vert^p dx = \int_{D \setminus D_\delta} e^{-p\tau (h_D(\rho)-x \cdot \rho )} dx = O(e^{-pc\tau} ),\; (\tau \rightarrow \infty).$

Assume now that D is of class C¹, therefore $\partial D$ has a well defined (exterior) unit normal vector field $\nu(\cdot)$. The plane $\{x \cdot \rho = h_D(\rho)\}$ intersects $\partial D$, at the points α_j 's, with ρ as a normal vector to it. Then $\nu(\alpha_j)$ and ρ are parallel, see below, namely $\nu(\alpha_j) = -\rho$ taking into account the orientation of the normal with respect to D. After the translation, of α_j to the origin, and the rotation, taking $\nu(\alpha_j)$ to $e_3: = (0, 0, 1)$, $D_{j, \delta}$ takes the form, with δ smaller if needed, $\{(y_1, y_2, y_3); y_1^2+y_2^2 \lt\delta^2 \mbox{and } l_j(y_1, y_2)\lt y_3\lt\delta\}$ with l_j as a local representation of $\partial D$ near α_j . Using these transformations and the property $\nu(\alpha_j) = -\rho$, we obtain

$$\begin{align*} \int_{D_{j, \delta}}\vert v\vert^p\mathrm dx = \int_{D_{j, \delta}} e^{p \tau \left(x\cdot \rho- h_D\left(\rho\right)\right)}\mathrm dx = \int_{y_1^2+y_2^2 \lt\delta^2}\left(\int^\delta_{l_j\left(y_1, y_2\right)}e^{-p\; \tau\: y_3}dy_3\right) \mathrm dy_1 \mathrm dy_2, \end{align*}$$

and then, with appropriate positive constants $C_{up}, c_{up}, C_{low}$ and c_low , we have

$$\begin{equation} \int_{D}\vert v\vert^p\mathrm dx \unicode{x2A7D} C_{up} \tau^{-1}\; \sum_{1\unicode{x2A7D} j \unicode{x2A7D} N}\int_{y_1^2+y_2^2 \lt\delta^2}e^{-p\; \tau\: l_j\left(y_1, y_2\right)} \mathrm dy_1 \mathrm dy_2 + O\left(e^{-c_{up}\;p\;\tau}\right) \end{equation} \tag{ 1 }$$

and

$$\begin{equation} \int_{D}\vert v\vert^p\mathrm dx \unicode{x2A7E} c_{low}\; \tau^{-1} \sum_{1\unicode{x2A7D} j \unicode{x2A7D} N}\int_{y_1^2+y_2^2 \lt\delta^2}e^{-p\; \tau\: l_j\left(y_1, y_2\right)} \mathrm dy_1 \mathrm dy_2 + O\left(e^{-C_{low}\;p\;\tau}\right).\end{equation} \tag{ 2 }$$

The estimates (1) and (2) are the ones in lemma 3.5 in [1]. The corresponding estimates for $\nabla v$ are deduced from the relation $\vert \nabla v\vert = \sqrt{2 \tau^2+\kappa^2} \vert v\vert$.

Let us now show why $\nu(\alpha_j)$ and ρ are parallel. Recall that, after translation and rotation, $\partial D$ is represented, near α_j , as $\{(y_1, y_2, l_{j}(y_1, y_2))\}$ with l_j of class C¹ and $l_j(0, 0) = \partial_{y_1} l_{j} (0, 0) = \partial_{y_2}l_{j} (0, 0) = 0$. Accordingly, the plane $\{x \cdot \rho = h_D(\rho)\}$ becomes, after translation and rotation, $\{y \cdot R \rho = 0\}$ where R is the rotating matrix. We set $b = (b_1, b_2, b_3): = R\rho$ and $f_j(y_1, y_2): = (y_1, y_2, l_{j}(y_1, y_2))\cdot b$. This is a C¹-regular function and it reaches its maximum, 0, at $(y_1. y_2) = (0, 0)$ (as $\alpha_j\cdot \rho = h_D(\rho)$). Therefore $(\nabla f_j)(0, 0) = 0$. This implies that $b_1 = b_2 = 0$ as $\partial_{y_1} l_{j} (0, 0) = \partial_{y_2}l_{j} (0, 0) = 0$, hence $R\rho$ is parallel to $e_3: = (0, 0, 1)$, i.e. ρ is parallel to $R^T e_3 = \nu(\alpha_j)$.

As we can see, the arguments in [1] to prove (1) and (2) are enough for a C¹-regular domain D. The analysis above is shown for a complex geometrical optics solution with a linear phase v. It applies the same way to the other complex geometrical optics (CGOs) as well (with logarithmic phase for instance) as the one used in [1]. In addition, the CGOs with linear phase used in electromagnetism and elasticity have a similar form as the one considered here. Therefore, the results in [1] and also the ones derived later for other models including the electromagnetism and elasticity are correct for a C¹-regular domain D.

Finally, we mention that in [2] the $C^{1, 1}$-regularity of the domain is required to justify this method.

The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.