Comment on ‘Revisiting the probe and enclosure methods’

In the recent manuscript (Ikehata 2022 Inverse Problems 38 075009), the author made some comments on our previous work (Sini and Yoshida 2012 Inverse Problems 28 055013) saying that few arguments in the proof of lemma 3.5 might be incomplete for C0,1 -regular domains. In this note, we reply to his comments by showing that, for C 1-regular domains, those arguments are correct.

√ τ 2 +k 2 x•ρ ⊥ , where ρ is a unit vector, with ρ • ρ ⊥ = 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 (ρ) where h D (ρ) := sup x∈D x • ρ and use the notation v(x, τ ) := v(x, τ, h D (ρ)) (as in lemma 3.5 of [1]).
Let D be a bounded and Lipschitz regular domain in R 3 .Following the notations in [1], section 3.2, we introduce the sets Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
As we can see, the arguments in [1] to prove (1) and ( 2) are enough for a C 1 -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 C 1 -regular domain D.
Finally, we mention that in [2] the C 1,1 -regularity of the domain is required to justify this method.

Data availability statement
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.