CORRIGENDUM: The inverse obstacle problem for nonlinear inclusions (2025 Inverse Problems 41 055002)

In [2], devoted to solving the inverse obstacle problem for nonlinear materials via the Monotonicity Principle, two issues were found in the formulation of some hypotheses. Specifically, the second assumption in (B2)/(4.29), i.e. (formula presented) is inconsistent for large s, when 1<2. Indeed, for those values of q, the upper bound vanishes as s→+∞, thus contradicting the assumed lower bound. Similarly, the second assumption of (B3)/(4.43), i.e. (formula presented) is inconsistent for large s, when q>2. Indeed, the lower bound diverges as s→+∞, which contradicts the existence of a finite upper bound γ―.Regarding the first issue, assumption (B2)/(4.29) for 1<2 is inconsistent and must be removed. Consequently, assumption (C2) for 1<2 is no longer required and must be removed.There exist two positive constants γ―⩽γ― such that (formula presented) There exist two positive constants γ― and s0 such that (formula presented) for a.e. x∈A― and for any s1,s2∈Rn.For fixed q⩾2, there exist three positive constants γ―⩽γ― and s0 such that (formula presented) for a.e. x∈A― and ∀s>0.For fixed q⩾2, there exist two positive constants γ― and s0 such that (formula presented) for a.e. x∈A― and for any s1,s2∈Rn.For fixed 1<2, there exists a positive constant γ― such that (formula presented) for a.e. x∈A― and ∀s>0.For fixed 1<2, there exist two positive constants γ― and s0 such that (formula presented) for a.e. x∈A― and for any s1,s2∈Rn.The revised assumptions impact two of the four converse results proven in [2]. Specifically, they impact the converse of the Monotonicity Principle when: (i) the anomaly is more conducting than the background and, moreover, its (nonlinear) electrical conductivity approaches +∞ (section 4.3 of [2]), and (ii) the anomaly is less conducting than the background and its (nonlinear) electrical conductivity approaches 0 (section 4.4 of [2]). The other two cases, where the electrical conductivity of the anomaly is bounded from zero and from +∞ (sections 4.1 and 4.2 of [2]), are unaffected by the present changes.Assumption (4.29) bounds the growth rate of the material property. The case q⩾2 corresponds to unbounded material properties increasing with the amplitude of s (γ→+∞, for s→+∞).In the present corrigendum we need to match the notation of [2] and that of [1]. Specifically, σ and E of [1] stand for γ and s of [2], respectively. The proofs in this corrigendum are given in the language used in [1].For fixed 1<+∞, there exists two positive constants σ― and E0 such that: (formula presented) for a.e. x∈B― and ∀E⩾0.For fixed 1<+∞, there exists two positive constants σ― and E0 such that: (formula presented) for a.e. x∈A and ∀E>0.For fixed 1<+∞, there exist two positive constants σ― and E0 such that: (formula presented) for a.e. x∈B, and for any E1,E2∈Rn.For fixed 1<+∞, there exist two positive constants σ― and E0 such that: (formula presented)  for a.e. x∈A, for any E1,E2∈Rn.Comparison of conductivity hypotheses for 1<2. Left: electrical conductivity fulfilling the hypothesis in [1]; the upper and lower bounds, and, hence, the electrical conductivity, are singular in E=0. Right: electrical conductivity fulfilling the new hypothesis; the upper bound only is singular in E=0. The lower bound is no longer approaching +∞ and, therefore, the electrical conductivity does not need to approach +∞ and may approach a finite limit. The figure is obtained for p=1.1, σ―=20 S m −1, σ―=10 S m −1, and E0=1 Vm>−1.Revised figure 8 in [2].The coercivity of the Dirichlet Energy density Q(⋅,⋅), as proved in theorem 2.7 of [1].The first inequality of chain (3.5) of [1] that appears in the proof of claim (i) of lemma 3.1 of [1], and the analogous inequality appearing in the proof of claim (ii) of the same lemma 3.1.Here, the estimate (2.2) of [1] is re-evaluated from below by using (2.1) and (2.2) of this corrigendum.Here, the leftmost inequality in (3.5) of [1] is re-evaluated by using (2.4) of this corrigendum. Regarding the second issue, i.e. assumption (B3)/(4.43), it has to be modified as follows: (formula presented) and for 1<2.Specifically, the case 1<2 in (B2)/(4.29) leads to an inconsistency, as clarified, and has been deleted, while the case q⩾2 is consistent and does not require any modification. Similarly, the case q⩾2 in (B3)/(4.43) leads to an inconsistency; however, it has been modified to account for 1<2.The modifications proposed in this section concern assumptions (P2) and (Q1) of [1], since it has been found that the lower bounds are no longer needed.We recall that the Dirichlet Energy density is defined as (formula presented) and its lower bound can be found by exploiting the new lower bounds given by (2.1) and (2.2). Specifically, a straightforward computation gives (formula presented) for a.e. x∈B and for any E>0. Finally, a further lower bound valid for any p>1 is given by (formula presented) for a.e. x∈B and for any E>0.Specifically, by using the first inequality of (2.4) (formula presented) where (formula presented) is a proper constant introduced in [1]. © 2026 The Author(s). Published by IOP Publishing Ltd. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.