Superconductive and insulating inclusions for linear and non-linear conductivity equations

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(\sigma\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $\sigma\colon\Omega\to[0,\infty]$ is zero or infinity in large sets and $1

1. Introduction. We study inverse boundary value problems for the partial differential equation (1) div(σ(x) |∇u(x)| p−2 ∇u(x)) = 0, where the measurable coefficient σ ≥ 0 is allowed to take the values 0 and ∞ in large sets and the exponent is in the range 1 < p < ∞. This includes the case p = 2 where our PDE becomes the linear conductivity equation div(σ∇u) = 0 which appears in Calderón's inverse problem [17].
Kirchhoff's law retains its linear form and consequently our PDE is of divergence form. This is convenient for the study of weak solutions and calculus of variations. Weak solutions of (1) in a domain Ω ⊂ R n are minimizers of the energy functional (2) E(u) = Ω σ(x) |∇u(x)| p dx in the space W 1,p (Ω) with some prescribed boundary values. This is true even if σ takes the values 0 and ∞ in non-empty open sets. This is our main result for the direct problem, and the details will be discussed in sections 1.1 and 3.
Our PDE can be classified as a quasilinear elliptic equation. We do not, however, bound the coefficient σ away from zero or infinity, so ellipticity holds in a weaker sense than usual. If σ ≡ 1, the solutions of (1) are known as p-harmonic functions. They have been studied extensively (see for instance [46,25,32,49] and the references therein), but inverse problems for elliptic equations of this type have received considerably less attention.
We assume our potential to be real-valued since this is physically most relevant. For complex-valued functions u one can obtain essentially the same results with the same tools, but we restrict our attention to the real case (apart from section 2.3).
Our goal is, given Dirichlet and Neumann boundary values of all solutions of (1), to reconstruct the shape of an unknown obstacle having zero or infinite conductivity. This data is encoded in the so-called Dirichlet-to-Neumann map which we will describe in section 1.1 and in more detail in section 3.2. When p = 2, there are very few results in this direction. When p = 2, this is a version of Calderón's famous inverse boundary value problem. We will summarize earlier results in section 1.2 and new results proven in this article in section 1.3. 1.1. The direct problem. Before embarking on a study of inverse problems, it is good to show that the direct problem is well-posed. The well-posedness result we present is, to the best of our knowledge, new in its generality.
Let Ω be a bounded domain and σ : Ω → [0, ∞] a measurable function. For simplicity, we make the standing assumption that the sets D 0 = σ −1 (0) and D ∞ = σ −1 (∞) are open and the three sets ∂Ω, D 0 and D ∞ are disjoint. We also assume that outside the sets D 0 and D ∞ the function σ is bounded away from both zero and infinity.
In this setting we look for minimizers of the energy (2) in W 1,p (Ω), given boundary values f ∈ W 1,p (Ω)/W 1,p 0 (Ω). Minimization of this energy corresponds to solving an Euler-Lagrange equation.
Usually such problems are posed on the domain Ω \ (D 0 ∪ D ∞ ) and one would impose suitable boundary conditions on ∂D 0 and ∂D ∞ . We prefer to work with all of Ω when possible, for in the inverse problem the function σ and therefore the sets D 0 and D ∞ are unknown. See remark 3.10 for the usual formulation. Let p ∈ (1, ∞). Fix any boundary value f ∈ W 1,p (Ω)/W 1,p 0 (Ω). The energy (2) has a minimizer u ∈ W 1,p (Ω) with boundary values u| ∂Ω = f . The minimizer is unique up to functions that have vanishing gradient outside D 0 and that vanish on ∂Ω.
A function u ∈ W 1,p (Ω) is such a minimizer if and only if it satisfies the Euler-Lagrange equation div(σ |∇u| p−2 ∇u) = 0 weakly in the sense that For a proof, see theorems 3.4 and 3.7 and their proofs. This theorem is true, in particular, in the linear case p = 2. Then the PDE is div(σ∇u) = 0.
We point out that even though the minimizer u of the energy is not unique, the vector-valued function σ |∇u| p−2 ∇u is unique. Since the exceptional sets D 0 and D ∞ do not reach the boundary, the boundary values can be interpreted in the Sobolev sense as usual.
1.2. Known results. Let us first fix p = 2. If σ is assumed to be bounded away from zero and infinity, the DN map Λ σ determines σ in two dimensions [5]. In higher dimensions this is true if σ is additionally assumed to be Lipschitz [19]. For more information about Calderón's problem, we refer to [17,66]. Astala, Lassas and Päivärinta [4] have investigated anisotropic conductivities that are not bounded from above or away from zero. A different kind of problem is to reconstruct an inclusion -a subdomain of Ω with zero, infinite or non-zero finite conductivity in a constant background conductivity -from the DN map. Problems of this kind are our object of study.
There are several methods for recovering inclusions with zero or finite conductivity by making boundary measurements of both voltages and currents [39,20,14,44,29,34,58,35,54,30]. For regions of infinite conductivity a recent article by Ramdani and Munnier [52] shows how to detect infinitely conductive bodies from the Dirichlet to Neumann map in two dimensional domain for the linear conductivity equation. Their method is based on geometry in the complex plane, in particular Riemann mappings. Brühl [ [21] have shown that one can recover the location and scale of a finite number of small inclusions with zero or infinite conductivity in an inhomogeneous background from the DN map. Superconductive but grounded inclusions have been detected in, for example, [45,8]. Moradifam, Nachman and Tamasan [51] consider a single interior measurement for conductivity equation and detect insulating or perfectly conducting inclusions, though their approach to the direct problem is not variational. Perfectly conducting inclusions in the context of the Maxwell equations have been detected by sampling methods [24,15].
Other results relating to infinitely conducting obstacles concern the situation of two such obstacles being close to each other and the main concern is the blow-up of the solutions [40,27].
For other values of p much less is known. We assume 1 < p < ∞ throughout this article. The p-conductivity equation (1) with infinite conductivity has been considered by Gorb and Novikov [27] for 2 ≤ p ∈ N in dimensions two and three.
We are not aware of a rigorous treatment of the forward problem in the literature, so we provide one in section 3 as summarized above in section 1.1.
The p-Calderón's problem, or Calderón's problem related to the p-conductivity equation, was introduced by Salo and Zhong [60]. They recover the conductivity on the boundary of the domain. Brander [9] improved the result to first order derivative of conductivity on the boundary, but with increased regularity assumptions. A recent result by Brander, Kar and Salo [10] shows that one can detect the convex hull of an inclusion with conductivity bounded away from zero and infinity. Further results related to the enclosure method and the monotonicity method for the p-Laplace equation can be found in [11]. Very recently Guo, Kar and Salo [28] proved that under a monotonicity assumption the DN map is injective for Lipschitz conductivites when n = 2 for general p, and when n ≥ 3 when one of the conductivities is almost constant. Their results assume the conductivity to be bounded from above and away from zero.
1.3. New results. First, we study the problem of recovering the set D ∞ from the DN map when p = 2, D 0 = ∅ and σ ≡ 1 outside D ∞ . Here we mainly follow the probe and enclosure methods of Ikehata [34,35] to reconstruct the unknown inclusion.
Let Ω ⊂ R n , n ≥ 2, be a bounded C 1 domain and let D be a Lipschitz domain compactly contained in Ω. Assume furthermore that each connected component of Ω \ D meets ∂Ω. Let If p = 2, then the Dirichlet-to-Neumann map determines the set D.
In the notation introduced for the direct problem, D = D ∞ . We will prove this theorem using the probe method. When we use the enclosure method instead, we can prove recoverability of the convex hull of D ∞ if D ∞ ⊂ R n is connected. For a precise statement of this result, see proposition 2.7. For more details and proofs, see section 2.
For general p ∈ (1, ∞) we only use the enclosure method. The Runge approximation property for the p-Laplace equation is not known, so the probe method is not applicable for this non-linear model. We assume that one of D 0 and D ∞ is empty and the other domain (called D) has Lipschitz boundary. We are unable to detect the convex hull of D, but we can recover a larger set, giving an estimate from above for the inclusion. We can also determine whether D = ∅ or not. See corollary 4.9 and the preceeding discussion for more details.
If we allow σ to take the values 0 or ∞ in large sets, the DN map Λ σ no longer determines σ uniquely. For example, consider the domain Ω = B(0, 3) ⊂ R n and a function σ : Ω → [0, ∞] which is zero or infinity on B(0, 2) \ B(0, 1). Then the values of σ in B(0, 1) have no effect on the DN map. We can only ever hope to recover σ up to the "outermost boundaries of D 0 and D ∞ " and whether or not these boundaries belong to D 0 or D ∞ . 2. Detecting perfectly conducting inclusions for p = 2. In this section we consider the linear case p = 2. Our problem is to detect an inclusion of superconductive material within a homogeneous background medium. We justify two types of reconstruction methods: the probe method and the enclosure method. These methods were proposed by Ikehata. In [35], he introduced the enclosure method where he first used the CGO solutions with linear phase to detect the convex hull of the obstacle with finite or zero conductivity. Regarding the probe method [34], he proposed to use the fundamental solution as a test function to detect the unknown inclusions of zero and finite conductivity for the conductivity equation. Since then a lot of work has been done in this direction for various linear PDEs, see [35,38] for an overview. We mention some examples: the Helmholtz model [53,55], Maxwell systems [42,69] and the linear elasticity equations [41,54].
In this article we apply these two methods to the linear conductivity model to detect the perfectly conductive obstacles and this is the first result which concerns the perfectly conductive case. In the zero conductivity case, estimates for the difference of the DN maps do not involve any boundary integrals [35,Lemma 4.1]. However, due to the presence of the boundary integral, see equation (10), it is more difficult to obtain the right kind of estimate in our situation. We overcome the technical difficulties by using the boundary layer potential theory for the Laplace operator. The rigorous analysis is in subsection 2.2.
Let us repeat the setting of theorem 1.2. Let Ω ⊂ R n , n ≥ 2, be a bounded domain with C 1 boundary and D ⊂ Ω be an open Lipschitz domain so that all components of Ω \ D meet ∂Ω. If ∂Ω is connected, this amounts to requiring that Ω \ D is connected. We consider the obstacle problem where ν is the outward unit normal to Ω \ D. The above problem formulates the situation where D is superconductive. To see why this formulation corresponds to σ ≡ ∞ in D, how to formulate the problem in Ω instead of Ω \ D and how to pose the problem in the weak form, see section 3 and especially remark 3.10. Notice that the constant value of u in the connected components depends on f and can be different in different components.
We index the Dirichlet to Neumann map by the domain D instead of the function σ. The DN map Λ D : H 1/2 (∂Ω) → H −1/2 (∂Ω) is defined by Λ D (f ) = ∂ ν u| ∂Ω . A weak version of Λ D is given by where g ∈ H 1/2 (∂Ω), φ ∈ H 1 (Ω) with φ| ∂Ω = g, ∇φ = 0 in D, and u satisfies (6). We define similarly a DN map in the case D = ∅; this is just the DN map for the Laplacian on Ω. We call it the free DN map and denote it by Λ ∅ . That is, in the weak sense.
Our goal is to reconstruct the shape of the superconductive inclusion D from the knowledge of the Dirichlet-to-Neumann map Λ D measured at ∂Ω.
Before presenting the enclosure and the probe method, we would like to state the following inequalities.
Proof. Proof of (10): Note that, where g ∈ H 1/2 (∂Ω), φ ∈ H 1 (Ω) with φ| ∂Ω = g, ∇φ = 0 in D, and u satisfies (6). Since u = u 0 = f on ∂Ω, and by taking φ = u, the left hand side of the above integral identity becomes Similarly, the free DN map can be written as where g ∈ H 1/2 (∂Ω) and φ ∈ H 1 (Ω) with g = φ| ∂Ω and u 0 satisfies (8). Since u = u 0 = f on ∂Ω, replacing φ = u or φ = u 0 in the above inequality gives us Using the inequality Recall that, w = u − u 0 satisfies (9). So, multiplying by u 0 on both sides of ∆w = 0 and integrating by parts we obtain Subtracting (18) from (20) multiplied by two, we obtain Proof of (11): We observe that where The third identity follows from equation (15) and the second last equality is due to the fact that ∇u = 0 in D. Since I ≤ 0, the required inequality now follows from (22).
There is a constant C independent of f so that and Proof. In this proof C will always denote a universal constant independent of f , but the constant may be different in different estimates.
Since w = u − u 0 , the triangle inequality gives is a classical continuity estimate for the solution operator of the Laplacian, so (25) implies (24). Since w has zero boundary values in the Sobolev sense, it suffices to establish Using the triangle inequality and the classical estimate for u 0 H 1 (Ω) again, we find that it suffices to prove We can establish (28) using the fact that u solves (6). We have ∆u = 0 in Ω \ D and ∇u = 0 in D, so The boundary term on ∂D and the interior term in Ω \ D vanish due to (6). Now estimating finishes the proof of (28) and therefore also those of (27), (25) and (24).
The following lemma is a consequence of the previous one. We do not need it, but we record it here for completeness. Proof. Linearity is an elementary observation, and continuity follows from lemma 2.2. Namely, estimate (24) yields Since u is constant in each D k , this proves the lemma.

2.2.
Layer potentials and a boundary integral estimate. In this subsection our main aim is to prove the following proposition. Let Ω ⊂ R n , n ≥ 2, be a bounded C 1 -smooth domain and D ⊂ R n be a bounded and connected subset of Ω with Lipschitz regular boundary.
Proposition 2.4. Let the reflected solution w = u − u 0 satisfies (9). We have the following integral estimate: where C > 0 be a constant independent of u 0 .
The main ingredient to show the above estimate is first to write the solution of the problem (9) in terms of the single layer potential and then to use the properties of the layer potential operator in the appropriate Sobolev spaces. Before going into details of the proof of this proposition, we would like to recall the integral operators of single layer type and their mapping properties.
Let E ⊂ R n , n ≥ 2, be an open bounded Lipschitz domain. We denote interior and exterior region by E + := E and E − := R n \ E respectively. We first define the single layer potential operator corresponding to the boundary ∂E as follows: and Φ is the fundamental solution for the Laplacian. If n = 2, we choose r > diam Ω and Φ = 1 2π log r |x| . For anyq ∈ H − 1 2 (∂E), the function S ∂Eq solves the Laplace equation ∆(S ∂Eq ) = 0 in the region R n \ ∂E. It is well known that the single layer potential operator defines a bounded linear operator from H − 1 2 (∂E) to H 1 loc (R n ), see for instance [50,Theorem 6.11]. We now define the function space, for n ≥ 2, The (interior/exterior) non-tangential boundary traces of S ∂Eq are given by (35) lim where S ∂Eq is the trace of the single layer operator S ∂Eq on ∂E, i.e., and Γ ± denote the interior of the two components (in E + and in E − ) of a regular family of circular doubly truncated cones {Γ(x); x ∈ ∂E} with vertex at x (for definition see for example [67, section 0.4]). In addition, for almost any x 0 ∈ ∂E, where K * is the adjoint of the double layer potential operator, see for instance [67]. We now define the equilibrium density for ∂E as in [50,Theorem 8.15]. There exists a unique distribution ψ eq ∈ H − 1 2 (∂E) so that S ∂E ψ eq is constant on ∂E and ψ eq , 1 − 1 2 , 1 2 ,∂E = 1. In addition, if n ≥ 3, then S ∂E ψ eq > 0. We now define the capacity of the boundary in terms of the equilibrium density. If n ≥ 3, the capacity of ∂E is denoted as Cap(∂E) and defined as Cap(∂E) = 1 S ∂E ψeq and Cap(∂E) = exp(−2πS ∂E ψ eq ) when n = 2. Following McLean [50], see also [52], we define is defined by (40) q := S ∂Eq .
For n = 2, the operator S ∂E is an isomorphism if Cap(∂E) = r, see [50,Theorem 8.16]. When n ≥ 3, S ∂E is injective [50, corollary 8.11]. Also, it is of Fredholm index zero [50, theorem 7.6]. Hence S ∂E is an isomorphism. Note that we do not need any assumption on the capacity of the domain for n ≥ 3. With respect to the inner products , the above isomorphism is actually an isometry through the identities The second equality can be realised by using the asymptotic behaviour of the single layer potential at infinity and Green's formula, see [50,Theorem 8.12]. We also need the following orthogonal decomposition in our analysis. Let us first define an orthogonal projection which is a bounded linear operator with the following unique decomposition [52]. Denote by Tr ∂E , the trace operator mapped into H 1 2 (∂E) and also Tr 0 ∂E := Π ∂E (Tr ∂E ). Let us assume that the diameter of Ω is less than r only when n = 2. In the higher dimensional case we do not need the assumption since the operator S ∂E is an isomorphism even without any assumptions on the capacity of the domain. In our setting D is a bounded Lipschitz domain included in Ω. So, the assumption on the diameter of Ω in R 2 implies that Cap(∂Ω) < r and Cap(∂D) < r [50, exercise 8.12]. Recall the model (9) in the case where D is connected. The reflected solution Here c f is a constant depending on f . (This dependence is linear and continuous by lemma 2.3.) To solve the problem (46), we use the integral equation method following [52,Proposition 2.12]. We represent the solution w as the superposition of single layer potentials on the boundaries ∂D and ∂Ω as follows: wherep ∈Ĥ(∂D) andq ∈ H − 1 2 (∂Ω). The density functionsp andq satisfy the following system of integral equations on ∂D and ∂Ω: We define here the boundary interaction operators K ∂D ∂Ω and K ∂Ω ∂D between ∂Ω and ∂D as follows: , for more details see [52].
Before presenting the proof of Proposition 2.4, we state the following lemma.
Lemma 2.5. The operators defined above have the following properties: (ii) The norms of the operators K ∂D ∂Ω and K ∂Ω ∂D are strictly less than 1. Proof. (i) The first part of the lemma is proved in [52, Proposition 2.9] for a planar domain. However, for n ≥ 3, we have the asymptotic behaviour of the single layer potential at infinity of the form (43). In this case, the proof goes similarly to [52,Proposition 2.9].
(ii) For n = 2, the proof follows from [52,Proposition 2.11]. In higher dimensions the proof goes along the same lines as in [52,Proposition 2.11]. For the convenience of the reader we present more details. To prove the appropriate norm bounds for the operators K ∂D ∂Ω and K ∂Ω ∂D , we first define certain kind of quotient weighted Sobolev spaces: Under the inner product the space W 1 0 (R n ) is a Hilbert space. For any q ∈ H(∂Ω), we have The above identity can be proved by considering an orthogonal decomposition of the Hilbert space W 1 0 (R n ) in terms of the single layer potential operator. See [52,Proposition 2.7] for the proof in R 2 ; the same technique works in higher dimensions. Therefore (57) is achieved for some q ∂Ω ∈ H(∂Ω). If we suppose Finally using the jump relation of the derivative of the single layer potential, we haveq ∂Ω = 0, which gives a contradiction. Thus, K ∂D ∂Ω < 1 and the proof is complete.
We are now in a position to present the details of the proof of proposition 2.4.
Proof of proposition 2.4. For convenience, we use the following notation: where ∂ ∂ν (S ∂Dp )| + x0 denotes the Neumann trace of the single layer potential at x 0 from the inside D. More precisely, for x 0 ∈ ∂D, Similarly, the Neumann trace of w − from the exterior of D at x 0 is where ∂ ∂ν (S ∂Dp )| − x0 denotes the Neumann trace of the single layer potential at x 0 from the inside Ω \ D. More precisely, for x 0 ∈ ∂D, ∂ν .
Therefore, the jump of the Neumann trace at the point x 0 is given by Finally, the boundary integral ∂D ∂w ∂ν u 0 dS can be estimated as follows: Note that ∂w + ∂ν so applying the trace theorem we estimate integral I 1 by Using (63) and the trace theorem, we obtain where C is a positive constant independent of u 0 .
2.3. The enclosure method. We start with the enclosure method. The basic idea of the this method is to use the complex geometrical optic (CGO) solution as a test function instead of using the fundamental solution of the Laplace operator.
We follow the approach of Ikehata [35]. We mainly use the CGO solution with linear phase function for detecting the convex hull of the obstacle. Here it is more convenient to work with complex-valued functions, but in the linear case p = 2 the real and complex versions of the inverse boundary value problem are trivially equivalent. CGO solutions with linear phase for the Laplacian can be constructed as follows. Take any unit vector ρ ∈ R n and an orthogonal unit vector ρ ⊥ ∈ R n . Let t and τ be real numbers with τ > 0. Then the function u 0 (x) = exp(τ (x · ρ + ix · ρ ⊥ − t)) is harmonic. Given ρ, we consider ρ ⊥ fixed although there is a freedom of choice; the results do not depend on the choice and we therefore omit ρ ⊥ in our notation.
We define an indicator function as where ρ ∈ S n−1 , τ > 0, t ∈ R and Λ D , Λ ∅ are the DN maps for the problems (6) and (8) respectively. For ρ ∈ S n−1 we define the convex support function h D (ρ) of D as One can reconstruct the convex hull of the superconductive material D from the following behavior of the indicator function.
where c, C > 0, and τ > 0 for the upper bound, and τ 1 for the lower bound. 3. When t < h D (ρ) we have and more precisely, where τ 1 and c, C > 0. Note that I ρ (u 0 , τ, t) = exp(2τ (h D (ρ)−t))I ρ (u 0 , τ, h D (ρ)). Therefore to prove the above properties of the indicator function it is enough to prove the inequality (78). To do that, we state the following lemma, see [10,Lemma 4.7] for the proof.
Therefore estimate (78) follows from proposition 2.4 and the properties of the CGO solutions as described in lemma 2.6. Using the above type of CGO solution, we describe a reconstruction procedure for the enclosure method as follows: Step 1. For each direction ρ ∈ S n−1 , we first define an indicator function I ρ (u 0 , τ, t) via the use of CGO solution and the corresponding measurement data (DN map) as in (73).
Step 2. When the hyperplane (level set) {x ∈ R n ; x · ρ = t} moves along ρ for each ρ and t, observe the asymptotic behavior of I ρ (u 0 , τ, t) as τ 1 as in (76), (78) and (80). Then one can know whether or not the level set of the phase function of the CGO solution touches the interface of the obstacle.
Step 3. Now, the support function h D (ρ) can be determined by the following formula: Step 4. Finally, the convex hull of D can be reconstructed by taking the intersection of {x ∈ S n ; x · ρ ≤ h D (ρ)} in a dense set of directions ρ ∈ S n−1 .
Using the enclosure method we have thus proven the following proposition. Note that it is weaker than theorem 1.2. Both results are constructive, as their proofs come with a method for finding the inclusion or its convex hull.

Proposition 2.7.
Let Ω ⊂ R n be a bounded Lipschitz domain and D ⊂ Ω a connected and compactly contained smaller Lipschitz domain. Consider a conductivity σ of the form and the corresponding DN map Λ D = Λ σ . The function h D of (74) is determined by the indicator function, which in turn is determined by the DN map Λ D . The convex hull of D can be found in terms of h D : Moreover, to recover the non-convex part of the superconductive obstacle, one can also use the complex geometrical optic solutions with the logarithmic phase for the Laplacian. CGOs with logarithmic phase have been used to reconstruct the inclusions with zero or finite conductivities for the Helmholtz equation [55,62] and for elastic model [41]. In the present situation one can also justify the asymptotic properties of the indicator function (76), (78) and (80) using proposition 2.4 and the properties of the CGO solutions with logarithmic phase.
2.4. The probe method for p = 2. In this subsection we start with the probe method, which we describe briefly here. For more details, consult the papers of Ikehata [35,34,37] and references therein.
We now introduce the "impact parameter" t(γ; D) which depends on a needle γ and the obstacle D. The impact parameter tells us the point in time when a needle hits the obstacle. We define the hitting time as (85) t(γ; D) = sup{τ ∈ (0, 1); γ(t) ∈ Ω \ D for all t < τ }.
We define an indicator function which indicates whether or not a needle touches ∂D. (86) where the sequence {f k (·, γ(t))} k is chosen according to the following proposition. We could also define the indicator function as limit inferior without changing anything.

Proposition 2.8 ([36, Proposition 2]).
Let γ be a needle as described above. Then for any t ∈ (0, 1) there exists a sequence of functions f k (·, γ(t)) ∈ H 1/2 (∂Ω) such that the solution v k of where Φ is the fundamental solution for the Laplacian. Moreover, for any fixed open Γ ⊂ ∂Ω with non-empty exterior we can assume that each f k (·, γ(t)) vanishes on Γ.
The proof of the proposition uses the Runge approximation property of −∆.
Notice that the sequence f k (·, γ(t)) does not depend on D. See [36, Proposition 2] for more details.
The set T (γ) can be calculated from the DN map. We will show that (89) T (γ) = (0, t(γ; D)), so the impact parameter can be reconstructed using t(γ; D) = sup T (γ). We also have that t(γ; D) < 1 if and only if lim t→t(γ;D) I(t, γ) = ∞. This will allow us to reconstruct the point γ(t(γ; D)) for any choice of γ. Under the topological assumptions of theorem 1.2 these points determine the set D. To justify these statements, we need some inequalities.
where C is a positive constant. Moreover, if t is such that γ(t) ∈ D, then I(t, γ) = ∞.
Proof. Consider a sequence of functions v k ∈ H 1 (Ω) such that Applying proposition 2.8 and trace theorem we obtain . Hence the estimates follow from lemma 2.1 and proposition 2.4. Suppose next that γ(t) ∈ D. For every ε > 0 define which is a compact subset of Ω \ γ ([0, γ(t)]). In particular, since the trace of γ is compact, each point in its complement has positive distance to it and hence Thus, by proposition 2.8 and lemma 2.1, we obtain (97) for every ε > 0. Letting ε → 0, we obtain The integrals converge as ε → 0 by the monotone convergence theorem, we can ignore the trace of γ since it has zero measure, and the last integral is infinite by Lipschitz regularity of D and the singularity of the fundamental solution Φ(·, γ(t)) at γ(t).
We are now ready to present a proof of theorem 1.2.

The direct problem.
3.1. Well-posedness. Let us now carefully formulate the direct problem and see that it is well-posed.
Proof. Let W 1,p 0 (Ω) be the closure under the W 1,p (Ω) norm of the space of smooth functions in Ω supported away from Γ. The prime reminds that the zero boundary value in the Sobolev sense is only assumed on Γ which may be a proper subset of ∂Ω.
Suppose there was no such constant C. Then there is a sequence of functions We may normalize this sequence so that u k L p (Ω) = 1 and ∇u k L p (Ω) < 1/k. By the Rellich-Kondrachov theorem there is a subsequence converging in L p (Ω). The Rellich-Kondrachov theorem holds in Sobolev extension domains, and a finite union of Sobolev extension domains with positive distance from each other also admits an extension operator. We denote the subsequence by (u k ) and the limit function by u.
For any test function η ∈ C ∞ 0 (Ω) we have k→∞ Ω η∇u k = 0, using W 1,p regularity of each u k and the norm bound on ∇u k . Therefore u is weakly differentiable and its weak gradient is identically zero. We have in fact u ∈ W 1,p (Ω) and the convergence u k → u happens also in W 1,p (Ω), so u ∈ W 1,p 0 (Ω). Since u has zero weak gradient, it must be constant on each connected component. The only possible constant value is zero due to the zero boundary value on Γ and the connectedness assumption. This contradicts the normalization u k L p (Ω) = 1 for all k.
Proof. This is a special case of lemma 3. Proof. We simply observe that since a p-harmonic function minimizes, by definition, the L p norm of the gradient. Fix p ∈ (1, ∞). Given any f ∈ W 1,p (Ω), there is a minimizer u ∈ f + W 1,p 0 (Ω) to the energy The minimal energy is finite and the minimizer is unique modulo functions that have zero gradient outside D 0 but still satisfy the Dirichlet boundary condition on ∂Ω.
Note that the essential boundedness of log σ is equivalent to the existence of c > 0 such that for almost all x we have 1/c < σ(x) < c.
Remark 3.5. The result which states that the minimizer is unique modulo functions that have zero gradient outside D 0 is simplified when the closure of each connected component of Ω \ D 0 intersects ∂Ω and in particular when Ω \ D 0 is connected. Under this assumption the uniqueness holds modulo W 1,p 0 (D 0 ). The minimizer can be made unique with a small number of adjustments. It is clear that changing the function u in D 0 does not change E(u). Therefore we consider the quotient space The energy functional E is well defined on this quotient space and E(u) < ∞ for all u ∈ S.
Since the only thing that matters about f are its boundary values and the sets ∂Ω, D 0 , and D ∞ are disjoint, we may assume that f vanishes on D 0 and D ∞ .
We denote the equivalence of u ∈ A by Notice that the truth value of u − f ∈ W 1,p 0 (Ω) does not depend on the choice of the representative of the equivalence class [u] ⊂ A, so S f is well-defined. We also use the space S 0 where the boundary value is zero instead of that of f . We shall show that there is a unique minimizer of E in S f .
Let U be a connected component of Ω \ D 0 so that U ∩ ∂Ω = ∅. Then one can shift the values of u ∈ S in U by a constant without changing u as an element of S. A minimizer must clearly have vanishing gradient in U , so we may assume that the minimizer (and all functions in a minimizing sequence) vanishes in U . We therefore assume, to the end of simplifying presentation, from now on that there are no such components U .
For any u ∈ S we pick a preferred representativeū ∈ W 1,p (Ω) by demandingū to be p-harmonic in D 0 . The boundary values ofū on ∂D 0 are determined by u. Let now (u k ) be a minimizing sequence of E in S f . Using the estimate (111) and u k − f ∈ S 0 , we get so for some other constant C we have It follows from the assumption on log σ that σ| Ω\D0 ≥ λ for some λ > 0. Therefore Since E(u k ) is bounded, this estimate guarantees that also the sequence (u k − f ) is bounded in W 1,p 0 (Ω). Therefore there is a subsequence (which we denote by the sequence itself) which converges weakly in W 1,p (Ω). Let u 0 − f be the limit function.
The energy functional E(u) is just a weighted Dirichlet energy of u| Ω\(D0∪D∞) with weight bounded away from zero and infinity. Such functionals are weakly lower semicontinuous and u k − f converges weakly to u 0 − f also in W 1,p (Ω \ (D 0 ∪ D ∞ )) (with the functions restricted appropriately). Therefore so u 0 indeed minimizes the energy. Suppose there are two minimizers u 0 and v 0 of E in S f . Since the functions are distinct, they must differ in Ω \ (D 0 ∪ D ∞ ), and so the set {x ∈ Ω \ (D 0 ∪ D ∞ ); ∇u 0 (x) = ∇v 0 (x)} must have positive measure. By strict convexity of t → t p this implies E( 1 . But this is impossible because u 0 and v 0 minimize the energy, so u 0 = v 0 as elements in S f .
The fact that our unique minimizer solves the PDE follows from standard techniques in the calculus of variations.
We remark that the weak limit of a locally uniformly bounded sequence of pharmonic functions is p-harmonic. 1 Therefore the minimizer constructed in the proof above is p-harmonic in D 0 at least if f is bounded. It is also p-harmonic (weighted with σ) in Ω \ (D 0 ∪ D ∞ ), but it need not be p-harmonic in any sense across ∂D 0 . Using p-harmonic extensions to D 0 was just a matter of convenience; the values in D 0 are irrelevant for the energy. All minimizers are p-harmonic in D ∞ since the gradient must vanish identically there.
The variational problem also leads to a PDE in the whole domain Ω despite σ being zero or infinite as we shall see next, as we provide a weak formulation for the equation div(σ |∇u| p−2 ∇u) = 0. for all φ ∈ W 1,p 0 (Ω) satisfying ∇φ = 0 on D ∞ . Proof. Let us again denote A = {u ∈ W 1,p (Ω); ∇u| D∞ = 0} and in addition A 0 = A ∩ W 1,p 0 (Ω). Suppose u is a minimizer of E inf + A 0 , where f −f ∈ W 1,p 0 (Ω) and ∇f = 0 in D ∞ . Take any φ ∈ A 0 . Now 1 Every locally uniformly bounded family of weak solutions to the p-Laplace equation is equicontinuous [32,Theorem 6.12] and thus by the Ascoli-Arzelà theorem has a uniformly convergent subsequence. A locally uniform limit of solutions is still a solution [32,Theorem 3.78]. A uniform limit is also a weak W 1,p limit. Commuting differentiation and integration is possible by the dominated convergence theorem and the mean value theorem for the map t → |∇(u + tφ)| p . Conversely, suppose (116) holds for all φ ∈ A 0 . The map R n ξ → |ξ| p ∈ R is convex and smooth outside the origin, so for all x ∈ Ω. Integrating this with weight σ and using (116), we have We also have a result for general φ -not just φ ∈ A 0 -by formal integration by parts.
Remark 3.9. Given the boundary values, a minimizer u of the energy E is not unique, but the function σ |∇u| p−2 ∇u is unique. This is the only quantity appearing in the weak Euler-Lagrange equation (116). In fact, this function is in L p (Ω; R n ) because ∇u = 0 whenever σ = ∞, with the conjugate exponent p defined by Remark 3.10. The partial differential equation div(σ |∇u| p−2 ∇u) = 0 (or in weak form (116)) can also be formulated in the domain Ω \ (D 0 ∪ D ∞ ). We only need to find the correct boundary conditions on ∂D 0 and ∂D ∞ . Let C be a connected component of D ∞ . Since ∇u must vanish in C, u is constant on C, but there is also another condition. We can choose a function φ ∈ C ∞ 0 so that φ ≡ 1 on C and φ ≡ 0 in D ∞ \ C. Comparing the two equations in theorem 3.7 (or integrating by parts), we observe that ∂C σ |∇u| p−2 ∂ ν u = 0. Let us then find the boundary conditions on ∂D 0 . To that end, we take an arbitrary test function φ ∈ C ∞ 0 (Ω) vanishing in D ∞ for weak Euler-Lagrange equation (116). Since σ vanishes in D 0 , the integral over Ω in (116) is in fact an integral over Ω \ D 0 . Integration by parts gives ∂D0 σ |∇u| p−2 (∂ ν u)φ = 0. If this is to hold for all such φ, we obtain the Neumann boundary condition σ |∇u| p−2 ∂ ν u = 0 on ∂D 0 . Therefore the equation of theorem 3.7 can be reformulated as The constant may be different for different components and the constant values depend on the boundary data f .
More well-posedness results for the direct problem when p = 2 can be found in lemmata 2.2 and 2.3.
The next lemma shows that the minimal energy depends monotonically on the conductivity σ. We remind the reader that the functions u σ and u γ in the lemma are not unique but the minimal energy is.
The minimizers exist by theorem 3.4.

3.2.
The weak Dirichlet-to-Neumann map. We are now ready to define the weak Dirichlet-to-Neumann map (DN map). We assume Ω and σ to be as in theorems 3.4 and 3.7 above. The simpler case where D 0 = D ∞ = ∅ was treated in [60,31]. Let X = W 1,p (Ω)/W 1,p 0 (Ω) and X be its dual. The DN map Λ σ : X → X is defined by wheref ∈ W 1,p (Ω) is any minimizer of the energy functional E with boundary values f ∈ X andḡ ∈ W 1,p (Ω) is an extension of g ∈ X with ∇ḡ = 0 in D ∞ . Since D ∞ ∩ ∂Ω = ∅, there always exists such an extensionḡ, and the existence of a minimizerf follows from theorem 3.4. Under the same assumptions we also have For a sufficiently nice conductivity σ we can use arbitrary extensionsḡ ∈ W 1,p (Ω); their gradient does not have to vanish in D ∞ . See remark 3.8 for more details. The DN map Λ σ : X → X has an additional term in such cases: wheref ∈ W 1,p 0 (Ω) is any minimizer of the energy functional E with boundary values f ∈ X andḡ ∈ W 1,p (Ω) is any extension of g ∈ X.
Let us see why the DN map is well-defined. As pointed out in remark 3.9, the minimizerf is not unique, but σ ∇f p−2 ∇f is. To see that the definition is independent of the choice ofḡ, we need to show that the above expression for Λ σ f, g vanishes whenḡ ∈ W 1,p 0 (Ω) and ∇ḡ = 0 in D ∞ or the extra integral over ∂D ∞ is present. But this is just the claim of theorem 3.7 and remark 3.8.
Linearity of Λ σ f as a functional on X is evident (although f → Λ σ f is linear if and only if p = 2). To see that Λ σ f is continuous consider the estimate which gives . The constant C depends on σ but not on f .
In the case p = 2 we also have the following ellipticity estimate for the DN map.
Proposition 3.12. For any f ∈ H 1/2 (∂Ω) = W 1,2 (Ω)/W 1,2 0 (Ω), letf ∈ W 1,2 (Ω) be the extension to Ω as a solution of the problem of theorems 3.4 and 3.7 with p = 2, and letḡ ∈ W 1,2 (Ω) be any extension of g with ∇ḡ = 0 in D ∞ . To make the extension unique, we demandf to be harmonic in D 0 . Then the weak DN map is given by σ∇f · ∇ḡ and satisfies for any constant c. There is a constant λ > 0 so that Proof. Sincef solves the problem, it is constant in all connected components of D ∞ . The properties of the DN map given in (130) are easy to see. Using (130) we observe that Λ σ f, f = Λ σ f + c, f + c for any c ∈ R. If V denotes the space of constant functions on ∂Ω, the DN map is a well-defined bilinear functional on the quotient Hilbert space H 1/2 (∂Ω)/V . If the equivalence class of f ∈ H 1/2 (∂Ω) in H 1/2 (∂Ω) is denoted by [f ], we have (132) Continuity of this bilinear map follows from (129), and this establishes the upper bound in (131). Suppose the lower bound did not hold. Then there is a sequence of functions We can shift the functions f k without changing their equivalence classes so that Since f k H 1/2 (∂Ω) = 1 for all k, the sequence (f k ) is bounded in W 1,2 (Ω). Therefore the sequence (f k ) k has a subsequence (denoted by (f k ) k itself) so thatf k converges weakly to some functionf in W 1,2 (Ω), f ∈ H 1/2 (∂Ω).
Since the Dirichlet energy is weakly lower semicontinuous, we have Λ σ This contradiction finishes the proof of the lower bound in (131).
If we have more control on the boundary values, the ellipticity result becomes nicer. The proof is similar to the previous one, so we give an abridged version. For any f ∈ H 1/2 (∂Ω) = W 1,2 (Ω)/W 1,2 0 (Ω), letf ∈ W 1,2 (Ω) be as in proposition 3.12. There is a constant λ > 0 so that for all f that vanish on Γ.
Proof. If A = {u ∈ W 1,2 (Ω); u| Γ = 0} (the closure of the space of smooth functions supported away from Γ) and B is the trace space of A, then we only consider boundary values f ∈ B. Notice that by lemma 3.1 the norms · H 1 (Ω) and ∇· L 2 (Ω) are comparable on A and thatf ∈ A for any f ∈ B.
The upper bound follows from continuity of the DN map as a bilinear form, so we focus on the lower bound. If the lower bound does not hold, there is a sequence of functions f k ∈ H 1/2 (∂Ω) so that f k H 1/2 (∂Ω) is bounded away from zero but Λ σfk ,f k → 0. The sequence (f k ) k is bounded in W 1,2 (Ω), so up to a subsequence it converges weakly to a limitf ∈ A.

4.
Enclosure method for p-Calderón problem. For any subset D ⊆ R n we define the (convex) support function h D : S n−1 → R as We will use specific solutions that are oscillating in one direction and have exponential behaviour in a perpendicular direction. The solutions were first introduced by Wolff [68, section 3] (see also [48]) and later applied to inverse problems by Salo and Zhong [60, section 3].
In this section we assume that the conductivity σ is constant 1 outside the possibly empty open Lipschitz sets D 0 = σ −1 ({0}) and D ∞ = σ −1 ({∞}). Note that this conductivity function satisfies the assumptions of theorem 3.4, so the forward problem is well-posed. For definition of the DN map Λ σ , see section 3.2.
where Λ ∅ is the DN map associated with no obstacle and conductivity 1. We use the shorthand notation (Λ σ − Λ ∅ )f := Λ σ f − Λ ∅ f although the DN maps are non-linear. As we keep the directions ρ and ρ ⊥ fixed, we often omit them from our notation and write the indicator function simply as I(t, τ ).
We record the following equality, which follows from the definition of the Wolff solutions (definition 4.1) and the indicator function (definition 4.3). Recall that h D (ρ) is the convex support function. The following lemma is crucial for the proof of the lower bound: Lemma 4.5. Suppose 1 < p < ∞ and D ⊂ Ω has Lipschitz boundary. Then, for sufficiently large τ > 0, we have For proof, see [10,Lemma 4.7].    Using inequality (147) for η = ∇u ∞ and ζ = ∇u, we obtain that I ≤ 0, that is,   The enclosure method requires both a lower bound (as provided in theorem 4.8) and an upper bound for the indicator function; see for example the reconstruction procedure in section 2.3 and also [11, proof of theorem 4.1]. The lower bound allows us to determine when a half-space intersects the inclusion; or, equivalently, when t < h D (ρ). The upper bound would allow us to determine when a half-space does not intersect the inclusion; or, equivalently, when t > h D (ρ). Since we only have the lower bound, we only know when we intersect the inclusion, and thus can only find a (convex) superset of the inclusion D. We have no control over how badly D in the corollary below overestimates D. Then we can find a convex set D , which is a superset of the convex hull of D. Furthermore, we can detect whether D = ∅ or not.