Enclosure method for the p-Laplace equation

We study the enclosure method for the p-Calder\'on problem, which is a nonlinear generalization of the inverse conductivity problem due to Calder\'on that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.


Introduction
We develop a reconstruction formula for identifying the shape or location of an obstacle, embedded in a known background medium, from the boundary measurements for an underlying non-linear PDE. There is a large literature concerning the case where the underlying equation is linear, and the related questions have applications to geophysical problems (detection of mines [11] or minerals [35] within the earth), bio-medical imaging (for example EIT [46] and coupled physics imaging methods [4]), radar technology etc.
Several methods have been proposed to reconstruct the shape of the obstacle in the linear case. We first mention the sampling and probing methods which are based on Isakov's idea of using singular solutions to reconstruct the obstacles included in a known background medium [27]. Among the sampling methods we refer to the work of Cakoni-Colton [10], Colton-Kirsch [12] for the linear sampling method, and Kirsch-Grinberg [29] and Harrach [17] for the factorization method. Related to probing methods we cite the probe method by Ikehata [22] and the singular source method by Potthast [38]. Due to the use of Green's type or singular solutions, an inconvenient step of probing methods is the need to use approximating domains isolating the source point of the use point sources. This step is quite inconvenient since to perform this method one requires to avoid the unknown obstacle, see [39] and references therein. To deal with this issue Ikehata proposed the enclosure method [23], which uses complex geometrical optics (CGO) solutions in place of point sources. Other methods include those based on oscillating-decaying solutions [33] or monotonicity arguments [18]. Our main concern in this paper is to study the enclosure method for a nonlinear equation.
Let us describe the enclosure method for the linear problem. Here we consider the conductivity problem with homogeneous background.
Let Ω ⊂ R n , n ≥ 2, be a bounded open set with Lipschitz boundary. Assume that the inclusion D ⊂ Ω is Lipschitz. The conductivity of the obstacle is taken to be jump discontinuous along ∂D, i.e., we consider σ(x) := 1 + σ D (x)χ D (x), a measurable function, where and χ D is the characteristic function D. Therefore, we formulate the following Dirichlet boundary value problem for the penetrable obstacle case: div(σ(x)∇u) = 0 in Ω, Given boundary data f ∈ H 1/2 (∂Ω), the above problem (1.2) is well posed in H 1 (Ω). Hence we define the voltage to current map, known also as the Dirichlet-to-Neumann map (DN map for short), formally by where u satisfies the conductivity problem (1.2). Using the weak definition (see Section 2), DN map becomes a bounded map Λ σ : The inverse problem in this setup is to reconstruct the shape and location of an unknown obstacle D from the knowledge of DN map Λ σ . The enclosure method introduced by Ikehata [23] uses CGO solutions with linear phase for the Laplace equation to detect the convex hull of the obstacle. The principal idea behind this method is to analyze the behavior of an indicator function, defined via the difference of DN maps Λ σ − Λ 1 and the boundary values of CGO solutions, to decide whether or not the level set of the linear phase function touches the surface of the obstacle. Taking all possible level sets touching the interface produces the convex hull of the obstacle.
There is an extensive literature in this direction for the linear models, see for instance [26,47] and the references therein, for an overview.
Here we would like to mention a few of the results. Using CGO solutions with spherical phase functions for the scalar Helmholtz model, a reconstruction scheme has been proposed by Nakamura and Yoshida [34] to detect some non-convex parts of the impenetrable obstacle from DN map in R n , n = 2, 3. In two dimensions the scalar problem has been studied by Nagayasu-Uhlmann-Wang [32], where they used CGO solutions with harmonic polynomial phases. In the recent work by Sini and Yoshida [42], both the penetrable and impenetrable obstacle cases were considered and some earlier curvature conditions on the boundary of the obstacles were removed. Concerning the Maxwell model, the enclosure method has been studied by Zhou [49] and Kar and Sini [28]. We also cite several other works related to the stationary models with fixed frequencies, see for instance [23,24,25] and references therein.
In analogy with the linear model, our interest in this paper is to consider the weighted p-harmonic model. In the linear model we have a linear Ohm's law (or Fourier's law of heat conduction); the current j is proportional to the conductivity σ and the gradient of the potential u: In the p-harmonic model the relation between current and potential is not linear; rather, we have with 1 < p < ∞. If p = 2, we recover the linear model. We combine the nonlinear Ohm's law and Kirchhoff's law, which states that the current j is divergence-free, to reach the weighted p-Laplace equation div(σ |∇u| p−2 ∇u) = 0.
If σ ≡ 1, this is the p-Laplace equation and solutions are called pharmonic functions.
Given a bounded open set Ω ⊂ R n , n ≥ 2, a subset D ⊂ Ω with Lipschitz boundary, and the conductivity σ(x) := 1+σ D (x)χ D (x), σ D ∈ L ∞ + (D), then for 1 < p < ∞ the Dirichlet problem for the weighted p-Laplace equation can be stated as div(σ(x) |∇u| p−2 ∇u) = 0 in Ω, u = f on ∂Ω. (1.5) The problem (1.5) is well posed in W 1,p (Ω) for a given Dirichlet boundary data f ∈ W 1,p (Ω) (the boundary values are understood so that u − f ∈ W 1,p 0 (Ω)), see for instance [13,20,31,41], and the solution u minimizes the p-Dirichlet energy As in the linear case, we formally define the non-linear DN map (keeping the same notations as the linear case), by where u ∈ W 1,p (Ω) satisfies (1.5). Using a natural weak definition (see Section 2), the DN map becomes a nonlinear map Λ σ : X → X where X is the abstract trace space X = W 1,p (Ω)/W 1,p 0 (Ω) and X denotes the dual of X (if ∂Ω has Lipschitz boundary, the trace space X can be identified with the Besov space B 1−1/p pp (∂Ω)). Physically Λ σ (f ) is the current flux density caused by the boundary potential f . See [19] for further properties of Λ σ .
The precise formulation of the inverse problem studied in this article is as follows.
Inverse Problem: Detect the shape and location of the obstacle D from the knowledge of the nonlinear DN map Λ σ .
As in the linear case, complex geometrical optics type solutions for the p-harmonic equation will make it possible to justify the enclosure method. Complex geometrical optics solutions for the p-harmonic equation of the form e ρ·x , ρ ∈ C n , with the condition (p − 1)|Re(ρ)| 2 = |Im(ρ)| 2 and Re(ρ) · Im(ρ) = 0, were used in [41] to prove a boundary determination result for the conductivities. Moreover, certain real valued exponential solutions to the p-Laplace equation were introduced by Wolff, see [48] and also Lemma 3.1 in Section 3, and these solutions were used in [41] to give a boundary uniqueness result for real valued data.
In order to deal with the enclosure method, both the complex exponentials and Wolff solutions could be used. We will only consider the Wolff solutions in this paper since they lead to a more general result and allow to detect the obstacle by using real valued boundary data alone. The main components in the proof are a suitable monotonicity inequality, see Lemma 2.1, and the properties of Wolff solutions. The monotonicity inequality of Lemma 2.1 is a nonlinear version of earlier inequalities in the linear case (see e.g. [17, Lemma 1] and references therein). The inequality might also be of interest for other purposes, for example to obtain boundary uniqueness for higher order derivatives of conductivities.
The first contribution to the inverse p-Laplace problem is due to Salo-Zhong [41], where a boundary uniqueness result for the conductivities was established by using CGO and Wolff type solutions. Recently Brander [8] gave a boundary uniqueness result for the first normal derivative of the conductivity.
We also mention the superficially related work of Bolanos and Vernescu [7]; they show that one can have ellipsoidal nonlinear inclusions that can be hidden from a single measurement with a layer of 2-harmonic material. The material in which they embed the inclusions is linear.
Part of the motivation for considering these problems comes from trying to understand inverse boundary value problems for strongly nonlinear equations. The p-Laplace type equations are a particular model where CGO type solutions can be used in a genuinely nonlinear way. Beyond the two results mentioned above and the enclosure method established in this paper, there are many open questions for p-Laplace type models (including boundary uniqueness for higher order derivatives, interior uniqueness, the validity of sampling or probe type methods, or even the enclosure method for impenetrable obstacles). We also remark that our results contain the linear case (p = 2) as a special case.
The paper is organized as follows: In Section 2 we prove the monotonicity inequality. Then in Section 3 we discuss the Wolff type solutions and their properties. The statement and proof of the main result are provided in Section 4. Some further remarks are given in Section 5.
Acknowledgements. All authors were partly supported by the Academy of Finland through the Finnish Centre of Excellence in Inverse Problems Research, and M.K. and M.S. were also supported in part by an ERC Starting Grant (grant agreement no 307023).

Monotonicity inequality
Let Ω ⊂ R n be a bounded open set. If σ ∈ L ∞ + (Ω), we define the DN map in the weak sense by where u ∈ W 1,p (Ω) is the unique solution of div(σ |∇u| p−2 ∇u) = 0 in Ω with u| ∂Ω = f , and v is any function in W 1,p (Ω) with v| ∂Ω = g.
Recall that X = W 1,p (Ω)/W 1,p 0 (Ω), so ( · , · ) is the duality between X and X. See [41,Appendix] for more details on the DN map. (Note that in this article we assume all functions are real valued.) The following monotonicity inequality will be crucial for the enclosure method.
We emphasise that if σ 1 ≥ σ 0 , then all the terms in the inequality are nonnegative, but if σ 1 ≤ σ 0 , they are nonpositive.
Proof. Let u 0 , u 1 ∈ W 1,p (Ω) be the solutions of the Dirichlet problem for the p-Laplace equation, div(σ |∇u| p−2 ∇u) = 0 in Ω, u = f on ∂Ω, (2.4) corresponding to the conductivities σ = σ 0 and σ = σ 1 respectively. Note that the solution of (2.4) can be characterized as the unique minimizer of the energy functional [41,Appendix]). Therefore, we obtain the following one sided inequality for the difference of DN maps: To obtain the other side of the inequality, we rewrite the difference of DN maps as follows: for some β > 0 to be specified later. Now, by applying Young's inequality |ab| ≤ |a| p p + |b| p p where 1/p + 1/p = 1, we have Note that (1+β) p β → ∞ as β → ∞ or β → 0. So, the function β → (1+β) p β attains its minimum at β = p − 1. Thus, we choose β = p − 1 so that from (2.5), we obtain the required inequality.

Wolff Solutions
The other main tool in justifying the enclosure method is the use of appropriate exponentially growing solutions. In the following lemma we describe real valued exponential solutions of the p-Laplace equation. The solutions are periodic in one direction and behave exponentially in a perpendicular direction. They were first introduced by Wolff [48, section 3] and later applied to inverse problems in [41, section 3].

2)
The function h is then p-harmonic. Given any initial conditions (a 0 , b 0 ) ∈ R 2 \{(0, 0)} there exists a solution a ∈ C ∞ (R) to the differential equation (3.1) which is periodic with period λ p > 0, satisfies the initial conditions (a(0), a (0)) = (a 0 , b 0 ), satisfies λp 0 a(s) ds = 0, and furthermore there exist constants c and C depending on a 0 , b 0 , p such that for all s ∈ R we have C > a(s) 2 + a (s) 2 > c > 0. Let τ ∈ R be a large parameter and t ∈ R be a constant. With the notation of Lemma 3.1, define u 0 : R n → R by We use a fixed function a (for some fixed initial data (a 0 , b 0 ) = (0, 0)) throughout the article. By Lemma 3.1, u 0 solves the p-Laplace equation in R n . Note that a is oscillating as a periodic function which integrates to zero over the period. Thus u 0 has exponential behavior in the ρ direction and oscillates rapidly in the ρ ⊥ direction if τ is large. The functions u 0 will be used as the complex geometrical optics solutions in the enclosure method. We record a formula for the gradient of u 0 , which will be used later: for all x ∈ R n by (3.3).

Main result and the proof
In this section we use the following standing assumptions, unless otherwise mentioned. Also, we use u 0 to denote the solutions (3.4), and use them in the definition of the indicator and support functions below.
Recall that we consider the set Ω ⊂ R n , n ≥ 2, to be a bounded domain. The inclusion D, D ⊂ Ω, is assumed to be a bounded open set with Lipschitz boundary. We furthermore assume that the conductivity σ has a jump discontinuity along the interface ∂D. In particular, we assume that σ(x) := 1 + σ D (x)χ D (x), where σ D ∈ L ∞ + (D) and χ D is the characteristic function of D. We let 1 < p < ∞ and consider the following Dirichlet problem for f ∈ W 1,p (Ω): div(σ(x) |∇u| p−2 ∇u) = 0 in Ω, u = f on ∂Ω. (4.1) Definition (Indicator function). We define the indicator function where ρ ∈ S n−1 , u 0 is the Wolff solution (3.4), τ > 0, t ∈ R, and Λ σ and Λ 1 are the non-linear DN maps corresponding to the conductivities σ, 1 respectively, defined by (2.1).
Now we state our main result. (1) When t > h D (ρ) we have and more precisely, for τ > 0, and where C > 0. (4.4) and more precisely when c, C > 0, and τ > 0 for the upper bound, and τ 1 for the lower bound.
From this theorem, we see that, for a fixed direction ρ, the behavior of the indicator function I ρ (u 0 , τ, t) changes drastically in terms of τ. Precisely, for t > h D (ρ) it is decaying exponentially, for t < h D (ρ) it is growing exponentially and for t = h D (ρ) it has a polynomial behavior. Using this property of the indicator function we can reconstruct the support function h D (ρ) from the non-linear DN map. holds.
The lemma is a straightforward consequence of the definitions of the indicator function and the Wolff solutions. In particular, we do not need any assumptions on the inclusion D.
Another way of reconstructing the support function from Theorem 4.1 is as follows.
Finally, from this support function we can estimate the convex hull of D, since for every direction ρ the support function determines a half-space that must contain the inclusion D, and so that boundary of the half-space intersects ∂D. The intersection of such half-spaces is the convex hull of the obstacle D. Thus we obtain If the domain Ω is not connected, then we can consider it componentwise by using a test function which equals the Wolff solution in the component under investigation and vanishes elsewhere. Hence, we can recover the convex hull of the inclusion within any fixed component.
We will now prove Theorem 4.1. To do this, we need only to show the estimate (4.5) as the other properties (4.3) and (4.7) will follow from (4.5) and the identity (4.8), as stated in the next lemma: Actually, we only need τ > 0 for the upper bound. This is clear from the proof of the next lemma. Proof. We have by Lemma 2.1 since t = h D (ρ). By assumption σ > 1 + ε on a set D of positive measure. This completes the proof.
We do not need any geometric assumptions on the inclusion D for the previous lemma (positive measure is sufficient). The proof of the lower bound in (4.5) is more difficult. Now, from Lemma 2.1 we can write (4.14) By (3.5) we obtain at t = h D (ρ) In preparation for the next lemma we define a cover of the set K := ∂D ∩ {x · ρ = h D (ρ)}. For any α ∈ K and δ > 0 we define B(α, δ) := {x ∈ R n ; |x − α| < δ}. (4.16) Then, K ⊂ ∪ α∈K B(α, δ). Since K is compact, there exist α 1 , · · · , α N ∈ K such that K ⊂ B(α 1 , δ) ∪ · · · ∪ B(α N , δ). We define D j,δ := D ∩ B(α j , δ), D δ := ∪ N j=1 D j,δ . Also with positive constant c as τ → ∞. Fix j ∈ {1, . . . , N }, so that α j ∈ K. By translation we may assume that α j = 0. Then there exists a hyperplane H j so that we can parametrise the boundary ∂D near α j as a Lipschitz function defined on H j , and with ρ / ∈ H j . 1 Let y = (y 1 , . . . , y n−1 ) be a basis of H j . Denote the parametrization of ∂D near α j by l j (y ). We select the remaining coordinate y n from the linear subspace spanned by w so that we have (4.18) Lemma 4.7. We assume the standing assumptions (see the beginning of Section 4). For 1 < p < ∞, the following estimate holds for τ 1: Proof. (4.20) Since we have l j (y ) ≤ C|y | as ∂D is Lipschitz, we have the following estimate Hence, combining the estimates (4.20) and (4.22), we obtain the required estimate for τ 1.
Finally, the proof of Theorem 4.1 for the penetrable obstacle case follows from equation (4.15) and Lemma 4.7.

Remarks
In this section we give some further remarks on the problem. These remarks refer to the proofs in Section 4.
Remark 5.1. We can also have inclusion with smaller conductivity than its surroundings, which corresponds to 0 < ε < σ D < 1 − ε for some ε > 0. The only difference in proofs is that the inequalities in the monotonicity inequality, Lemma 2.1, have reversed roles in the following proofs: • At the very beginning of the proof of Lemma 4.6.
Remark 5.2. The inclusion D does not need to have Lipschitz boundary everywhere. The regularity of the boundary ∂D is only used in proving Lemma 4.7.
As a first observation, we only consider the points of ∂D that are also boundary points of the convex hull co D, so we only need to impose restrictions there. In particular, the set D does not need to be open near points that are not near ∂ (co D). As a second observation, we need to be able to parametrise ∂D near every point x 0 ∈ ∂D ∩ ∂ (co D) as a function defined on some hyperspace H, so that ρ / ∈ H. To have the estimate (4.21) it is sufficient to have Lipschitz boundary near those points.
We also get partial results even when we do not have an inclusion, but do have monotonicity: Remark 5.3. Suppose σ ≥ 1 and the inequality is strict on a set of positive measure. Write D = {x ∈ Ω; σ(x) > 1}. Then we have lemma 4.6. By the identity (4.8) we know that for t > h D (ρ) |I ρ (u 0 , τ, t)| ≤ Ce −cτ (5.1) for τ > 0, and with C > 0.
That is, we know what must happen if t > h D (ρ). If the absolute value of the indicator function does not vanish in the limit, then we must have t ≤ h D (ρ). This is a sufficient condition, so it finds a subset of co D, which might be empty. The remark also applies to σ ≤ 1; see remark 5.1.