Calder\'on problem for the p-Laplacian: First order derivative of conductivity on the boundary

We recover the gradient of a scalar conductivity defined on a smooth bounded open set in $\mathbb{R}^d$ from the Dirichlet to Neumann map arising from the $p$-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when $p \neq 2$. In the $p = 2$ case boundary determination plays a role in several methods for recovering the conductivity in the interior.


Introduction
Throughout the article we assume 1 < p < ∞, unless explicitly written otherwise. We investigate a generalisation of the Calderón problem to the case of the p-Laplace equation: Given a bounded open set Ω ⊂ R d and a bounded conductivity γ > 0 on Ω we have the Dirichlet problem (1.1) ∆ γ p (u) = div γ(x)|∇u| p−2 ∇u = 0 in Ω, u = v on ∂Ω.
For v ∈ W 1,p (Ω) there exists a unique weak solution that satisfies u W 1,p ≤ C v W 1,p , where the constant C does not depend on v. This is completely standard, see e.g. [41,Proposition A.1]. The equation arises from the variational problem of minimising the energy´Ω γ|∇u| p dx. We next define the Dirichlet to Neumann map (hereafter DN map). In the p = 2 case the DN map gives the stationary measurements of electrical current for given voltage v. Definition 1.1 (Dirichlet to Neumann map). Suppose Ω ⊂ R d is bounded open C 1 -set and 0 < γ 0 < γ ∈ L ∞ (Ω) for some constant γ 0 . The weak DN map (1.2) Λ w γ : W 1,p (Ω)/W 1,p 0 (Ω) → W 1,p (Ω)/W 1,p 0 (Ω) is defined by where u solves the boundary value problem (1.1) with boundary values v, andg ∈ W 1,p (Ω) with trace g on the boundary. The strong DN map is defined pointwise on ∂Ω by the expression We also write Λ γ (v) when the function v is continuous and defined on a superset of ∂Ω. With sufficient regularity we can recover the values of the strong DN map from the weak DN map (see lemma 4.4).
Given knowledge of the DN map we determine ∇γ| ∂Ω , which in the p = 2 case stands for the gradient of conductivity on the boundary.
is a bounded open C 2,β set for some 0 < β < 1 and the conductivities γ 1 and γ 2 are bounded from below by a positive constant and continuously differentiable with Hölder-continuous derivatives in Ω.
We use an explicit sequence of boundary values to reconstruct ∇γ(x 0 ) at a boundary point x 0 . The sequence is supported in arbitrarily small neighbourhood of x 0 . The boundary values we use were first introduced by Wolff [46] and used by Salo and Zhong [41] to recover the conductivity on the boundary.
Salo and Zhong [41] show that for all boundary points x 0 ∈ ∂Ω there is a sequence of solutions (u M ) M such that as M, N (M ) → ∞, with explicitly computable constant c p (see equation (3.9)). They recover the integrals with g = γ from the DN map. We recover the integrals with g = α·∇γ from a Rellich-type identity, theorem 4.1, for arbitrary direction α ∈ R n . The Calderón problem was first introduced in [13]. For a review see [45]. There have been several boundary determination results in the p = 2 case. Boundary uniqueness for derivatives of the conductivity was first proven by Kohn and Vogelius [26]. Sylvester and Uhlmann in [44] recovered, for smooth conductivity in smooth domain, conductivity and all its derivatives on the boundary by considering Λ γ as a pseudodifferential operator. Nachman [34] recovered γ| ∂Ω in Lipschitz domain when γ ∈ W 1,q , q > d, and its first derivative when γ ∈ W 2,q , q > d/2. Alessandrini [1] used singular solutions with singularity near boundary to recover all derivatives of γ| ∂Ω with less regularity assumptions on γ. There have been several local boundary determination results, e.g. [36,23]. Other reasonably recent boundary determination results include [2,12].
A Rellich identity was used by Brown, Garcia and Zhang [16, appendix] to recover the gradient of conductivity on the boundary in the p = 2 case. The Rellich identity was, to the best of our knowledge, introduced in [38].
Several results (e.g. [1]) rely on investigating the difference Λ γ1 −Λ γ2 of DN maps at different conductivities. This is difficult in the present setting due to non-linearity of the p-Laplace equation. We use Rellich identity, theorem 4.1, to avoid this problem.
Electrical impedance tomography has applications in medical (e.g. [11]) and industrial imaging (e.g. [24]), and geophysics and environmental sciences; see e.g. the review [9] and references therein. There are practical numerical algorithms for boundary determination in the p = 2 case, e.g. [35]. Boundary determination is used in recovering the conductivity in the interior, e.g. [19]. For example, the algorithm in [43] uses the values of conductivity and its first derivative on the boundary to extend the conductivity, thence applying to conductivities that are not constant on the boundary.
For more on the p-Laplace equation see e.g. [31,20,15]. The equation has applications in e.g. image processing [27], fluid mechanics [6], plastic moulding [4], and modelling of sand-piles [5]. The Calderón problem for the p-Laplace equation was first introduced in [41]. The authors consider the real and the complex case separately. In the complex case they define p-harmonic versions of complex geometrical optics solutions to create highly oscillating functions focused around a given boundary point and use them as Dirichlet data, thus recovering conductivity at the boundary point in question. In the real case they replace the p-harmonic CGO solutions with real-valued functions having similar behaviour. The real-valued functions were originally introduced by Wolff [46].
One method of investigating the Calderón-type inverse problems for non-linear equations is based on studying the Gâteaux derivatives of the map Λ γ at constant boundary values a. In our case this does not work [41, appendix]: for positive t. For p < 2 the Gâteaux derivates do not exist and for p > 2 the higher derivates fail to exist, or vanish, though one of them might equal Λ γ (f ). Hence nonlinear methods are necessary.
In section 2 we introduce our notation and state lemmata. In section 3 we restate boundary determination results we use in this article. In the final section 4 we reconstruct ∇γ| ∂Ω from the DN map.

Preliminaries
We use the following notation: We write ∂ α = α · ∇ for vectors α. On the C 1 boundary ∂Ω of an open set Ω we decompose the gradient of a function f defined in Ω as ∇f where ∇ ν is the normal component of the derivative, i.e. the orthogonal projection of the gradient to the normal space of the boundary, and ∇ T is the tangential derivative, i.e. the orthogonal projection of the gradient on the tangent space of the boundary. We have the following identity: We denote the outer unit normal by ν.
We write the Hölder seminorm of order k as |f | k,β . We need the following inequalities: Then for some C > 0 we have from which the first estimate follows by observing that 0 ≤ A+t(B−A) ≤ A+B.
It is well-known that the solutions of the p-Laplace equation are in C 1,β . For proof of the following lemma see e.g. [29].

Lemma 2.2 (Regularity result).
Suppose Ω is a bounded open C 1,β1 set with 0 < β 1 ≤ 1, and suppose the conductivity 0 < γ ∈ C 0,β2 (Ω) is bounded from above and away from zero. Consider the weighted p-Laplace equation (1.1) with boundary values v ∈ C 1,β1 (∂Ω). Then the solution u of the weighted p-Laplace equation In general the solutions are not twice continuously differentiable [32]. To establish Rellich identity, theorem 4.1, we use the following interpolation lemma similar to [30, lemma A.1]: Suppose Ω is a bounded open set in R d , such that |B(y, δ)| is proportional to |B(y, δ) ∩ Ω| uniformly for sufficiently small radii δ and all y ∈ Ω. This holds for bounded Lipschitz domains, for example. Also The proof is almost the same as in [30].
Proof. Let y ∈ Ω and write B δ = B(y, δ). For δ > 0 we have whereby it follows that In our proofs we use the ε-perturbed p-Laplace equation. Let ε > 0 and define u ε as the solution of the boundary value problem By calculus of variations there exists a unique solution to (2.10) when Ω is a bounded open set and v ∈ W 1,p (Ω). The energy corresponding to (2.10) is
Proof. We first show that the energy (2.11) of the perturbed equation (2.10) converges to the energy of the non-perturbed equation (1.1): The solution u minimises energy, and so we havê where the latter integrals vanish as ε → 0 since |∇u| is bounded (lemma 2.2). We used the fact that u ε minimises the perturbed energy (2.11), and also lemma 2.1 with q = 1 + p/2 > 2 when p > 2, and the fact that for non-negative numbers (a + b) q ≤ a q + b q when 0 ≤ q = p/2 ≤ 1, in the final inequality.
We established that the energy of u ε converges to the energy of u as ε → 0. Since u ε L ∞ are uniformly bounded, it follows that u ε are bounded in W 1,p with weight. This implies that the sequence has a weakly converging subsequence with limit that has lesser or equal energy than u. But u uniquely minimises energy, so u ε → u weakly in weighted W 1,p . Since the weighted L p norms of ∇u ε converge to weighted L p norm of ∇u, and the sequence converges weakly, we get that ∇u ε → ∇u strongly in weighted L p space; this is the Radon-Riesz property proven in [37, 39,40]. Since the weighted and standard norms are equivalent, ∇u ε → ∇u strongly in standard L p .
Since ∇u ε are uniformly Hölder in Ω (lemma 2.4) we can use the interpolation lemma 2.3 to deduce convergence of ∇u ε to ∇u in C(Ω). The same argument holds for u ε , since by Friedrichs-Poincaré inequality u ε → u strongly in L p .

Previous results on boundary determination for p-Laplacian
We use several results in the paper [41] and restate them here for convenience. Suppose Ω ⊂ R d is an open bounded C 1 set and suppose ρ ∈ C 1 (R d ) is its boundary defining function; that is, and ∇ρ = 0 on ∂Ω. By translation we may assume that we are trying to recover ∇γ at the origin 0 ∈ ∂Ω, and by rotation and scaling we may assume that the dth coordinate vector. We define the map f : for x = (x , x d ). Since ρ is a C 1 function, the map is close to the identity near the origin, and hence invertible in some neighbourhood of the origin. We now introduce the p-harmonic oscillating functions of Wolff [46].
is the solution to the differential equation Then ∆ 1 p (h) = 0 and the function a is smooth and periodic with period λ(p), so that We define a smooth positive cutoff function ζ ∈ C ∞ 0 (R d ), such that ζ(x) = 1 when |x| < 1/2 and supp ζ ⊂ B(0, 1). We write where M and N = N (M ) are large positive numbers, M = o(N ). We define u M to be the solutions of the initial value problem In [41, lemma 3.3] Salo and Zhong show that as M → ∞. The constant c p is explicit: The proof also holds when γ is replaced by any other function continuous at 0, so as M → ∞ we have Using [41, lemma 3.4] we get the following result: Suppose Ω ⊂ R d is a bounded open set with C 1 boundary, x 0 ∈ ∂Ω, d ≥ 2, and g is continuous. Then as M → ∞.

Proof of the main result
In this section we establish that the quantity Theorem 4.1 (Rellich identity). Suppose 1 < p < ∞ and that Ω ⊂ R d is a bounded open set with C 2,β boundary for some 0 < β < 1. Let 0 < γ ∈ C 1 (Ω) and let u be a weak solution of the equation ∆ γ p u = 0. Let α ∈ R d . Then Proof. Suppose we have the following identity for the solution u ε of the perturbed p-Laplace equation (2.10): Then, when ε → 0, ∇u ε → ∇u uniformly (by lemma 2.5), so we get the claimed identity.
We now prove the perturbed identity, integrating by parts twice: We need to recover γ| ∂Ω and ∇u| ∂Ω to recover´Ω ∂ α γ|∇u| p dx from the DN map.
Supposing Ω is an open set with C 1 boundary and γ is continuous, we can recover γ(x 0 ) for all boundary points x 0 . This follows directly from [41].
To calculate ∂ ν u we use C 1,β boundary values v so that the solution u of the boundary value problem (1.1) is continuously differentiable up to the boundary (by lemma 2.2), so ∂ ν u is defined pointwise. Recovering ∂ ν u from the strong definition of the DN map is straightforward, and we do so in lemma 4.2. However, we do not a priori have access to the strong DN map, and must recover it from the weak definition 1.1. This we do in lemma 4.4, which requires more smoothness on Ω and boundary values.
We have (4.10) Λ s γ (v) = γ|∇u| p−2 ∂ ν u, from which by expanding ∇u we get the equation (4.11) γ|∇ T u + ∇ ν u| p−2 ∂ ν u = Λ s γ (v). Taking absolute values and reorganising we have We write the left hand side as a function of |∂ ν u| = |∇ ν u| = t: There is a unique To see this, observe that lim t→0 F (t) = 0, lim t→∞ F (t) = ∞, and that the function F is strictly increasing and continuous.
The rate of convergence is uniform in ε when δ is so small that |∇u ε (x)| is bounded away from zero for x ∈ B(x 0 , δ), since then the following ε-indexed family of functions (4.31) are equicontinuous with parameter ε: The family is Hölder-continuous on ∂Ω when x − x 0 is so small that |∇u ε (x)| ≥ m > 0 for some m and all ε < a: By lemma 2.4 the functions x → ∇u ε (x) are equicontinuous with parameter ε, and the functions w → γ |w| 2 + ε p−2 2 w · ν are equicontinuous when |w| is bounded from above and away from zero. Since we know γ| ∂Ω by [41, theorem 1.1], ∇u| ∂Ω by lemma 4.2 and the strong DN map by lemma 4.4, we also know´Ω(∂ α γ)|∇u| p dx.
Remark 4.6. In using the Rellich identity we need to know the values of the conductivity γ on the entire boundary ∂Ω. The gradient ∇u M should be large near x 0 and small elsewhere (we have not made this precise), so our proof is essentially local in nature.
The main theorem 1.2 now follows, since the value of the integral (4.32)ˆΩ ∂ α γ|∇u M | p dx