We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the p-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.

We recover the gradient of a scalar conductivity defined on a smooth bounded open set in 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 = 2. In the p = 2 case boundary determination plays a role in several methods for recovering the conductivity in the interior.

Abstract. We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón 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.

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 div(σ|∇u| p−2 ∇u) = 0 where the measurable conductivity σ : Ω → [0, ∞] is zero or infinity in large sets and 1 < p < ∞.

We consider one-dimensional Calderón's problem for the variable exponent p (·)-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted p(·)-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in L ∞ restricted to the coarsest sigma-algebra that makes the exponent p (·) measurable.Calderón's problem [10] asks if an electric conductivity γ in an object Ω can be reconstructed from boundary measurements of current and voltage, which are given by the Dirichlet-to-Neumann map (DN map) u| ∂Ω → γ∇u · ν| ∂Ω , where ν is the unit outer normal. In one dimension, the answer to Calderón's problem is negative; only the resistance, that is, the total resistivity´I γ −1 dx, can be recovered [17, section 1.1], where I ⊂ R is an open bounded interval. A similar result holds for p-Calderón's problem, where the forward model is given by the weighted p-Laplace equation − div γ |u | p−2 u = 0: it is only possible to recover the value of the integral I γ 1/(1−p) dx from the DN map [4, theorem 2.2]. This is true for any constant 1 < p < ∞, and is also a special case of corollary 10 in this paper. We describe what can be recovered in the case of a variable exponent p(·). This is the first investigation of an inverse problem related to the variable exponent p(·)-Laplacian. The problem in the constant exponent case was introduced by Salo and Zhong [31] in 2012, after which other theoretical results have been published [3,5,7,8,18,24], as well as a numerical study [19]. The works of Sun and others consider the problem of recovering the dependence of A in div (A(x, u, ∇u)∇u) = 0 on all of x, u, and ∇u, but they either do not admit degenerate equations [29,34] or assume the equivalent of constant p [21].One can think of the variable exponent conductivity equation − div γ |∇u| p(x)−2 ∇u = 0 as arising from a non-linear Ohm's lawwhich has a power law dependence between the current j and the gradient of the electric potential u at every point, but where the exponent in the power law varies from point to point. We use this terminology and intuition in the article. An example of a power-law type Ohm's law is certain polycrystalline compounds near the transition to superconductivity [9,15], where the exponent p is a function of temperature. However, it is not clear how relevant electrical impedance tomography of such materials would be.