Calderón's problem for p-laplace type equations
We investigate a generalization of Calderón’s problem of recovering the conductivity coeﬃcient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation div σ |∇u|p−2 ∇u = 0 with 1 < p < ∞, which reduces to the standard conductivity equation when p = 2. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity σ may be zero or inﬁnity in large sets. As a boundary determination result we recover the ﬁrst order derivative of a smooth conductivity on the boundary. We use the enclosure method of Ikehata to recover the convex hull of an inclusion of ﬁnite conductivity and ﬁnd an upper bound for the convex hull if the conductivity within an inclusion is zero or inﬁnite.
PublisherUniversity of Jyväskylä
- Article I: Brander, T., Kar, M., & Salo, M. (2015). Enclosure method for the p-Laplace equation. Inverse Problems, 31(4), 045001. DOI: 10.1088/0266-5611/31/4/045001
- Article II: Brander, T. (2016). Calderón problem for the 𝑝-Laplacian: First order derivative of conductivity on the boundary. Proceedings of the American Mathematical Society, 144(1), 177-189. DOI: 10.1090/proc/12681
- Article III: Brander, T., Ilmavirta, J., & Kar, M. Superconductive and insulating inclusions for non-linear conductivity equations. Preprint.
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