Calderón problem for the 𝑝-Laplacian: First order derivative of conductivity on the boundary

Brander, Tommi

doi:10.1090/proc/12681

Calderón problem for the $p$-Laplacian: First order derivative of conductivity on the boundary
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Proc. Amer. Math. Soc. 144 (2016), 177-189 Request permission

Abstract:

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.

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