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


Author: Tommi Brander
Journal: Proc. Amer. Math. Soc. 144 (2016), 177-189
MSC (2010): Primary 35R30, 35J92
DOI: https://doi.org/10.1090/proc/12681
Published electronically: July 24, 2015
MathSciNet review: 3415587
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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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Tommi Brander
Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD) FI-40014 University of Jyväskylä, Finland
Email: tommi.o.brander@jyu.fi

DOI: https://doi.org/10.1090/proc/12681
Keywords: Calder\'on's problem, inverse problems, $p$-Laplace equation, boundary determination
Received by editor(s): March 4, 2014
Received by editor(s) in revised form: November 26, 2014
Published electronically: July 24, 2015
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

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