Vol. 10, No. 1, 2017

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Partial data inverse problems for the Hodge Laplacian

Francis J. Chung, Mikko Salo and Leo Tzou

Vol. 10 (2017), No. 1, 43–93
Abstract

We prove uniqueness results for a Calderón-type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth-order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometrical optics solutions which reduce the Calderón-type problem to a tomography problem for 2-tensors. The arguments in this paper allow us to establish partial data results for elliptic systems that generalize the scalar results due to Kenig, Sjöstrand and Uhlmann.

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Keywords
inverse problems, Hodge Laplacian, partial data, absolute and relative boundary conditions, admissible manifolds, Carleman estimates
Mathematical Subject Classification 2010
Primary: 35R30
Secondary: 58J32
Milestones
Received: 9 March 2016
Revised: 10 May 2016
Accepted: 19 September 2016
Published: 30 January 2017
Authors
Francis J. Chung
Department of Mathematics
University of Kentucky
Lexington, KY 40506
United States
Mikko Salo
Department of Mathematics and Statistics
University of Jyväskylä
FI-40014 Jyväskylä
Finland
Leo Tzou
School of Mathematics and Statistics
University of Sydney
Sydney NSW 2006
Australia