Determining an unbounded potential from Cauchy data in admissible geometries
Ferreira, D. D. S., Kenig, C. E., & Salo, M. (2013). Determining an unbounded potential from Cauchy data in admissible geometries. Communications in Partial Differential Equations, 38(1), 50-68. https://doi.org/10.1080/03605302.2012.736911
Published inCommunications in Partial Differential Equations
DisciplineMatematiikkaInversio-ongelmien huippuyksikköMathematicsCentre of Excellence in Inverse Problems
© Taylor & Francis Group, LLC, 2013. This is a final draft version of an article whose final and definitive form has been published by Taylor & Francis Group, LLC. Published in this repository with the kind permission of the publisher.
In [4 Dos Santos Ferreira , D. , Kenig , C.E. , Salo , M. , Uhlmann , G. ( 2009 ). Limiting Carleman weights and anisotropic inverse problems . Invent. Math. 178 : 119 – 171 . [Crossref], [Web of Science ®], [Google Scholar] ] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n ≥ 3. In this article we extend this result to the case of unbounded potentials, namely those in L n/2. In the process, we derive L p Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison and Kenig [8 Jerison , D. , Kenig , C.E. ( 1985 ). Unique continuation and absence of positive eigenvalues for Schrödinger operators . Ann. Math. 121 : 463 – 494 . [Crossref], [Web of Science ®], [Google Scholar] ] and Kenig et al. [9 Kenig , C.E. , Ruiz , A. , Sogge , C.D. ( 1987 ). Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators . Duke Math. J. 55 : 329 – 347 . [Crossref], [Web of Science ®], [Google Scholar] ]. ...
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