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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An inverse problem for an inhomogeneous conformal Killing field equation


Author: Ziqi Sun
Journal: Proc. Amer. Math. Soc. 131 (2003), 1583-1590
MSC (2000): Primary 35R30, 53C21
Published electronically: December 16, 2002
MathSciNet review: 1949889
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Abstract: Let $g$ be a $C^{2,\alpha}$ Riemannian metric defined on a bounded domain $\Omega\subset R^2$ with $C^{3,\alpha}$ boundary and let $X$be a $C^{2,\alpha}$ vector field on $\bar{\Omega}$ satisfying $X\vert _{\partial\Omega}=0$. We show that if $l$ is a gradient field of a solution $u$ to the equation $\triangle_gu-\bigl\langle\nabla_{g\,}\sigma,\,\nabla_gu\bigr\rangle_g=0$ on $\Omega$, then both inner products $\bigl\langle l,X\bigr\rangle_g\,$ and $\bigl\langle l^\perp,X\bigr\rangle_g\,$ are uniquely determined by the restriction of the tensor ${\mathcal L}_X(g)-(e^\sigma\,\nabla_{g}\cdot(e^{-\sigma}X)) g$ to the gradient field $l$, where ${\mathcal L}_X(g)$ is the Lie derivative of the metric tensor $g$ under the vector field $X$ and $\sigma=log\sqrt{det(g)}$. This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.


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Additional Information

Ziqi Sun
Affiliation: Department of Mathematics, Wichita State University, Wichita, Kansas 67260-0033
Email: ziqi.sun@wichita.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06973-3
PII: S 0002-9939(02)06973-3
Received by editor(s): January 8, 2002
Published electronically: December 16, 2002
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society