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

 

Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding


Authors: Gui-Qiang Chen, Marshall Slemrod and Dehua Wang
Journal: Proc. Amer. Math. Soc. 138 (2010), 1843-1852
MSC (2010): Primary 53C42, 53C21, 53C45, 35L65, 35M10, 35B35; Secondary 53C24, 57R40, 57R42, 58J32
Published electronically: December 31, 2009
MathSciNet review: 2587469
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Abstract: We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $ L^p$-bounded solution sequence for $ p>2$, which implies that the weak limit of the isometric embeddings of the manifold in a fixed coordinate chart is an isometric immersion. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $ L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $ H^{-1}_{\text{loc}}$), the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made.


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Additional Information

Gui-Qiang Chen
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China – and – Department of Mathematics, Northwestern University, Evanston, Illinois 60208 – and – Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom
Email: gqchen@math.northwestern.edu

Marshall Slemrod
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 – and – Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 – and – Department of Mathematical Sciences, Korean Advanced Institute for Science and Technology, Daejeon, Republic of Korea
Email: slemrod@math.wisc.edu

Dehua Wang
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: dwang@math.pitt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10187-9
PII: S 0002-9939(09)10187-9
Keywords: Weak continuity, Gauss-Codazzi-Ricci system, isometric embedding, weak convergence, approximate solutions, compensated compactness, Div-Curl lemma, minimization problem, selection criterion, Riemann curvature tensor
Received by editor(s): August 14, 2009
Received by editor(s) in revised form: August 27, 2009
Published electronically: December 31, 2009
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.