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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lower semicontinuity of parametric integrals


Author: Edward Silverman
Journal: Trans. Amer. Math. Soc. 175 (1973), 499-508
MSC: Primary 49F20; Secondary 49A50
MathSciNet review: 0310744
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Abstract: It has been known for a long time that the usual two-dimensional parametric integrals in three-space are lower semicontinuous with respect to uniform convergence. In an earlier paper we saw that an easy argument extends this result to all parametric integrals generated by simply-convex integrands, with no restrictions on the dimension of the surfaces or the containing space. By using these techniques again, and generalizing to surfaces a result concerning convergent sequences of closed curves we show that a parametric integral generated by a parametric integrand which is convex in the Jacobians is lower semicontinuous with respect to uniform convergence provided all of the functions lie in a bounded subset of the Sobolev space $ H_s^1$ where $ s + 1$ exceeds the dimension of the parametric integral.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0310744-6
PII: S 0002-9947(1973)0310744-6
Keywords: Topological index, essential multiplicity, Peano area, Lebesgue area, Sobolev space
Article copyright: © Copyright 1973 American Mathematical Society