Lower semicontinuity of parametric integrals

Silverman, Edward

doi:10.1090/S0002-9947-1973-0310744-6

Lower semicontinuity of parametric integrals
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Trans. Amer. Math. Soc. 175 (1973), 499-508 Request permission

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.

References

Topologie

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Surface area

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