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Online ISSN 1552-4485; Print ISSN 0033-569X
Towards a theory of multi-parameter geometrical variational problems: Fibre bundles, differential forms, and Riemannian quasiconvexity
Author:
Siran Li
Journal:
Quart. Appl. Math. 78 (2020), 469-483
MSC (2010):
Primary 49J45, 49J10, 49Q20
DOI:
https://doi.org/10.1090/qam/1557
Published electronically:
August 29, 2019
MathSciNet review:
4100289
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We are concerned with the existence and associated gauge problems of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions which give rise to an associated gauge theory. Moreover, we propose the notion of “Riemannian quasiconvexity” for cost functions whose variables are differential forms on Riemannian manifolds, which extends the classical quasiconvexity condition in the Euclidean settings. The existence of minimisers under the Riemannian quasiconvexity condition has been established. This work may serve as a tentative generalisation of the framework developed in the recent paper [Arch. Ration. Mech. Anal. 234 (2019), pp. 317–349] by Dacorogna–Gangbo.
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Additional Information
Siran Li
Affiliation:
Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston, Texas, 77251-1892; and Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, H3A 0B9, Canada.
MR Author ID:
1231491
Email:
siran.li@rice.edu
Keywords:
Calculus of variations; fibre bundles; existence of minimiser; differential forms; gauges; quasiconvexity; direct method for calculus of variations.
Received by editor(s):
May 3, 2019
Received by editor(s) in revised form:
July 19, 2019
Published electronically:
August 29, 2019
Article copyright:
© Copyright 2019
Brown University