Hyperjacobians, Determinantal Ideals and Weak Solutions to Variational Problems

Peter J. Olver

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


The problem of classifying homogeneous null Lagrangians satisfying an nth order divergence identity is completely solved. All such differential polynomials are affine combinations of higher order Jacobian determinants, called hyperjacobians, which can be expressed as higher dimensional determinants of higher order Jacobian matrices. Special cases, called transvectants, are of importance in classical invariant theory. Transform techniques reduce this question to the characterization of the symbolic powers of certain determinantal ideals. Applications to the proof of existence of minimizers of certain quasi-convex variational problems with weakened growth conditions are discussed.

Original languageEnglish (US)
Pages (from-to)317-340
Number of pages24
JournalProceedings of the Royal Society of Edinburgh: Section A Mathematics
Issue number3-4
StatePublished - 1983

Bibliographical note

Funding Information:
The research in this paper was for the most part accomplished under a United Kingdom SRC research grant during 1980 at the University of Oxford. It is a pleasure to thank John Ball for helpful suggestions, comments and much needed encouragement.

Funding Information:
t Research supported in part by NSF grant MCS 81-00786 and a United Kingdom SRC research grant.


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