Hyperjacobians, determinantal ideals and weak solutions to variational problems

Abstract

SynopsisThe 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.

Invariant theory and differential equations

“…The present version is essentially the same as that discussed by Ball, Currie and Olver, [2], in the solution of the first and fourth problems of section 1. Subsequently, [14], this transform was, in the special case of polynomial functions u, recognized to be equivalent to the standard symbolic method of classical invariant theory.…”

“…The isomorphism theorem, with the proper interpretation of "symmetric", works just as before. (This is equivalent to, but slightly different from, the procedure used in [2], [14]. )…”

“…The answer is given by the theory of hyperjacobians, [14]. Rather than give the most general definition of a hyperjacobian, which is a generalization of the classical concept of a Jacobian determinant, we present the second degree examples, from which the general definition can easily be guessed.…”

“…Subsequently the transform method has been applied to a wide range of problems arising in the study of differential equations and the calculus of variations. It has also been recognized, [14], that when the functions are homogeneous polynomials, the transform method is equivalent to the classical symbolic method of invariant theory. This paper will review the transform method, its relationship and application to classical invariant theory, and its application to problems arising in the calculus of variations.…”

“…This result was recently used by Dunn and Serrin, [6], in their theory of interstitial working. Finally, problem 4, which is the most interesting from the point of view of classical invariant theory, arose in generalizations of the applications of problem 1 to the variational problems of elasticity, and was used to produce nonconvex variational problems with rather weak coercivity conditions for which it was still possible to prove the existence of weak minimizers, [14]. The solution to this last problem, to be explained in more detail below, is that such a differential polynomial must be a linear combination of k* order transvectants of the functions u and their derivatives, the Hessian being a multiple of the second order transvectant (u,u)' ', in the case that u is a homogeneous polynomial function.…”