A Characterization of the Weakly Continuous Polynomials in the Method of Compensated Compactness

Abstract

We present a sufficient condition for weak continuity in the method of compensated compactness. The condition links weak continuity to the structure of the wave cone and the characteristic set for polynomials of degree greater than two. The condition applies to all the classical examples of weakly continuous functions and generalizes the Quadratic Theorem and the Wedge Product Theorem. In fact, the condition reduces to the Legendre-Hadamard Necessary Condition when the polynomial is quadratic, and also whenever a certain orthogonality condition is satisfied. The condition is derived by isolating conditions under which the quadratic theorem can be iterated.