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MiS Preprint
29/1999
Rank-one convexity implies quasiconvexity on diagonal matrices
Stefan Müller
Abstract
We prove a conjecture of Tartar regarding weak lower semicontinuity of functionals on sequences $u_j , v_j: \Omega \subset R^2 \to R $ which satisfy $\partial_2 u_j \to 0 , \partial _1 v_j \to 0 $ in $H^{-1}$. This is the simplest example in the theory of compensated compactness for which the constant rank condition fails. The proof uses the fact that certain coefficients in the Haar basis expansion can be estimated in terms of the Riesz transform which seems to be of independent interest. Applications to the relation between rank-1 convexity and quasiconvexity are indicated.
