Rank-.n 1/ convexity and quasiconvexity for divergence free fields
نویسندگان
چکیده
We prove that rank-.n 1/ convexity does not imply quasiconvexity with respect to divergence free fields (so-called S-quasiconvexity) in M n for m > n, by adapting the well-known Šverák’s counterexample to the solenoidal setting. On the other hand, we also remark that rank-.n 1/ convexity and S-quasiconvexity turn out to be equivalent in the space of n n diagonal matrices.
منابع مشابه
ON THE RELATIONSHIP BETWEEN RANK-(n − 1) CONVEXITY AND S-QUASICONVEXITY
Abstract. We prove that rank-(n−1) convexity does not imply S-quasiconvexity (i.e., quasiconvexity with respect to divergence free fields) in M for m > n, by adapting the well-known Šverák’s counterexample [5] to the solenoidal setting. On the other hand, we also remark that rank-(n − 1) convexity and Squasiconvexity turn out to be equivalent in the space of n×n diagonal matrices. This follows ...
متن کاملRelaxation of Three Solenoidal Wells and Characterization of Extremal Three-phase H-measures
We fully characterize quasiconvex hulls for three arbitrary solenoidal (divergence free) wells in dimension three. With this aim we establish weak lower semicontinuity of certain functionals with integrands restricted to generic twodimensional planes and convex in (up to three) rank-2 directions within the planes. Within the framework of the theory of compensated compactness, the latter represe...
متن کاملMajorisation with Applications to the Calculus of Variations
This paper explores some connections between rank one convexity, multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant. Theorem 6.2 describes a class of non-polyconvex but multiplicative quasiconvex isotropic functions. Relevance of...
متن کاملFour applications of majorization to convexity in the calculus of variations
The resemblance between the Horn-Thompson theorem and a recent theorem by Dacorogna-Marcellini-Tanteri indicates that Schur convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank-one convexity or quasiconvexity. We give in theorem 6.6 simple necessary and sufficient conditions for an isotropic object...
متن کاملRank-one Convexity Implies Quasiconvexity on Diagonal Matrices
The question of whether such a result holds was raised inTartar’s seminal paper [T1], where he proved estimate (1.3) under the stronger condition that ∂2uj and ∂1vj are bounded in Lloc. In this case, he even showed that the Young measure generated by the pair (uj, vj) (see below for definitions) is a tensor product, and he gave an example that this need no longer be the case if only (1.2) holds...
متن کامل