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We compute the quasiconvex envelope of certain functions defined on the space $M_{m n}$ of real $m \times n$ matrices. These functions are basically functions of a quadratic form on $M_{m n}$ . The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.

Remarks on the regularity of the minimizers of certain degenerate functionals

Mariano Giaquinta, Giuseppe Modica (1986/1987)

Manuscripta mathematica

Rigidity for the hyperbolic Monge-Ampère equation

Chun-Chi Lin (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Some properties of nonlinear partial differential equations are naturally associated with the geometry of sets in the space of matrices. In this paper we consider the model case when the compact set $K$ is contained in the hyperboloid $ℋ_{- 1}$ , where $ℋ_{- 1} \subset 𝕄_{sym}^{2 \times 2}$ , the set of symmetric $2 \times 2$ matrices. The hyperboloid $ℋ_{- 1}$ is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation $det \nabla^{2} u = - 1$ . For some compact subsets $K \subset ℋ_{- 1}$ containing a rank-one connection, we show the rigidity property of $K$ by imposing...

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