# American Institute of Mathematical Sciences

January  2008, 21(1): 353-366. doi: 10.3934/dcds.2008.21.353

## On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets

 1 Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom

Received  December 2006 Revised  August 2007 Published  February 2008

Let $M^{N\times n}$ be the space of real $N\times n$ matrices. We construct non-negative quasiconvex functions $F:M^{N\times n}\to R_+$ of quadratic growth whose zero sets are the graphs $\Gamma_f$ of certain Lipschitz mappings $f:K\subset E\to$ $E^$⊥, where $E\subset M^{N\times n}$ is a linear subspace without rank-one matrices, $K$ a compact subset of $E$ with $E^$⊥ its orthogonal complement. We show that the gradients $DF:M^{N\times n}\to M^{N\times n}$ are strictly quasimonotone mappings and satisfy certain growth and coercivity conditions so that the variational integrals $u\to \int_{\Omega}F(Du(x))dx$ satisfy the Palais-Smale compactness condition in $W^{1,2}$. If $K$ is a smooth compact manifold of $E$ without boundary and the Lipschtiz mapping $f$ is of class $C^2$, then the closed $\epsilon$-neighbourhoods $(\Gamma_f)_\epsilon$ for small $\epsilon>0$ are quasiconvex sets.
Citation: Kewei Zhang. On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 353-366. doi: 10.3934/dcds.2008.21.353
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