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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

Kewei Zhang 1,

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.
Keywords: quasiconvex function, prescribed zero set, quasimonotone gradient mapping, squared-distance function, smooth manifold, local behaviour., translation method.
Mathematics Subject Classification: Primary: 49J10,49J45; Secondary: 58E05,74P1.
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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