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

Previous Article
Solutions with interior bubble and boundary layer for an elliptic problem
DCDS Home
This Issue
Next Article

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

Abstract
Related Papers

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

[1]	Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[2]	Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial & Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115
[3]	Y. Goto, K. Ishii, T. Ogawa. Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature. Communications on Pure & Applied Analysis, 2005, 4 (2) : 311-339. doi: 10.3934/cpaa.2005.4.311
[4]	Yubo Yuan, Weiguo Fan, Dongmei Pu. Spline function smooth support vector machine for classification. Journal of Industrial & Management Optimization, 2007, 3 (3) : 529-542. doi: 10.3934/jimo.2007.3.529
[5]	Peter Frolkovič, Karol Mikula, Jozef Urbán. Distance function and extension in normal direction for implicitly defined interfaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 871-880. doi: 10.3934/dcdss.2015.8.871
[6]	Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[7]	Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
[8]	Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
[9]	Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381
[10]	Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
[11]	Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1399-1406. doi: 10.3934/mbe.2017072
[12]	Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.
[13]	Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
[14]	Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial & Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705
[15]	Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895
[16]	Zhichuan Zhu, Bo Yu, Li Yang. Globally convergent homotopy method for designing piecewise linear deterministic contractual function. Journal of Industrial & Management Optimization, 2014, 10 (3) : 717-741. doi: 10.3934/jimo.2014.10.717
[17]	Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39
[18]	Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa. Average optimal strategies for zero-sum Markov games with poorly known payoff function on one side. Journal of Dynamics & Games, 2014, 1 (1) : 105-119. doi: 10.3934/jdg.2014.1.105
[19]	Jianling Li, Chunting Lu, Youfang Zeng. A smooth QP-free algorithm without a penalty function or a filter for mathematical programs with complementarity constraints. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 115-126. doi: 10.3934/naco.2015.5.115
[20]	Zeng Zhang, Zhaoyang Yin. Global existence for a two-component Camassa-Holm system with an arbitrary smooth function. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5523-5536. doi: 10.3934/dcds.2018243

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences