# American Institute of Mathematical Sciences

July  2009, 25(2): 687-699. doi: 10.3934/dcds.2009.25.687

## Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian

 1 College of Mathematics, Jilin University, Changchun 130012 2 Department of Mathematics, Jilin University, Changchun 130023

Received  August 2008 Revised  January 2009 Published  June 2009

A variational problem

Minimize $\quad \int_a^bL(x,u(x),u'(x),u''(x))dx,\quad u\in\Omega$

with a second-order Lagrangian is considered. In absence of any smoothness or convexity condition on $L$, we present an existence theorem by means of the integro-extremal technique. We also discuss the monotonicity property of the minimizers. An application to the extended Fisher-Kolmogorov model is included.

Citation: Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687
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