Existence and monotonicity property of minimizers of a nonconvexvariational problem with a second-order Lagrangian

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.