Calculus of Variations and Geometric Measure Theory

G. Cupini - C. Marcelli

Monotonicity properties of minimizers and relaxation for autonomous variational problems

created by cupini on 08 Jan 2010
modified on 25 Feb 2011


Published Paper

Inserted: 8 jan 2010
Last Updated: 25 feb 2011

Journal: ESAIM Control Optim. Calc. Var.
Volume: 17
Pages: 222-242
Year: 2011


We consider the following classical autonomous variational problem \[\textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ dx\,:\,v\in AC([a,b]), v(a)=\alpha, v(b)=\beta \right\},\] where the Lagrangian $f$ is possibly neither continuous, nor convex, nor coercive.

We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

Keywords: existence of minimizers, nonconvex variational problems, autonomous variational problems, DuBois-Reymond necessary condition