Calculus of Variations and Geometric Measure Theory

G. Cupini - M. Guidorzi - C. Marcelli

Existence of minimizers of free autonomous variational problems via solvability of constrained ones.

created by cupini on 08 Jan 2010


Published Paper

Inserted: 8 jan 2010

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Volume: 26
Pages: 1183-1205
Year: 2009


We consider the following autonomous variational problem \[ \mbox{minimize } \ \left\{ \int_a^b f(v(x),v'(x)) \ dx \ : \ v\in W^{1,1}(a,b),\ v(a)=\alpha, \ v(b)=\beta\right\}\] where the Lagrangian $f$ is assumed to be continuous, but not necessarily coercive, nor convex. We show that the existence of the minimum is linked to the solvability of certain constrained variational problems. This allows us to derive existence theorems covering a wide class of nonconvex noncoercive problems.

Keywords: relaxation, Nonconvex problems, noncoercive problems, autonomous Lagrangians