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Zbl 1167.49002
Cupini, Giovanni; Guidorzi, Marcello; Marcelli, Cristina
Existence of minimizers of free autonomous variational problems via solvability of constrained ones. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 4, 1183-1205 (2009). ISSN 0294-1449

Summary: We consider the following autonomous variational problem $$\text {minimize } \bigg\{\int_a^b f(v(x),v'(x))\,dx: v \in W^{1,1}(a,b),\ v(a) = \alpha,\ v(b)=\beta\bigg\},$$ 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.

MSC 2000:: *49J05 Free problems in one independent variable (existence)
49K05 Free problems in one independent variable (nec./ suff.)

Keywords: nonconvex problems; noncoercive problems; autonomous Lagrangians; constrained problems; relaxation; DuBois-Reymond condition

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