A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

Mariconda, Carlo; Treu, Giulia

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 201-210

Let $L:{ℝ}^N\times {ℝ}^N\to ℝ$ be a borelian function and consider the following problems

$$inf\left\{F\left(y\right)={\int }_a^{b}L(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}\left(P\right)$$

$$inf\left\{F^{**}\left(y\right)={\int }_a^bŁ(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}·\left(P^{**}\right)$$

We give a sufficient condition, weaker then superlinearity, under which $infF=inf{F}^{**}$ if $L$ is just continuous in $x$. We then extend a result of Cellina on the Lipschitz regularity of the minima of $\left(P\right)$ when $L$ is not superlinear.

Classification : 37N35
Keywords: Lipschitz, regularity, non-coercive, discontinuous, calculus of variations

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