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Let $u\in\phi+ W_0^{1,1}(\Omega)$ be a minimum for ...

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.

Let $L:{ℝ}^N\times {ℝ}^N\to ℝ$ be a Borelian function and consider the following problems $$inf\left\{F\left(y\right)={\int }_a^{b}L(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}\left(P\right)$$
 $$inf\left\{F^{**}\left(y\right)={\int }_a^b{L}^{**}(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}·\left(P^{**}\right)$$ We give a sufficient condition, weaker then superlinearity, under which $infF=inf{F}^{**}$ if is just continuous in . We then extend a result of Cellina on the Lipschitz regularity of the minima of when is not superlinear.

Let $u\in\phi+ W_0^{1,1}(\Omega)$ be a minimum for ...