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We consider the following classical autonomous variational problem $$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$ where the Lagrangianf 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...

We prove the equiabsolute integrability of a class of gradients, for functions in $W^{1,1}$. The present result appears as the localized version of well-known classical theorems.

We consider the following classical autonomous variational problem $$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$ 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...

We consider general second order boundary value problems on the whole line of the type $u^{\text{'}\text{'}}=h(t,u,u^{\text{'}})$, $u(-\infty )=0,u(+\infty )=1$, for which we provide existence, non-existence, multiplicity results. The solutions we find can be reviewed as heteroclinic orbits in the $(u,u^{\text{'}})$ plane dynamical system.