Advanced Search

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 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 show that local minimizers of functionals of the form ${\int }_{\Omega }\left[f\left(Du\right(x\left)\right)+g(x,u(x\left)\right)\right]\mathrm{d}x$,  $u\in u_0+W_0^{1,p}\left(\Omega \right)$, are locally Lipschitz continuous provided is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and is Lipschitz continuous in . As a consequence of this, we obtain an existence result for a related nonconvex functional.