We consider the following classical autonomous variational problem: minimize {F(v)=\int_a^b f(v(x),v′(x)) dx : v ∈ AC([a,b]), v(a) = α, v(b) = β}, 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 necessary condition and some existence or non-existence criteria.
Monotonicity properties of minimizers and relaxation for autonomous variational problems / Cupini, G.; Marcelli, Cristina. - In: ESAIM. COCV. - ISSN 1292-8119. - STAMPA. - 17:1(2011), pp. 222-242. [10.1051/cocv/2010001]
Monotonicity properties of minimizers and relaxation for autonomous variational problems
MARCELLI, Cristina
2011-01-01
Abstract
We consider the following classical autonomous variational problem: minimize {F(v)=\int_a^b f(v(x),v′(x)) dx : v ∈ AC([a,b]), v(a) = α, v(b) = β}, 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 necessary condition and some existence or non-existence criteria.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.