Geodesic
Monotonicity properties of minimizers and relaxation for autonomous variational problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 222-242

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We consider the following classical autonomous variational problem

minimize F(v)=∫ a b f(v(x),v ' (x))x̣: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.

DOI : 10.1051/cocv/2010001
Classification : 49K05, 49J05
Keywords: nonconvex variational problems, autonomous variational problems, existence of minimizers, Dubois-Reymond necessary condition, relaxation 
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     title = {Monotonicity properties of minimizers and relaxation for autonomous variational problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {222--242},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     doi = {10.1051/cocv/2010001},
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     zbl = {1213.49028},
     language = {en},
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Cupini, Giovanni; Marcelli, Cristina. Monotonicity properties of minimizers and relaxation for autonomous variational problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 222-242. doi: 10.1051/cocv/2010001

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