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Calculus of Variations with Classical and Fractional Derivatives

Odzijewicz, Tatiana, Torres, Delfim F. M. (2012)

Mathematica Balkanica New Series

MSC 2010: 49K05, 26A33We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both classical and fractional derivatives are proved. The fundamental problem of the calculus of variations with mixed integer and fractional order derivatives as well as isoperimetric problems are considered.

Curvature functionals for curves in the equi-affine plane

Steven Verpoort (2011)

Czechoslovak Mathematical Journal

After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail. Notice. An extended version of this article is available on arXiv:0912.4075.

Monotonicity properties of minimizers and relaxation for autonomous variational problems

Giovanni Cupini, Cristina Marcelli (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the following classical autonomous variational problem minimize F ( v ) = a b f ( v ( x ) , v ' ( x ) ) x ̣ : v A C ( [ a , b ] ) , v ( a ) = α , v ( b ) = β , 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 or non-existence criteria.

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