In this monograph, in order to avoid technicalities, we mainly deal with a simple nonlinear second order periodic boundary problem, which is treated in order to illustrate the following methods. (a) The direct method of the Calculus of Variations. (b) The dual action principal of the Calculus of Variations introduced in the 70’s by Clarke and Ekeland for convex Hamiltonian systems. This principle consists in transforming a Lagrangian system into its Hamiltonian form, and then applying the direct method of the Calculus of Variations to the dual functional χ of the indefinite functional φ via convexity theory (Legendre transform, etc. . . . ). (c) Mini–max theorems (e.g. the Mountain Pass Theorem of Ambrosetti and Rabinowitz, and the Saddle Point Theorem of Rabinowitz) which can be applicable in important cases when (a) and (b) fail, or when (a) and (b) produce only the trivial solution of the underlying variational problem which is of course not of interest.
A first course in Calculus of Variations
PUCCI, Patrizia
1995
Abstract
In this monograph, in order to avoid technicalities, we mainly deal with a simple nonlinear second order periodic boundary problem, which is treated in order to illustrate the following methods. (a) The direct method of the Calculus of Variations. (b) The dual action principal of the Calculus of Variations introduced in the 70’s by Clarke and Ekeland for convex Hamiltonian systems. This principle consists in transforming a Lagrangian system into its Hamiltonian form, and then applying the direct method of the Calculus of Variations to the dual functional χ of the indefinite functional φ via convexity theory (Legendre transform, etc. . . . ). (c) Mini–max theorems (e.g. the Mountain Pass Theorem of Ambrosetti and Rabinowitz, and the Saddle Point Theorem of Rabinowitz) which can be applicable in important cases when (a) and (b) fail, or when (a) and (b) produce only the trivial solution of the underlying variational problem which is of course not of interest.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
