A formulation of Noether's theorem for fractional problems of the calculus of variations

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.     

Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case

Under general growth assumptions, that include some cases of linear growth, we prove existence of Lipschitzian solutions to the problem of minimizing ∫_a^bL(x(s),x′(s)) ds with the boundary conditions x(a)=A, x(b)=B.     

Duality theorems for convex problems with time delay

In this paper, we examine a class of convex problems of Bolza type, involving a time delay in the state. It encompasses a variety of time-delay problems arising in the calculus of variations and optimal control. A duality analysis is carried out which, among other things, leads to a characterization of minimizers in terms of the Euler-Lagrange inclusion. The results obtained improve in significant respects on what is achievable by techniques previously employed, based on elimination of the time delay by introduction of an infinite-dimensional state space or on the method of steps.     

An existence theorem in the calculus of variations

In this paper, a nonparametric variational problem is considered in the setting of the theory of generalized curves. It is assumed that the integrand of the problem does not grow at infinity faster than the norm of the variable , for all values of the other variablest andx (which take their values in a compact product set). It is shown that a generalized curve exists such that the minimum of the functional over an appropriate set is achieved. This generalized curve does not in general have compact support.     

Existence and uniqueness results for minimization problems with nonconvex functionals

In some physical problems (mechanical problems, optimal control problems, phase transition problems, etc.), we have to minimize a functionalJ over a topological spaceU for whichJ is not sequentially lower semicontinuous. In this article, we prove new existence results for general one-dimensional vector problems of calculus of variations without any convexity condition on the integrand of the problem. In particular, we do not suppose that the integrand is split in two parts, one part depending on the gradient variable and the other part depending on the state variable, as is often supposed in recent results. In the case where the integrand is the sum of two functions, the first one depending on the gradient variable and the second one depending on the state variable, we also prove a uniqueness result without any convexity assumption with respect to the gradient variable.A preliminary version of some results given in this article was presented at the Workshop on Calculus of Variations and Nonlinear Elasticity organized at Cortona, Italy, 27–31 May 1991 by B. Dacorogna, P. Marcellini, and C. Sbordone. The author would like to thank the organizers of this workshop for their invitation.     

Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian

This paper proves new results of existence of minimizers for the nonconvex integral , among the AC functions with x(a)=A, x(b)=B. Our Lagrangian L() is e.g. lsc with superlinear growth, assuming +∞ values freely. We replace convexity by almost convexity, a hypothesis which in the radial superlinear case L(s,ξ)=f(s,|ξ|) is automatically satisfied provided f(s,) is convex at zero.     

Stochastic extensions to necessary conditions in the theory of the calculus of variations

A stochastic version of the modified Young's generalized necessary conditions in the calculus of variations is given in this paper. It is based on an extension of Minkowski's theorem on the existence of a flat support for a convex figure, and it generalizes the necessary conditions of Weierstrass and Euler in the classical theory of the calculus of variations to a class of admissible curves which are expressible in terms of a finite number of random parameters. The integrals which we consider here are in the general Denjoy sense, except those with respect to the random parameters, which exist in the Lebesgue sense defined on a probability space. The importance of our stochastic analysis lies in the completion that a minimum not attained in the classical sense may be, and frequently is, attained in the stochastic case.This research was supported in part by the National Science Foundation under Grants Nos. GK-1834X and GK-31229     

An example of stability for the minima of a sequence of dc functions: Homogenization for a class of nonlinear Sturm-Liouville problems

Combining the Clarke-Ekeland dual least action principle and the epi-convergence, we state an existence result and study the asymptotic behaviour for the periodic solution of a nonlinear Sturm-Liouville problem deriving from a convex subquadratic potential, when the data are perturbed in a suitable sense. The result appears like a stability result for the minimizers of a sequence of DC functions.     

Radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations

In this paper, we study the existence of radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations. By applying a well‐known fixed point theorem in cones, we shall establish several new criteria for the existence of nontrivial radially symmetric convex solutions for the systems of Monge‐Ampère equations with or without an eigenvalue parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

On the existence in a problem of the calculus of variations without convexity assumptions

Summary We obtain new sufficient conditions for the existence in a problem of the calculus of variations without convexity assumptions.     

Contributions to the calculus of variations

In this paper, the theory of the calculus of variations for the simplest problem is reviewed. Simpler proofs are given for the classical conditions and a new necessary condition is presented which allows strong, necessary, and sufficient conditions to be stated for the first time.The major part of this research was conducted in the Department of Electrical Engineering, University of the West Indies, St. Augustine, Trinidad, West Indies. The author is indebted to Dr. B. Bhatt for interesting discussions.     

Optimality conditions for discrete calculus of variations problems

Nonlinear discrete calculus of variations problems with variable endpoints and with equality type constraints on trajectories are considered. We derive new nontrivial first- and second-order necessary optimality conditions.     

Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.     

Existence of multiple solutions with precise sign information for superlinear Neumann problems

We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator and having a p-superlinear nonlinearity. Using truncation techniques combined with the method of upper–lower solutions and variational arguments based on critical point theory, we prove the existence of five nontrivial smooth solutions, two positive, two negative and one nodal. For the semilinear (i.e., p = 2) problem, using critical groups we produce a second nodal solution. This paper was completed while N.S. Papageorgiou was visiting the University of Aveiro as an invited scientist. The hospitality and financial support of the host institution are gratefully acknowledged. V. Staicu acknowledges partial financial support from the Portuguese Foundation for Sciences and Technology (FCT) under the project POCI/MAT/55524/2004.