Existence theorems for parametric problems in the calculus of variations and approximation (original) (raw)

The Existence Question in the Calculus of Variations: A Density Result

Proceedings of the American Mathematical Society, 1994

We show the existence of a dense subset 3! of W(R) such that, for g in it, the problem minimum / g(x(t))dt+ f h(x'{t))dt , x{0) = a , x(T) = b Jo Jo admits a solution for every lower semicontinuous h satisfying growth conditions

Regularity properties of solutions to the basic problem in the calculus of variations

Transactions of the American Mathematical Society, 1985

This paper concerns the basic problem in the calculus of variations: minimize a functional J J defined by \[ J ( x ) = ∫ a b L ( t , x ( t ) , x ˙ ( t ) ) d t J(x) = \int _a^b {L(t,x(t),\dot x(t))\;dt} \] over a class of arcs x x whose values at a a and b b have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution x x , there exists a relatively open subset Ω \Omega of [ a , b ] [a,b] , of full measure, on which x x is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the globa...

Existence and regularity in the small in the calculus of variations

Journal of Differential Equations, 1985

A local existence theorem is proved for the basic problem in the calculus of variations, that of minimizing SL(t, X, n) dt over a class of functions x assuming given boundary conditions. The Lagrangian L is only assumed to be locally Lipschitz and strictly convex in its i variable. 8 1985 Academic PIES, IIIC. * The support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

Some Regularity Properties on Bolza problems in the Calculus of Variations

Comptes Rendus Mathematique, 2022

The paper summarizes the main core of the last results that we obtained in [4, 8, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann-Du-Bois Reymond equation. Under our assumptions, there exists a minimizing sequence of Lipschitz functions. A first consequence is that we can exclude the presence of the Lavrentiev phenomenon. Moreover, under a further mild growth assumption satisfied by the minimal length functional, fully described in the paper, the above sequence may be taken with the same Lipschitz rank, even when the initial datum and initial value vary on a compact set. The Lipschitz regularity of the value function follows.