A Straightforward Sufficiency Proof for a Nonparametric Problem of Bolza in the Calculus of Variations (original) (raw)
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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...
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