Intermediate existence and regularity in the calculus of variations

AbstractConsider the basic problem in the calculus of variations—given a Langrangian L: [a,b] x Rn × Rn → R and a pair of endpoints α, β ∈ Rn, choose x∈ W1,1 ([a,b],Rn) to minimize Λ [x]≔∫ab L(t, x(t), x∣(t)) dt subject to x(a) = α, x(b) = β. (P) In 1915, Tonelli [5] put forward hypothesis under which the functional Λ is weakly lower semicontinuous on W1,1 and the level set {x : Λ[x] ≤ inf (P) + ε, x(a) = α, x(b) = β} is weakly compact for some ε > 0. He deduced that any minimizing sequence in (P) has a subsequence converging weakly to a solution to (P), and thereby established the topological approach which remains the state of the art. (Cesari [1] presents an up-to-date survey of the field.) Our approach, however, is quite different, being based on the properties of extremal arcs in certain auxiliary problems. It applies to certain problems with non-compact level sets, and offers a unified treatment of local and global existence theory for both fast-and slow-growth Lagrangians. Moreover, our results include the desirable “regularity” assertion that all solutions of (P) are Lipschitzian. They rely on modest conditions on L, together with a system of inequalities relating the data of the problem