A new variational inequality in the calculus of variations and Lipschitz regularity of minimizers

Abstract

We consider a problem of the calculus of variations of the form(P){MinimizeI(x):=∫abΛ(t,x(t),x′(t))dt+Ψ(x(a),x(b))Subject to:x∈W1,m([a,b];Rn),x′(t)∈C a.e., x(t)∈Σ∀t∈[a,b], where Λ:[a,b]×Rn×Rn→R∪{+∞} is Borel measurable, C is a cone, Σ is a nonempty subset of Rn and Ψ is an arbitrary extended valued function. We prove that if t↦Λ(t,x,ξ) satisfies a nonsmooth version of Cesari's Condition (S), then any local minimizer of (P) in the AC norm is Lipschitz whenever Λ is coercive. The proof is obtained via a new variational inequality formulated here, that holds under the extended Condition (S) (just Borel measurability if Λ is autonomous): For a.e. t in [a,b],(W)Λ(t,x⁎(t),x⁎′(t)v)v−Λ(t,x⁎(t),x⁎′(t))≥p(t)(v−1),∀v>0, where p(t) is an absolutely continuous arc, the derivative of which belongs to a suitable subdifferential of Λ(⋅,x⁎(t),x⁎′(t)) a.e. on [a,b]. The proof of the directional Weierstrass type condition (W) is based on Clarke's nonsmooth recent versions of the Maximum Principle.