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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Bettiol, Piernicola¹  Mariconda, Carlo² 

[1] Univ Brest, UMR CNRS 6205, F-29200 Brest, France

[2] Univ Padua, Dipartimento Matemat Tullio Levi Civita, I-35121 Padua, Italy

关键词: Weierstrass;    Tonelli-Morrey;    Proximal;    Maximum Principle;    Calculus of Variations;    Du Bois-Reymond;    Erdmann;    Regularity;    Lipschitz;    Nonautonomous;   

DOI  :  10.1016/j.jde.2019.09.011

来源: Elsevier

【 摘 要 】

We consider a problem of the calculus of variations of the form {Minimize I(x) := integral(b)(a) Lambda(t, x(t), x'(t))dt + Psi(x(a), x(b)) Subject to: x is an element of W-1,W-m ([a, b]; R-n), (P) x'(t) is an element of C a.e., x(t) is an element of Sigma (sic) is an element of[a, b], where Lambda :[ a, b] x R-n x R-n -> R boolean OR{+infinity} is Borel measurable, C is a cone, Sigma is a nonempty subset of R-n and Psi is an arbitrary extended valued function. We prove that if t -> Lambda(t, x, xi) satisfies a nonsmooth version of Cesari's Condition (S), then any local minimizer of (P) in the AC norm is Lipschitz whenever Lambda is coercive. The proof is obtained via a new variational inequality formulated here, that holds under the extended Condition (S) (just Borel measurability if Lambda is autonomous): For a.e. t in [a, b], Lambda(t, x(*)(t), x(*)'(t)/upsilon)upsilon - Lambda(t, x(*)(t), x(*)'(t)) >= p(t)(upsilon-1), (sic)upsilon>0, (W) where p(t) is an absolutely continuous arc, the derivative of which belongs to a suitable subdifferential of Lambda(center dot, 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. (C) 2019 Published by Elsevier Inc.

【 授权许可】

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