A Du Bois-Reymond Convex Inclusion for Nonautonomous Problems of the Calculus of Variations and Regularity of Minimizers - Université de Bretagne Occidentale Access content directly
Journal Articles Applied Mathematics and Optimization Year : 2019

A Du Bois-Reymond Convex Inclusion for Nonautonomous Problems of the Calculus of Variations and Regularity of Minimizers

Abstract

We consider a local minimizer of the classical problem of the calculus of variations, where the Lagrangian is just Borel measurable. When is real valued, we merely assume a local Lipschitz condition on with respect to t, allowing to be discontinuous and nonconvex in x or . In the case of an extended valued Lagrangian, we impose the lower semicontinuity of the Lagrangian, and a condition on the effective domain of the Lagrangian. We use a recent variational Weierstrass type inequality to show that the minimizers satisfy a relaxation result and an Erdmann – Du Bois-Reymond convex inclusion which, remarkably, holds whenever is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer the Lipschitz continuity of minimizers.
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Dates and versions

hal-04315969 , version 1 (30-11-2023)

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Piernicola Bettiol, Carlo Mariconda. A Du Bois-Reymond Convex Inclusion for Nonautonomous Problems of the Calculus of Variations and Regularity of Minimizers. Applied Mathematics and Optimization, 2019, 83 (3), pp.2083-2107. ⟨10.1007/s00245-019-09620-y⟩. ⟨hal-04315969⟩
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