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

• Published in: Applied mathematics & optimization Vol. 83; no. 3; pp. 2083 - 2107
• Main Authors: Bettiol, Piernicola, Mariconda, Carlo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V; Springer Verlag (Germany)
• Subjects: Calculus of variations; Calculus of Variations and Optimal Control; Optimization; Control; Lipschitz condition; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Simulation; Systems Theory; Theoretical; Proximal; Erdmann; Tonelli–Morrey; Growth; Maximum principle; Weierstrass, directional; Lipschitz; Nonautonomous Lagrangian; Du Bois-Reymond; 49N60; 90C25; Slow growth; 49K05; Calculus of variations; Regularity; Slow growth condition; Regularity of minimizers; Du Bois-Reymond condition; Weierstrass condition
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-019-09620-y

We consider a local minimizer, in the sense of the W 1 , m norm ( m ≥ 1 ), of the classical problem of the calculus of variations P Minimize I ( x ) : = ∫ a b Λ ( t , x ( t ) , x ′ ( t ) ) d t + Ψ ( x ( a ) , x ( b ) ) subject to: x ∈ W 1 , m ( [ a , b ] ; R n ) , x ′ ( t ) ∈ C a.e., x ( t ) ∈ Σ ∀ t ∈ [ a , b ] . where Λ : [ a , b ] × R n × R n → R ∪ { + ∞ } is just Borel measurable, C is a cone, Σ is a nonempty subset of R n and Ψ is an arbitrary possibly extended valued function. When Λ is real valued, we merely assume a local Lipschitz condition on Λ with respect to t , allowing Λ ( t , x , ξ ) to be discontinuous and nonconvex in x or ξ . In the case of an extended valued Lagrangian, we impose the lower semicontinuity of Λ ( · , x , · ) , and a condition on the effective domain of Λ ( t , x , · ) . 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 Λ ( x , ξ ) is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer the Lipschitz continuity of minimizers.