Autonomous Integral Functionals with Discontinuous Nonconvex Integrands: Lipschitz Regularity of Minimizers, DuBois—Reymond Necessary Conditions, and Hamilton—Jacobi Equations
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois—Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally Lipschitz and satisfies (in a generalized sense) a Hamilton—Jacobi equation.
- Autonomous Integral Functionals with Discontinuous Nonconvex Integrands: Lipschitz Regularity of Minimizers, DuBois—Reymond Necessary Conditions, and Hamilton—Jacobi Equations
Applied Mathematics & Optimization
Volume 48, Issue 1 , pp 39-66
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Key words. Discontinuous Lagrangians, Nonconvex integrands, Lipschitz minimizers, DuBois—Reymond necessary conditions, Hamilton—Jacobi equations. AMS Classification. Primary 49N60, Secondary 49K05, 49L25.