The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In Zaslavski (Nonlinear Analysis 43 (200l) 339), a generic well-posedness result (with respect to variations of the integrand of the integral functional) without the convexity condition was established for a class of optimal control problems satisfying the Cesari growth condition. In Zaslavski (Communications in Applied Analysis, to appear) we extended this generic well-posedness result to a class of variational problems in which the values at the end points are also subject to variations. More precisely, we established a generic well-posedness result for a class of variational problems (without convexity assumptions) over functions with values in a Banach space E which is identified with the corresponding complete metric space of pairs (f,(ξ1,ξ2)) (where f is an integrand satisfying the Cesari growth condition and ξ1,ξ2∈E are the values at the end points) denoted by script A sign. We showed that for a generic (f,(ξ1,ξ2))∈ script A sign the corresponding variational problem is well posed. In this paper we study the set of all pairs (f,(ξ1,ξ2))∈ script A sign for which the corresponding variational problem is well posed. We show that the complement of this set is not only of the first category but also a σ-porous set. © 2003 Elsevier Science Ltd. All rights reserved.
Zaslavski, A. J. (2003). Well posedness and porosity in the calculus of variations without convexity assumptions. Nonlinear Analysis, Theory, Methods and Applications, 53(1), 1–22. https://doi.org/10.1016/S0362-546X(02)00180-3