Well posedness and porosity in the calculus of variations without convexity assumptions

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Abstract

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.

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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

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