Table of Contents
Conference Papers in Mathematics
Volume 2013, Article ID 353460, 4 pages
Conference Paper

Overview on the Pointwise Constrained Liapunov Vectorial Convexity Theorem

Cima-ue, Rua Romão Ramalho 59, 7000-671 Évora, Portugal

Received 17 May 2013; Accepted 18 June 2013

Academic Editors: B. Datta, G. S. F. Frederico, N. Martins, D. F. M. Torres, and A. J. Zaslavski

This Conference Paper is based on a presentation given by Clara Carlota and Sílvia Chá at “The Cape Verde International Days on Mathematics 2013” held from 22 April 2013 to 25 April 2013 in Praia, Cape Verde.

Copyright © 2013 Clara Carlota et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In applications of the Calculus of Variations, Optimal Control and Differential Inclusions, very important real-life problems are nonconvex vectorial and subject to pointwise constraints. The classical Liapunov convexity theorem is a crucial tool allowing researchers to solve nonconvex vectorial problems involving single integrals. However, the possibility of extending such theorem so as to deal with pointwise constraints has remained an open problem for two decades, in the more realistic case using variable vectorial velocities. We have recently solved it, in the sense of proving necessary conditions and sufficient conditions for solvability of such problem. A quick overview of our results is presented here, the main point being that, somehow, convex constrained nonuniqueness a.e. implies nonconvex constrained existence.