Abstract

We consider the following classical autonomous variational problem: Minimize {F(u)=∫abf(u(x),u′(x)) dx : u∈AC([a,b]),u(a)=α,u(b)=β, u([a,b])⊆I}\left\{F(u)=\int_a^b f(u(x),u'(x))\,dx\,:\,u\in AC([a,b]), u(a)=\alpha, u(b)=\beta,\,u([a,b]) \subseteq I \right\} where II is a real interval, α,β∈I\alpha, \beta\in I, and f:I×R→[0,+∞)f:I\times \mathbb{R}\to [0,+\infty) is possibly neither continuous, nor coercive, nor convex; in particular f(s,⋅)f(s,\cdot) may be not convex at 00. Assuming the solvability of the relaxed problem, we prove under mild assumptions that the above variational problem has a solution, too.

Author information

Contact details are reproduced from the original publication and may be historical.

Giovanni Cupini

Dip. di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

giovanni.cupini@unibo.it

Keywords

  • Non-convex variational problem
  • non-coercive variational problem
  • autonomous variational problem
  • relaxation result

Mathematics Subject Classification

49K05,49J05

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Suggested citation

M. Bianchini, G. Cupini. “A Relaxation Result for Non-Convex and Non-Coercive Simple Integrals.” Journal of Convex Analysis 19 (2012), No. 1, 225–248. https://doi.org/10.68381/jca19014

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.