A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

Carlo Mariconda; Giulia Treu

doi:10.1051/cocv:2004004

All issues

Volume 10 / No 2 (April 2004)

ESAIM: COCV, 10 2 (2004) 201-210

Abstract

Free Access

Issue		ESAIM: COCV Volume 10, Number 2, April 2004


Page(s)		201 - 210
DOI		https://doi.org/10.1051/cocv:2004004
Published online		15 March 2004

ESAIM: COCV 10 (2004) 201-210

A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

Carlo Mariconda and Giulia Treu

Dipartimento di Matematica pura e applicata, Università di Padova, 7 via Belzoni, 35131 Padova, Italy; maricond@math.unipd.it.; treu@math.unipd.it.

Received: 27 June 2003

Abstract

Let $L:\Bbb R^N\times\Bbb R^N\rightarrow\Bbb R$ be a Borelian function and consider the following problems $\inf\left\{F(y)=\int_a^bL(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\} \qquad\quad\! (P)$   $\inf\left\{F^{**}(y)=\int_a^bL^{**}(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\}\cdot \quad\;\ \! (P^{**})$ We give a sufficient condition, weaker then superlinearity, under which $\inf F=\inf F^{**}$ if L is just continuous in x. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) when L is not superlinear.

Mathematics Subject Classification: 37N35

Key words: Lipschitz / regularity / non-coercive / discontinuous / calculus of variations.

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