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

Published online by Cambridge University Press:  15 March 2004

Carlo Mariconda  and
Giulia Treu
Carlo Mariconda
Affiliation:
Dipartimento di Matematica pura e applicata, Università di Padova, 7 via Belzoni, 35131 Padova, Italy; maricond@math.unipd.it.; treu@math.unipd.it.
Giulia Treu
Affiliation:
Dipartimento di Matematica pura e applicata, Università di Padova, 7 via Belzoni, 35131 Padova, Italy; maricond@math.unipd.it.; treu@math.unipd.it.
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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.

Keywords

Lipschitzregularitynon-coercivediscontinuouscalculus of variations.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 2 , April 2004 , pp. 201 - 210
Copyright
© EDP Sciences, SMAI, 2004

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References

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