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

Carlo Mariconda; Giulia Treu

ESAIM: Control, Optimisation and Calculus of Variations (2004)

  • Volume: 10, Issue: 2, page 201-210
  • ISSN: 1292-8119

Abstract

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Let L : ℝ N × ℝ N → ℝ be a borelian function and consider the following problems inf F ( y ) = ∫ a b L ( y ( t ) , y ' ( t ) ) d t : y ∈ A C ( [ a , b ] , ℝ N ) , y ( a ) = A , y ( b ) = B ( P ) inf F * * ( y ) = ∫ a b Ł ( y ( t ) , y ' ( t ) ) d t : y ∈ A C ( [ a , b ] , ℝ N ) , y ( a ) = A , y ( b ) = B · ( 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.

How to cite

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Mariconda, Carlo, and Treu, Giulia. "A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2004): 201-210. <http://eudml.org/doc/245476>.

@article{Mariconda2004,
abstract = {Let $L:\mathbb \{R\}^N\times \mathbb \{R\}^N\rightarrow \mathbb \{R\}$ be a borelian function and consider the following problems\[ \inf \left\lbrace F(y)=\int \_a^bL(y(t),y^\{\prime \}(t))\,\{\rm d\}t:\,y\in AC([a,b],\mathbb \{R\}^N), y(a)=A,\,y(b)=B\right\rbrace \qquad \quad \! (P) \]\[ \hspace*\{-17.07182pt\}\inf \left\lbrace F^\{**\}(y)=\int \_a^bŁ(y(t),y^\{\prime \}(t))\,\{\rm d\}t:\,y\in AC([a,b],\mathbb \{R\}^N), y(a)=A,\,y(b)=B\right\rbrace \cdot \quad \;\ \! (P^\{**\}) \]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.},
author = {Mariconda, Carlo, Treu, Giulia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Lipschitz; regularity; non-coercive; discontinuous; calculus of variations; non-coercivity; discontinuous integrand; relaxation; integral functional},
language = {eng},
number = {2},
pages = {201-210},
publisher = {EDP-Sciences},
title = {A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand},
url = {http://eudml.org/doc/245476},
volume = {10},
year = {2004},
}

TY - JOUR
AU - Mariconda, Carlo
AU - Treu, Giulia
TI - A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2004
PB - EDP-Sciences
VL - 10
IS - 2
SP - 201
EP - 210
AB - Let $L:\mathbb {R}^N\times \mathbb {R}^N\rightarrow \mathbb {R}$ be a borelian function and consider the following problems\[ \inf \left\lbrace F(y)=\int _a^bL(y(t),y^{\prime }(t))\,{\rm d}t:\,y\in AC([a,b],\mathbb {R}^N), y(a)=A,\,y(b)=B\right\rbrace \qquad \quad \! (P) \]\[ \hspace*{-17.07182pt}\inf \left\lbrace F^{**}(y)=\int _a^bŁ(y(t),y^{\prime }(t))\,{\rm d}t:\,y\in AC([a,b],\mathbb {R}^N), y(a)=A,\,y(b)=B\right\rbrace \cdot \quad \;\ \! (P^{**}) \]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.
LA - eng
KW - Lipschitz; regularity; non-coercive; discontinuous; calculus of variations; non-coercivity; discontinuous integrand; relaxation; integral functional
UR - http://eudml.org/doc/245476
ER -

References

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  4. [4] G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Res. Notes Math. Ser. 207 (1989). Zbl0669.49005MR1020296
  5. [5] A. Cellina, The classical problem of the calculus of variations in the autonomous case: Relaxation and lipschitzianity of solutions. Preprint (2001). Zbl1064.49027MR2020039
  6. [6] G. Dal Maso and H. Frankowska, Autonomous Integral Functionals with Discontinuous Nonconvex Integrands: Lipschitz Regularity of Minimizers, DuBois-Reymond Necessary Conditions, and Hamilton-Jacobi Equations. Preprint (2002). Zbl1035.49035MR1977878
  7. [7] I. Ekeland and R. Témam, Convex analysis and variational problems. Classics Appl. Math. 28 (1999). Zbl0939.49002MR1727362
  8. [8] C. Mariconda and G. Treu, Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth. Dipartimento di Matematica pura e applicata, Università di Padova 10 (2003) preprint. Zbl1112.49020MR2305479
  9. [9] W. Rudin, Functional analysis. International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York (1991). Zbl0867.46001MR1157815

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