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

Carlo Mariconda; Giulia Treu

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 L * * ( 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 (2010): 201-210. <http://eudml.org/doc/90725>.

@article{Mariconda2010,
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. },
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},
month = {3},
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/90725},
volume = {10},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 2
SP - 201
EP - 210
AB - 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.
LA - eng
KW - Lipschitz; regularity; non-coercive; discontinuous; calculus of variations.; non-coercivity; discontinuous integrand; relaxation; integral functional
UR - http://eudml.org/doc/90725
ER -

References

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  2. M. Amar, G. Bellettini and S. Venturini, Integral representation of functionals defined on curves of W1,p. Proc. R. Soc. Edinb. Sect. A128 (1998) 193-217.  
  3. L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl.142 (1989) 301-316.  
  4. G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Res. Notes Math. Ser.207 (1989).  
  5. A. Cellina, The classical problem of the calculus of variations in the autonomous case: Relaxation and lipschitzianity of solutions. Preprint (2001).  
  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).  
  7. I. Ekeland and R. Témam, Convex analysis and variational problems. Classics Appl. Math.28 (1999).  
  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.  
  9. W. Rudin, Functional analysis. International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York (1991).  

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