Optimality conditions for nonconvex variational problems relaxed in terms of Young measures

Tomáš Roubíček

Kybernetika (1998)

  • Volume: 34, Issue: 3, page [335]-347
  • ISSN: 0023-5954

Abstract

top
The scalar nonconvex variational problems of the minimum-energy type on Sobolev spaces are studied. As the Euler–Lagrange equation dramatically looses selectivity when extended in terms of the Young measures, the correct optimality conditions are sought by means of the convex compactification theory. It turns out that these conditions basically combine one part from the Euler–Lagrange equation with one part from the Weierstrass condition.

How to cite

top

Roubíček, Tomáš. "Optimality conditions for nonconvex variational problems relaxed in terms of Young measures." Kybernetika 34.3 (1998): [335]-347. <http://eudml.org/doc/33358>.

@article{Roubíček1998,
abstract = {The scalar nonconvex variational problems of the minimum-energy type on Sobolev spaces are studied. As the Euler–Lagrange equation dramatically looses selectivity when extended in terms of the Young measures, the correct optimality conditions are sought by means of the convex compactification theory. It turns out that these conditions basically combine one part from the Euler–Lagrange equation with one part from the Weierstrass condition.},
author = {Roubíček, Tomáš},
journal = {Kybernetika},
keywords = {nonconvex variational problem; Sobolev space; Young measure; convex compactification theory; Euler-Lagrange equation; Weierstrass condition; minimum-energy type; optimality conditions; nonconvex variational problem; Sobolev space; Young measure; convex compactification theory; Euler-Lagrange equation; Weierstrass condition; minimum-energy type; optimality conditions},
language = {eng},
number = {3},
pages = {[335]-347},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Optimality conditions for nonconvex variational problems relaxed in terms of Young measures},
url = {http://eudml.org/doc/33358},
volume = {34},
year = {1998},
}

TY - JOUR
AU - Roubíček, Tomáš
TI - Optimality conditions for nonconvex variational problems relaxed in terms of Young measures
JO - Kybernetika
PY - 1998
PB - Institute of Information Theory and Automation AS CR
VL - 34
IS - 3
SP - [335]
EP - 347
AB - The scalar nonconvex variational problems of the minimum-energy type on Sobolev spaces are studied. As the Euler–Lagrange equation dramatically looses selectivity when extended in terms of the Young measures, the correct optimality conditions are sought by means of the convex compactification theory. It turns out that these conditions basically combine one part from the Euler–Lagrange equation with one part from the Weierstrass condition.
LA - eng
KW - nonconvex variational problem; Sobolev space; Young measure; convex compactification theory; Euler-Lagrange equation; Weierstrass condition; minimum-energy type; optimality conditions; nonconvex variational problem; Sobolev space; Young measure; convex compactification theory; Euler-Lagrange equation; Weierstrass condition; minimum-energy type; optimality conditions
UR - http://eudml.org/doc/33358
ER -

References

top
  1. Aubin J.-P., Ekeland I., Applied Nonlinear Analysis, Wiley, New York 1984 Zbl1115.47049MR0749753
  2. Aubin J.-P., Frankowska H., Set-valued Analysis, Birkhäuser 1990 Zbl1168.49014MR1048347
  3. Ball J. M., James R. D., 10.1007/BF00281246, Arch. Rational Mech. Anal. 100 (1988), 13–52 (1988) MR0906132DOI10.1007/BF00281246
  4. Ball J. M., James R. D., 10.1098/rsta.1992.0013, Philos. Trans. Roy. Soc. London Ser. A 338 (1992), 389–450 (1992) Zbl0758.73009DOI10.1098/rsta.1992.0013
  5. Chipot M., Kinderlehrer D., 10.1007/BF00251759, Arch. Rational Mech. Anal. 103 (1988), 237–277 (1988) Zbl0673.73012MR0955934DOI10.1007/BF00251759
  6. DiPerna R. J., Majda A. J., 10.1007/BF01214424, Commun. Math. Physics 108 (1987), 667–689 (1987) Zbl0626.35059MR0877643DOI10.1007/BF01214424
  7. Gamkrelidze R. V., Principles of Optimal Control Theory (in Russian), Tbilisi Univ. Press, Tbilisi 1977. (English Translation: Plenum Press, New York 1978) (1977) MR0686793
  8. Hoffmann K.-H., Roubíček T., 10.1007/BF01189449, Appl. Math. Optim. 30 (1994), 113–126 (1994) Zbl0812.49009MR1284322DOI10.1007/BF01189449
  9. Ioffe A. D., Tihomirov V. M., Extensions of variational problems, Trans. Moscow Math. Soc. 18 (1968), 207–273 (1968) MR0254702
  10. Kinderlehrer D., Pedregal P., 10.1007/BF00375279, Arch. Rational Mech. Anal. 115 (1991), 329–365 (1991) Zbl0754.49020MR1120852DOI10.1007/BF00375279
  11. Kinderlehrer D., Pedregal P., 10.1137/0523001, SIAM J. Math. Anal. 23 (1992), 1–19 (1992) Zbl0757.49014MR1145159DOI10.1137/0523001
  12. Kinderlehrer D., Pedregal P., 10.1007/BF02921593, J. Geom. Anal. 4 (1994), 59–90 (1994) Zbl0828.46031MR1274138DOI10.1007/BF02921593
  13. Kružík M., Roubíček T., 10.1006/jmaa.1996.0115, J. Math. Anal. Appl. 198 (1996), 830–843 (198) MR1377827DOI10.1006/jmaa.1996.0115
  14. McShane E. J., Necessary conditions in the generalized-curve problems of the calculus of variations, Duke Math. J. 7 (1940), 1–27 (1940) MR0003478
  15. Roubíček T., 10.1016/0362-546X(91)90199-B, Nonlinear Anal. 16 (1991), 1117–1126 (1991) MR1111622DOI10.1016/0362-546X(91)90199-B
  16. Roubíček T., A note about optimality conditions for variational problems with rapidly oscillating solutions, In: Progress in Partial Differential Equations: Calculus of Variations, Applications (C. Bandle et al, eds., Pitman Res. Notes in Math. Sci. 267), Longmann, Harlow, Essex 1992, pp. 312–314 (1992) MR1194208
  17. Roubíček T., Nonconcentrating generalized Young functionals, Comment. Math. Univ. Carolin. 38 (1997), 91–99 (1997) MR1455472
  18. Roubíček T., Hoffmann K.-H., 10.1002/mma.1670180902, Math. Methods Appl. Sci. 18 (1995), 671–685 (1995) Zbl0828.35024MR1339929DOI10.1002/mma.1670180902
  19. Roubíček T., Hoffmann K.-H., 10.1016/0362-546X(94)00150-G, Nonlinear Anal. 25 (1995), 607–628 (1995) MR1338806DOI10.1016/0362-546X(94)00150-G
  20. Weierstraß K., Mathematische Werke, Teil VII “Vorlesungen über Variationsrechnung”, (Nachdr. d. Ausg., Leipzig, 1927) Georg Olms, Verlagsbuchhandlung, Hildesheim 1967 (1927) 
  21. Young L. C., 10.1007/BF02547714, Acta Math. 69 (1938), 239–258 (1938) Zbl0019.26702MR1555440DOI10.1007/BF02547714
  22. Young L. C., Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia 1969 Zbl0289.49003MR0259704
  23. Zeidler E., Nonlinear Functional Analysis and its Applications III, Variational Methods and Optimization. Springer, New York 1985 Zbl0583.47051MR0768749

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.