On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)

Marcus Wagner

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 1, page 68-101
  • ISSN: 1292-8119

Abstract

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Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K ⊂ ℝ n m instead of the whole space ℝ n m as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous quasiconvex envelope f ( q c ) ( v ) = sup { g ( v ) | g : ℝ n m → ℝ ∪ { + ∞ } quasiconvex and lower semicontinuous, g ( v ) ≤ f ( v ) ∀ v ∈ ℝ n m } . Our main result is a representation theorem for f ( 𝑞𝑐 ) which generalizes Dacorogna's well-known theorem on the representation of the quasiconvex envelope of a finite function. The paper will be completed by the calculation of f ( 𝑞𝑐 ) in two examples.


How to cite

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Wagner, Marcus. "On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 68-101. <http://eudml.org/doc/250566>.

@article{Wagner2009,
abstract = { Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K $\subset \mathbb\{R\}^\{nm\}$ instead of the whole space $\mathbb\{R\}^\{nm\}$ as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous quasiconvex envelope $f^\{(qc)\} (v) = \{\rm sup\} \\{ \,g(v)\, \vert \,g : \mathbb\{R\}^\{nm\} \rightarrow \mathbb\{R\} \cup \\{ + \infty \\}$ quasiconvex and lower semicontinuous, $g(v) \leq f(v) \,\,\,\,\forall v \in \mathbb\{R\}^\{nm\}\,\\}.$ Our main result is a representation theorem for $f^\{(\{\it qc\})\}$ which generalizes Dacorogna's well-known theorem on the representation of the quasiconvex envelope of a finite function. The paper will be completed by the calculation of $f^\{(\{\it qc\})\}$ in two examples.
},
author = {Wagner, Marcus},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Unbounded function; quasiconvex function; quasiconvex envelope; Morrey's integral inequality; representation theorem; unbounded function},
language = {eng},
month = {1},
number = {1},
pages = {68-101},
publisher = {EDP Sciences},
title = {On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)},
url = {http://eudml.org/doc/250566},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Wagner, Marcus
TI - On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2009/1//
PB - EDP Sciences
VL - 15
IS - 1
SP - 68
EP - 101
AB - Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K $\subset \mathbb{R}^{nm}$ instead of the whole space $\mathbb{R}^{nm}$ as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous quasiconvex envelope $f^{(qc)} (v) = {\rm sup} \{ \,g(v)\, \vert \,g : \mathbb{R}^{nm} \rightarrow \mathbb{R} \cup \{ + \infty \}$ quasiconvex and lower semicontinuous, $g(v) \leq f(v) \,\,\,\,\forall v \in \mathbb{R}^{nm}\,\}.$ Our main result is a representation theorem for $f^{({\it qc})}$ which generalizes Dacorogna's well-known theorem on the representation of the quasiconvex envelope of a finite function. The paper will be completed by the calculation of $f^{({\it qc})}$ in two examples.

LA - eng
KW - Unbounded function; quasiconvex function; quasiconvex envelope; Morrey's integral inequality; representation theorem; unbounded function
UR - http://eudml.org/doc/250566
ER -

References

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  1. J.A. Andrejewa and R. Klötzler, Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. Teil I. Z. Angew. Math. Mech.64 (1984) 35–44.  
  2. J.A. Andrejewa and R. Klötzler, Zur analytischen Lösung geometrischer Optimierungsaufgaben mittels Dualität bei Steuerungsproblemen. Teil II. Z. Angew. Math. Mech.64 (1984) 147–153.  
  3. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. 2nd Edn., Springer, New York etc. (2006).  
  4. J.M. Ball and F. Murat, W 1 , p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253.  
  5. A. Brøndsted, An Introduction to Convex Polytopes. Springer, New York - Heidelberg - Berlin (1983).  
  6. C. Brune, H. Maurer and M. Wagner, Edge detection within optical flow via multidimensional control. BTU Cottbus, Preprint-Reihe Mathematik, Preprint Nr. M-02/2008 (submitted).  
  7. C. Carathéodory, Vorlesungen über reelle Funktionen. 3rd Edn., Chelsea, New York (1968).  
  8. E. Casadio Tarabusi, An algebraic characterization of quasi-convex functions. Ricerche di Mat.42 (1993) 11–24.  
  9. F.H. Clarke, Optimization and Nonsmooth Analysis. 2nd Edn., SIAM, Philadelphia (1990).  
  10. L. Collatz and W. Wetterling, Optimierungsaufgaben, 2nd Edn., Heidelberger Taschenbücher15. Springer, Berlin - Heidelberg - New York (1971).  
  11. B. Dacorogna, Quasiconvexity and relaxation of nonconvex problems in the calculus of variations. J. Funct. Anal.46 (1982) 102–118.  
  12. B. Dacorogna, Direct Methods in the Calculus of Variations. 2nd Edn., Springer, New York etc. (2008).  
  13. B. Dacorogna and N. Fusco, Semi-continuité des fonctionnelles avec contraintes du type “ det ∇ u > 0 ". Boll. Un. Mat. Ital. B (6)4 (1985) 179–189.  
  14. B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math.178 (1997) 1–37.  
  15. B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems. J. Funct. Anal. 152 (1998) 404–446.  
  16. B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations. Birkhäuser, Boston - Basel - Berlin (1999).  
  17. B. Dacorogna and A.M. Ribeiro, On some definitions and properties of generalized convex sets arising in the calculus of variations, in Recent Advances on Elliptic and Parabolic Issues, M. Chipot and H. Ninomiya Eds., Proceedings of the 2004 Swiss-Japanese Seminar: Zurich, Switzerland, 6–10 December 2004, World Scientific, Singapore (2006) 103–128.  
  18. R. De Arcangelis and E. Zappale, The relaxation of some classes of variational integrals with pointwise continuous-type gradient constraints. Appl. Math. Optim.51 (2005) 251–257.  
  19. R. De Arcangelis, S. Monsurrò and E. Zappale, On the relaxation and the Lavrentieff phenomenon for variational integrals with pointwise measurable gradient constraints. Calc. Var. Partial Differential Equations21 (2004) 357–400.  
  20. I. Ekeland and R. Témam, Convex Analysis and Variational Problems. 2nd Edn., SIAM, Philadelphia (1999).  
  21. J. Elstrodt, Maß- und Integrationstheorie. Springer, New York - Heidelberg - Berlin (1996).  
  22. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton etc. (1992).  
  23. A.D. Ioffe and V.M. Tichomirow, Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979).  
  24. B. Kawohl, From Mumford-Shah to Perona-Malik in image processing. Math. Meth. Appl. Sci.27 (2004) 1803–1814.  
  25. D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rat. Mech. Anal.115 (1991) 329–365.  
  26. J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire16 (1999) 1–13.  
  27. J.B. Kruskal, Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping. Proc. Amer. Math. Soc.23 (1969) 697–703.  
  28. M. Kružík, Bauer's maximum principle and hulls of sets. Calc. Var. Partial Differential Equations11 (2000) 321–332.  
  29. M. Kružík, Quasiconvex extreme points of convex sets, in Elliptic and Parabolic Problems, J. Bemelmans, B. Brighi, A. Brillard, M. Chipot, F. Conrad, I. Shafrir, V. Valente and G. Vergara-Caffarelli Eds., World Scientific Publishing, River Edge (2002) 145–151.  
  30. K.A. Lur'e, Hayka, Moscow (1975).  
  31. C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math.2 (1952) 25–53.  
  32. S. Pickenhain and M. Wagner, Piecewise continuous controls in Dieudonné-Rashevsky type problems. J. Optim. Theory Appl.127 (2005) 145–163.  
  33. R.T. Rockafellar, Convex Analysis. 2nd Edn., Princeton University Press, Princeton (1972).  
  34. R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, Grundlehren317. Springer, Berlin etc. (1998).  
  35. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993).  
  36. K. Schulz and B. Schwartz, Finite extensions of convex functions. Math. Operationsforschung Statist. Ser. Optimization10 (1979) 501–509.  
  37. V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Ser. A120 (1992) 185–189.  
  38. T.W. Ting, Elastic-plastic torsion of convex cylindrical bars. J. Math. Mech.19 (1969) 531–551.  
  39. T.W. Ting, Elastic-plastic torsion problem III. Arch. Rat. Mech. Anal.34 (1969) 228–244.  
  40. M. Wagner, Erweiterungen des mehrdimensionalen Pontrjaginschen Maximumprinzips auf meßbare und beschränkte sowie distributionelle Steuerungen. Ph.D. thesis, Universität Leipzig, Germany (1996).  
  41. M. Wagner, Nonconvex relaxation properties of multidimensional control problems, in Recent Advances in Optimization, A. Seeger Ed., Lecture Notes in Economics and Mathematical Systems563, Springer, Berlin etc. (2006) 233–250.  
  42. M. Wagner, Mehrdimensionale Steuerungsprobleme mit quasikonvexen Integranden. Habilitation thesis, Brandenburgische Technische Universität Cottbus, Cottbus, Germany (2006).  
  43. M. Wagner, Pontryagin's maximum principle for multidimensional control problems in image processing. J. Optim. Theory Appl. (to appear).  
  44. K. Zhang, On the structure of quasiconvex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire15 (1998) 663–686.  
  45. K. Zhang, On the quasiconvex exposed points. ESAIM: COCV6 (2001) 1–19 (electronic).  

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