On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 68-101.

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 nm instead of the whole space 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)= sup {g(v)|g: nm {+} quasiconvex and lower semicontinuous, g(v)f(v)v nm }. 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.

DOI : https://doi.org/10.1051/cocv:2008067
Classification : 26B25,  26B40,  49J45,  52A20
Mots clés : unbounded function, quasiconvex function, quasiconvex envelope, Morrey's integral inequality, representation theorem
@article{COCV_2009__15_1_68_0,
     author = {Wagner, Marcus},
     title = {On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {68--101},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {1},
     year = {2009},
     doi = {10.1051/cocv:2008067},
     zbl = {1173.26009},
     mrnumber = {2488569},
     language = {en},
     url = {www.numdam.org/item/COCV_2009__15_1_68_0/}
}
Wagner, Marcus. On the lower semicontinuous quasiconvex envelope for unbounded integrands (I). ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 68-101. doi : 10.1051/cocv:2008067. http://www.numdam.org/item/COCV_2009__15_1_68_0/

[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. | MR 736678

[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. | MR 748301

[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). | MR 2244145 | Zbl 1110.35001

[4] J.M. Ball and F. Murat, W 1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR 759098 | Zbl 0549.46019

[5] A. Brøndsted, An Introduction to Convex Polytopes. Springer, New York - Heidelberg - Berlin (1983). | MR 683612 | Zbl 0509.52001

[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). | MR 225940

[8] E. Casadio Tarabusi, An algebraic characterization of quasi-convex functions. Ricerche di Mat. 42 (1993) 11-24. | MR 1283802 | Zbl 0883.26011

[9] F.H. Clarke, Optimization and Nonsmooth Analysis. 2nd Edn., SIAM, Philadelphia (1990). | MR 1058436 | Zbl 0696.49002

[10] L. Collatz and W. Wetterling, Optimierungsaufgaben, 2nd Edn., Heidelberger Taschenbücher 15. Springer, Berlin - Heidelberg - New York (1971). | MR 347131 | Zbl 0224.90004

[11] B. Dacorogna, Quasiconvexity and relaxation of nonconvex problems in the calculus of variations. J. Funct. Anal. 46 (1982) 102-118. | MR 654467 | Zbl 0547.49003

[12] B. Dacorogna, Direct Methods in the Calculus of Variations. 2nd Edn., Springer, New York etc. (2008). | MR 2361288 | Zbl 1140.49001

[13] B. Dacorogna and N. Fusco, Semi-continuité des fonctionnelles avec contraintes du type 4 (1985) 179-189. | MR 783338 | Zbl 0564.49005

[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. | MR 1448710 | Zbl 0901.49027

[15] B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems. J. Funct. Anal. 152 (1998) 404-446. | MR 1607932 | Zbl 0911.35034

[16] B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations. Birkhäuser, Boston - Basel - Berlin (1999). | MR 1702252 | Zbl 0938.35002

[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. | Zbl pre05258462

[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. | MR 2148926 | Zbl 1100.49015

[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 Equations 21 (2004) 357-400. | MR 2098073 | Zbl 1062.49012

[20] I. Ekeland and R. Témam, Convex Analysis and Variational Problems. 2nd Edn., SIAM, Philadelphia (1999). | MR 1727362 | Zbl 0939.49002

[21] J. Elstrodt, Maß- und Integrationstheorie. Springer, New York - Heidelberg - Berlin (1996). | MR 2257838 | Zbl 0861.28001

[22] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton etc. (1992). | MR 1158660 | Zbl 0804.28001

[23] A.D. Ioffe and V.M. Tichomirow, Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979). | MR 527119 | Zbl 0403.49001

[24] B. Kawohl, From Mumford-Shah to Perona-Malik in image processing. Math. Meth. Appl. Sci. 27 (2004) 1803-1814. | MR 2087298 | Zbl 1060.35054

[25] D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rat. Mech. Anal. 115 (1991) 329-365. | MR 1120852 | Zbl 0754.49020

[26] J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 1-13. | Numdam | MR 1668552 | Zbl 0932.49015

[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. | MR 259752 | Zbl 0184.47401

[28] M. Kružík, Bauer's maximum principle and hulls of sets. Calc. Var. Partial Differential Equations 11 (2000) 321-332. | MR 1797873 | Zbl 0981.49010

[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. | MR 1937535 | Zbl 1033.52007

[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. | MR 54865 | Zbl 0046.10803

[32] S. Pickenhain and M. Wagner, Piecewise continuous controls in Dieudonné-Rashevsky type problems. J. Optim. Theory Appl. 127 (2005) 145-163. | MR 2175074 | Zbl pre05353255

[33] R.T. Rockafellar, Convex Analysis. 2nd Edn., Princeton University Press, Princeton (1972). | MR 1451876 | Zbl 0193.18401

[34] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, Grundlehren 317. Springer, Berlin etc. (1998). | MR 1491362 | Zbl 0888.49001

[35] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993). | MR 1216521 | Zbl 0798.52001

[36] K. Schulz and B. Schwartz, Finite extensions of convex functions. Math. Operationsforschung Statist. Ser. Optimization 10 (1979) 501-509. | MR 568627 | Zbl 0439.26007

[37] V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Ser. A 120 (1992) 185-189. | MR 1149994 | Zbl 0777.49015

[38] T.W. Ting, Elastic-plastic torsion of convex cylindrical bars. J. Math. Mech. 19 (1969) 531-551. | MR 250546 | Zbl 0197.23301

[39] T.W. Ting, Elastic-plastic torsion problem III. Arch. Rat. Mech. Anal. 34 (1969) 228-244. | MR 264889 | Zbl 0179.53903

[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 Systems 563, Springer, Berlin etc. (2006) 233-250. | MR 2191161 | Zbl 1108.49028

[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). | MR 2481615 | Zbl 1159.49033

[44] K. Zhang, On the structure of quasiconvex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 663-686. | Numdam | MR 1650974 | Zbl 0917.49014

[45] K. Zhang, On the quasiconvex exposed points. ESAIM: COCV 6 (2001) 1-19 (electronic). | Numdam | MR 1804495 | Zbl 0970.49013