Existence of minimizers for non-quasiconvex integrals. (English) Zbl 0837.49002

Dacorogna, Bernard; Marcellini, Paolo

Arch. Ration. Mech. Anal. 131, No. 4, 359-399 (1995).

In this paper the authors give conditions for existence and non-existence of solutions of a problem of the type \[ \inf\Biggl\{F(u)= \int_\Omega f(Du(x))dx,\quad u\in u_0+ W^{1,\infty}_0(\Omega, \mathbb{R}^N)\Biggr\} \] with linear boundary data \(u_0\) and non-quasiconvex functions \(f\). Among others, their detailed analysis shows that there is more hope to solve the minimization problem in the vectorial case \((N> 1)\) than in the scalar case \((N= 1)\).
People interested in these topics can find in this paper a rich bibliography and a variety of tools and examples.

Reviewer: R.Schianchi (Roma)

MSC:

49J10	Existence theories for free problems in two or more independent variables
74S30	Other numerical methods in solid mechanics (MSC2010)
74P10	Optimization of other properties in solid mechanics
49J45	Methods involving semicontinuity and convergence; relaxation

Keywords:

quasiconvexity; existence of minimizers

References:

[1]	J. J. Alibert & B. Dacorogna, An example of a quasiconvex function that is not polyconvex in two dimensions, Arch. Rational Mech. Anal., 117 (1992), 155-166. · Zbl 0761.26009
[2]	G. Aubert & R. Tahraoui, Théorèmes d’existence pour des problèmes du calcul des variations, J. Diff. Eqs., 33 (1979), 1-15. · Zbl 0404.49001
[3]	G. Aubert & R. Tahraoui, Sur la minimisation d’une fonctionelle non convexe, non différentiable en dimension 1, Boll. Un. Mat. Ital., 17 (1980), 244-258. · Zbl 0435.49001
[4]	G. Aubert & R. Tahraoui, Sur quelques résultats d’existence en optimisation non convexe, C.R. Acad. Sci. Paris, 297 (1983), 287-289. · Zbl 0534.49004
[5]	G. Aubert & R. Tahraoui, Théorèmes d’existence en optimisation non convexe, Appl. Anal., 18 (1984), 75-100. · Zbl 0563.49003
[6]	J. M. Ball & R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), 15-52. · Zbl 0629.49020
[7]	J. M. Ball & R. D. James, Proposed experimental tests of a theory of fine microstructure and the two well problem, Phil. Trans. Roy. Soc. London, A 338 (1991), 389-450. · Zbl 0758.73009
[8]	K. Bhattacharya, N. B. Firoozye, R. D. James & R. V. Kohn, Restrictions on microstructure, Proc. Royal Soc. Edinburgh, 124 (1994), 843-878. · Zbl 0808.73063
[9]	P. Bauman & D. Phillips, A nonconvex variational problem related to change of phase, J. Appl. Math. Optimization, 21 (1990), 113-138. · Zbl 0686.73018
[10]	H. Brezis, Analyse fonctionnelle, théorié et applications, Masson, Paris. 1983.
[11]	G. Buttazzo, V. Ferone & B. Kawohl, Minimum problems over sets of concave functions and related questions, Math. Nachrichten, to appear. · Zbl 0835.49001
[12]	A. Cellina, On minima of a functional of the gradient: necessary conditions, Nonlinear Anal, Theory Meth. Appl, 20 (1993), 337-341. · Zbl 0784.49021
[13]	A. Cellina, On minima of a functional of the gradient: sufficient conditions, Nonlinear Anal, Theory Meth. Appl, 20 (1993), 343-347. · Zbl 0784.49022
[14]	A. Cellina & G. Colombo, On a classical problem of the calculus of variations without convexity assumptions, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 7 (1990), 97-106. · Zbl 0702.49002
[15]	A. Cellina & S. Zagatti, An existence result for a minimum problem in the vectorial case of the calculus of variations, S.I.A.M J. Control Optimization, to appear. · Zbl 0822.49009
[16]	A. Cellina & S. Zagatti, A version of Olech’s lemma in a problem of the calculus of variations, S.I.A.M. J. Control Optimization, 32 (1994), 1114-1127. · Zbl 0874.49013
[17]	L. Cesari, An existence theorem without convexity conditions, S.I.A.M. J. Control, 12 (1974), 319-331. · Zbl 0255.49008
[18]	L. Cesari, Optimization ? Theory and applications, Springer-Verlag, 1983. · Zbl 0506.49001
[19]	M. Chipot & D. Kinderlehrer, Equilibrium configurations of crystals, Arch. Rational Mech. Anal., 103 (1988), 237-277. · Zbl 0673.73012
[20]	P. G. Ciarlet, Mathematical elasticity, Volume 1: Three-dimensional elasticity, North-Holland, 1988. · Zbl 0648.73014
[21]	P. G. Ciarlet & G. Geymonat, Sur les lois de comportement en élasticité non line?aire compressible, C. R. Acad. Sci. Paris, 295 (1982), 423-426. · Zbl 0497.73017
[22]	A. Cutrì, Some remarks on Hamilton-Jacobi equations and non convex minimization problems, Rendiconti Mat., 13 (1993), 733-749. · Zbl 0806.49021
[23]	B. Dacorogna, A relaxation theorem and its applications to the equilibrium of gases, Arch. Rational Mech. Anal., 77 (1981), 359-386. · Zbl 0492.49002
[24]	B. Dacorogna, Direct methods in the calculus of variations, Springer-Verlag, 1989. · Zbl 0703.49001
[25]	I. Ekeland, Discontinuités de champs hamiltoniens et existence de solutions optimales en calcul des variations, Publ Math. (IHES), 47 (1977), 5-32. · Zbl 0447.49015
[26]	I. Ekeland & R. Temam, Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars, 1974. · Zbl 0281.49001
[27]	N. B. Firoozye & R. V. Kohn, Geometric parameters and the relaxation of multiwell energies, in Microstructure and Phase Transition, edited by D. Kinderlehrer et al, Springer, 1992. · Zbl 0797.73051
[28]	I. Fonseca, Variational methods for elastic crystals, Arch. Rational Mech. Anal., 97 (1987), 189-220. · Zbl 0611.73023
[29]	I. Fonseca & L. Tartar, The gradient theory of phase transition for systems with two potential wells, Proc. Royal Soc. Edinburgh, 111A (1989), 89-102. · Zbl 0676.49005
[30]	R. L. Fosdick & G. Macsithigh, Helical shear of an elastic, circular tube with a nonconvex stored energy, Arch. Rational Mech. Anal., 84 (1983), 31-53. · Zbl 0529.73007
[31]	G. Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Royal Soc. Edinburgh, 124 (1994), 437-471. · Zbl 0809.49017
[32]	D. Giachetti & R. Schianchi, Minima of some non convex non coercive problems, Ann. Mat. Pura Appl., 165 (1993), 109-120. · Zbl 0807.49017
[33]	M. E. Gurtin & R. Temam, On the anti-plane shear problem in finite elasticity, J. Elasticity, 11 (1981), 197-206. · Zbl 0496.73036
[34]	Z. Hashin & S. Shtrikman, A variational approach to the theory of effective magnetic permeability of multiphase materials, J. Applied Phys., 33 (1962), 3125-3131. · Zbl 0111.41401
[35]	G. Hellwig, Partial differential equations, Teubner, 1977.
[36]	D. Kinderlehrer & P. Pedregal, Remarks about the analysis of gradient Young measures, Proc. Partial Differential Equations and Related Subjects, edited by M. Miranda, Pitman Research Notes in Math., 262 (1992), Longman, 125-150. · Zbl 0833.49012
[37]	R. Klötzler, On the existence of optimal processes, Banach Center Publications, Vol. 1, Warszawa, 1976, 125-130.
[38]	R. V. Kohn, The relaxation of a double-well energy, Continuum Mech. Thermodynamics, 3 (1991), 918-1000.
[39]	R. V. Kohn & G. Strang, Optimal design and relaxation of variational problems I, II, III, Comm. Pure Appl. Math., 39 (1986), 113-137, 139-182, 353-377. · Zbl 0609.49008
[40]	H. Le Dret & A. Raoult, Enveloppe quasi-convexe de la densité d’énergie de Saint Venant-Kirchhoff, C. R. Acad. Sci Paris, 318 (1994), 93-98. · Zbl 0792.73034
[41]	H. Le Dret & A. Raoult, The quasiconvex envelope of the Saint Venant-Kirchhoff energy function, preprint, 1994. · Zbl 0792.73034
[42]	P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Pitman Research Notes in Math. 69, 1982.
[43]	P. Marcellini, Alcune osservazioni sull’esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessita, Rendiconti di Matematica, 13 (1980), 271-281. · Zbl 0454.49015
[44]	P. Marcellini, A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations, Mathematical Theories of Optimization, Proceedings, edited by J. P. Cecconi & T. Zolezzi, Springer Lecture Notes in Math., 979, 1983, 216-231. · Zbl 0505.49009
[45]	P. Marcellini, Some remarks on uniqueness in the calculus of variations, Collège de France Seminar, edited by H. Brezis & J.-L. Lions, Pitman Research Notes in Math., 84, 1983, 148-153.
[46]	P. Marcellini, Non convex integrals of the calculus of variations, Methods of nonconvex Analysis, edited by A. Cellina, Springer Lecture Notes in Math., 1446, 1990, 16-57. · Zbl 0735.49002
[47]	P. Marcellini & C. Sbordone, Semicontinuity problems in the calculus of variations, Nonlinear Anal., Theory Meth. Appl., 4 (1980), 241-257. · Zbl 0537.49002
[48]	E. Mascolo, Some remarks on non-convex problems, Material Instabilities in Continuum Mechanics, edited by J. M. Ball, Oxford Univ. Press, 1988, 269-286.
[49]	E. Mascolo & R. Schianchi, Existence theorems for nonconvex problems, J. Math. Pures Appl., 62 (1983), 349-359. · Zbl 0522.49001
[50]	E. Mascolo & R. Schianchi, Nonconvex problems of the calculus of variations, Nonlinear Anal., Theory. Meth. Appl., 9 (1985), 371-379.
[51]	E. Mascolo & R. Schianchi, Existence theorems in the calculus of variations, J. Diff. Eqs., 67 (1987), 185-198.
[52]	C. B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, 1966. · Zbl 0142.38701
[53]	S. Müller, Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations, Proceedings, Problems involving change of phase, Stuttgart, 1988, edited by K. Kirchgässner, Springer Lecture Notes in Physics, 1990, 31-44.
[54]	C. Olech, Integrals of set valued functions and linear optimal control problems, Colloque sur la théorie mathématique du contrôle optimal, C.B.R.M. Louvain, 1970, 109-125.
[55]	A. C. Pipkin, Elastic materials with two prefered states, Quart. J. Mech. Appl. Math., 44 (1991), 1-15. · Zbl 0735.73032
[56]	J. P. Raymond, Champs Hamiltoniens, relaxation et existence de solutions en calcul des variations, J. Diff. Eqs., 70 (1987), 226-274. · Zbl 0643.49022
[57]	J. P. Raymond, Conditions nécessaires et suffisantes d’existence de solutions en calcul des variations, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 4 (1987), 169-202.
[58]	J. P. Raymond, Théorème d’existence pour des problèmes variationnels non convexes, Proc. Royal Soc. Edinburgh, 107A (1987), 43-64. · Zbl 0628.49006
[59]	J. P. Raymond, Existence of minimizers for vector problems without quasiconvexity condition, Nonlinear Anal., Theory, Methods Appl., 18 (1992), 815-828. · Zbl 0762.49001
[60]	P. Sternberg, Vector valued local minimizers of nonconvex variational problems, Rocky Mountain J. Math., 21 (1991), 799-807. · Zbl 0760.49008
[61]	V. Sverak, On the problem of two wells, Geometric parameters and the relaxation of multiwell energies, in Microstructure and Phase Transition, edited by D. Kinderlehrer et al., Springer, 1992.
[62]	R. Tahraoui, Théorèmes d’existence en calcul des variations et applications à l’élasticité non linéarie, C. R. Acad. Sci. Paris, 302 (1986), 495-498. Also in Proc. Royal Soc. Edinburgh, 109 A (1988), 51-78. · Zbl 0587.49008
[63]	R. Tahraoui, Sur une classe de fonctionnelles non convexes et applications, S.I.A.M. J. Math. Anal., 21 (1990), 37-52.
[64]	L. Tartar, Estimations fines des coefficients homogénéisés, Ennio De Giorgi Colloquium, edited by P. Krée, Pitman Research Notes in Math. 125, 1985, 168-187.
[65]	C. Truesdell & W. Noll, The nonlinear field theories of mechanics, in Handbuch der Physik, III/3, Springer-Verlag, 1965. · Zbl 0779.73004