Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The -dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.
Mots-clés : nonconvex variational problems, uniform convexity, regularity, implicit differential equations
@article{COCV_2002__7__69_0,
author = {Fonseca, Irene and Fusco, Nicola and Marcellini, Paolo},
title = {An existence result for a nonconvex variational problem via regularity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {69--95},
publisher = {EDP-Sciences},
volume = {7},
year = {2002},
doi = {10.1051/cocv:2002004},
mrnumber = {1925022},
zbl = {1044.49011},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2002004/}
}
TY - JOUR AU - Fonseca, Irene AU - Fusco, Nicola AU - Marcellini, Paolo TI - An existence result for a nonconvex variational problem via regularity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 69 EP - 95 VL - 7 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002004/ DO - 10.1051/cocv:2002004 LA - en ID - COCV_2002__7__69_0 ER -
%0 Journal Article %A Fonseca, Irene %A Fusco, Nicola %A Marcellini, Paolo %T An existence result for a nonconvex variational problem via regularity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 69-95 %V 7 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2002004/ %R 10.1051/cocv:2002004 %G en %F COCV_2002__7__69_0
Fonseca, Irene; Fusco, Nicola; Marcellini, Paolo. An existence result for a nonconvex variational problem via regularity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 69-95. doi : 10.1051/cocv:2002004. http://www.numdam.org/articles/10.1051/cocv:2002004/
[1] and, Regularity for minimizers of nonquadratic functionals: The case . J. Math. Anal. Appl. 140 (1989) 115-135. | MR | Zbl
[2] , and, Special functions of bounded variation and free discontinuity problems. Oxford University Press (2000). | MR | Zbl
[3] and, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 15-52. | MR | Zbl
[4] and, Proposed experimental tests of a theory of fine microstructure and the two wells problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1991) 389-450. | Zbl
[5] and, Minimizing non convex, multiple integrals: A density result. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 721-741. | MR | Zbl
[6] , On minima of a functional of the gradient: Necessary conditions. Nonlinear Anal. 20 (1993) 337-341. | MR | Zbl
[7] , On minima of a functional of the gradient: Sufficient conditions. Nonlinear Anal. 20 (1993) 343-347. | MR | Zbl
[8] and, Existence of minimizers for non quasiconvex integrals. Arch. Rational Mech. Anal. 131 (1995) 359-399. | MR | Zbl
[9] and, Théorème d'existence dans le cas scalaire et vectoriel pour les équations de Hamilton-Jacobi. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 237-240. | MR | Zbl
[10] and, Sur le problème de Cauchy-Dirichlet pour les systèmes d'équations non linéaires du premier ordre. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 599-602. | Zbl
[11] and, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math. 178 (1997) 1-37. | Zbl
[12] and, Implicit partial differential equations. Birkhäuser, Boston (1999). | MR | Zbl
[13] and, Attainment of minima and implicit partial differential equations. Ricerche Mat. 48 (1999) 311-346. | MR | Zbl
[14] and, On the Dirichlet problem for first order partial differential equations. A Baire category approach. NoDEA Nonlinear Differential Equations Appl. 6 (1999) 13-34. | MR | Zbl
[15] ,, and, The two-well problem in three dimensions. Calc. Var. Partial Differential Equations 10 (2000) 21-40. | MR | Zbl
[16] , Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986) 227-252. | MR | Zbl
[17] and, Blowup, compactness and partial regularity in the calculus of variations. Indiana Univ. Math. J. 36 (1987) 361-371. | MR | Zbl
[18] and, asymptotic analysis of an optimal design problem for thin films. J. Reine Angew. Math. 505 (1998) 173-202. | MR | Zbl
[19] and, Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997) 463-499. | Numdam | MR | Zbl
[20] and, Bulk and contact energies: Nucleation and relaxation. SIAM J. Math. Anal. 30 (1998) 190-219. | MR | Zbl
[21] , A necessary and sufficient condition for non attainment and formation of microstructure almost everywhere in scalar variational problems. Proc. Royal Soc. Edinburgh Sect. A 124 (1994) 437-471. | MR | Zbl
[22] , A relation between existence of minima for nonconvex integrals and uniqueness for not strictly convex integrals of the calculus of variations, Math. Theories of Optimization, edited by J.P. Cecconi and T. Zolezzi. Springer-Verlag, Lecture Notes in Math. 979 (1983) 216-231. | MR | Zbl
[23] and, Existence theorems for nonconvex problems. J. Math. Pures Appl. 62 (1983) 349-359. | MR | Zbl
[24] and, Nonconvex problems in the calculus of variations. Nonlinear Anal. 9 (1985) 371-379. | MR
[25] and, Existence theorems in the calculus of variations. J. Differential Equations 67 (1987) 185-198. | MR
[26] and, Attainment results for the two-well problem by convex integration, edited by J. Jost. International Press (1996) 239-251. | MR | Zbl
[27] , Existence of minimizers for vector problems without quasiconvexity conditions. Nonlinear Anal. 18 (1992) 815-828. | MR | Zbl
[28] , Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 611-631. | MR | Zbl
[29] , Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000) 384-399. | Zbl
[30] , Weakly differentiable functions. Springer-Verlag, New York, Grad. Texts in Math. (1989). | MR | Zbl
Cité par Sources :
