An existence result for a nonconvex variational problem via regularity
ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 69-95.

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 x-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

DOI: 10.1051/cocv:2002004
Classification: 49J45, 49K20, 35F30, 35R70
Keywords: nonconvex variational problems, uniform convexity, regularity, implicit differential equations
Fonseca, Irene ; Fusco, Nicola 1; Marcellini, Paolo 

1 Dipartimento di Matematica e Applicazioni Monte Sant’Angelo, via Cintia, 80126 Napoli, Italy;
@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://archive.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://archive.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://archive.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, Volume 7 (2002), pp. 69-95. doi : 10.1051/cocv:2002004. http://archive.numdam.org/articles/10.1051/cocv:2002004/

[1] E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case 1<p<2. J. Math. Anal. Appl. 140 (1989) 115-135. | MR | Zbl

[2] L. Ambrosio, N. Fusco and D. Pallara, Special functions of bounded variation and free discontinuity problems. Oxford University Press (2000). | MR | Zbl

[3] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 15-52. | MR | Zbl

[4] J.M. Ball and R.D. James, 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] P. Celada and S. Perrotta, Minimizing non convex, multiple integrals: A density result. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 721-741. | MR | Zbl

[6] A. Cellina, On minima of a functional of the gradient: Necessary conditions. Nonlinear Anal. 20 (1993) 337-341. | MR | Zbl

[7] A. Cellina, On minima of a functional of the gradient: Sufficient conditions. Nonlinear Anal. 20 (1993) 343-347. | MR | Zbl

[8] B. Dacorogna and P. Marcellini, Existence of minimizers for non quasiconvex integrals. Arch. Rational Mech. Anal. 131 (1995) 359-399. | MR | Zbl

[9] B. Dacorogna and P. Marcellini, 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] B. Dacorogna and P. Marcellini, 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] B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math. 178 (1997) 1-37. | Zbl

[12] B. Dacorogna and P. Marcellini, Implicit partial differential equations. Birkhäuser, Boston (1999). | MR | Zbl

[13] B. Dacorogna and P. Marcellini, Attainment of minima and implicit partial differential equations. Ricerche Mat. 48 (1999) 311-346. | MR | Zbl

[14] F.S. De Blasi and G. Pianigiani, 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] G. Dolzmann, B. Kirchheim, S. Müller and V. Šverák, The two-well problem in three dimensions. Calc. Var. Partial Differential Equations 10 (2000) 21-40. | MR | Zbl

[16] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986) 227-252. | MR | Zbl

[17] L.C. Evans and R.F. Gariepy, Blowup, compactness and partial regularity in the calculus of variations. Indiana Univ. Math. J. 36 (1987) 361-371. | MR | Zbl

[18] I. Fonseca and G. Francfort, 3D-2D asymptotic analysis of an optimal design problem for thin films. J. Reine Angew. Math. 505 (1998) 173-202. | MR | Zbl

[19] I. Fonseca and N. Fusco, Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997) 463-499. | Numdam | MR | Zbl

[20] I. Fonseca and G. Leoni, Bulk and contact energies: Nucleation and relaxation. SIAM J. Math. Anal. 30 (1998) 190-219. | MR | Zbl

[21] G. Friesecke, 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] P. Marcellini, 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] E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems. J. Math. Pures Appl. 62 (1983) 349-359. | MR | Zbl

[24] E. Mascolo and R. Schianchi, Nonconvex problems in the calculus of variations. Nonlinear Anal. 9 (1985) 371-379. | MR

[25] E. Mascolo and R. Schianchi, Existence theorems in the calculus of variations. J. Differential Equations 67 (1987) 185-198. | MR

[26] S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, edited by J. Jost. International Press (1996) 239-251. | MR | Zbl

[27] J.P. Raymond, Existence of minimizers for vector problems without quasiconvexity conditions. Nonlinear Anal. 18 (1992) 815-828. | MR | Zbl

[28] M.A. Sychev, Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 611-631. | MR | Zbl

[29] S. Zagatti, Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000) 384-399. | Zbl

[30] W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York, Grad. Texts in Math. (1989). | MR | Zbl

Cited by Sources: