An existence result for a nonconvex variational problem via regularity
ESAIM: Control, Optimisation and Calculus of Variations, Tome 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 : https://doi.org/10.1051/cocv:2002004
Classification : 49J45,  49K20,  35F30,  35R70
Mots clés : nonconvex variational problems, uniform convexity, regularity, implicit differential equations
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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] 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 997847 | Zbl 0686.49004

[2] L. Ambrosio, N. Fusco and D. Pallara, Special functions of bounded variation and free discontinuity problems. Oxford University Press (2000). | MR 1857292 | Zbl 0957.49001

[3] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 15-52. | MR 906132 | Zbl 0629.49020

[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 0758.73009

[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 1776673 | Zbl 0963.49001

[6] A. Cellina, On minima of a functional of the gradient: Necessary conditions. Nonlinear Anal. 20 (1993) 337-341. | MR 1206422 | Zbl 0784.49021

[7] A. Cellina, On minima of a functional of the gradient: Sufficient conditions. Nonlinear Anal. 20 (1993) 343-347. | MR 1206423 | Zbl 0784.49022

[8] B. Dacorogna and P. Marcellini, Existence of minimizers for non quasiconvex integrals. Arch. Rational Mech. Anal. 131 (1995) 359-399. | MR 1354700 | Zbl 0837.49002

[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 1378259 | Zbl 0846.35028

[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 0860.35020

[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 0901.49027

[12] B. Dacorogna and P. Marcellini, Implicit partial differential equations. Birkhäuser, Boston (1999). | MR 1702252 | Zbl 0938.35002

[13] B. Dacorogna and P. Marcellini, Attainment of minima and implicit partial differential equations. Ricerche Mat. 48 (1999) 311-346. | MR 1765691 | Zbl 0939.49013

[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 1674778 | Zbl 0922.35039

[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 1803972 | Zbl 0956.74039

[16] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986) 227-252. | MR 853966 | Zbl 0627.49006

[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 891780 | Zbl 0626.49007

[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 1662252 | Zbl 0917.73052

[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 1612389 | Zbl 0899.49018

[20] I. Fonseca and G. Leoni, Bulk and contact energies: Nucleation and relaxation. SIAM J. Math. Anal. 30 (1998) 190-219. | MR 1656999 | Zbl 0924.49010

[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 1286914 | Zbl 0809.49017

[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 713812 | Zbl 0505.49009

[23] E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems. J. Math. Pures Appl. 62 (1983) 349-359. | MR 718948 | Zbl 0522.49001

[24] E. Mascolo and R. Schianchi, Nonconvex problems in the calculus of variations. Nonlinear Anal. 9 (1985) 371-379. | MR 783584

[25] E. Mascolo and R. Schianchi, Existence theorems in the calculus of variations. J. Differential Equations 67 (1987) 185-198. | MR 879692

[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 1449410 | Zbl 0930.35038

[27] J.P. Raymond, Existence of minimizers for vector problems without quasiconvexity conditions. Nonlinear Anal. 18 (1992) 815-828. | MR 1162475 | Zbl 0762.49001

[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 1769245 | Zbl 0971.49004

[29] S. Zagatti, Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000) 384-399. | Zbl 0948.49004

[30] W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York, Grad. Texts in Math. (1989). | MR 1014685 | Zbl 0692.46022

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