Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T15:16:44.717Z Has data issue: false hasContentIssue false
  • English
  • Français

An existence result for a nonconvex variational problem via regularity

Published online by Cambridge University Press:  15 September 2002

Irene Fonseca ,
Nicola Fusco  and
Paolo Marcellini
Irene Fonseca
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.; fonseca@andrew.cmu.edu.
Nicola Fusco
Affiliation:
Dipartimento di Matematica “R. Caccioppoli”, Università di Napoli, Via Cintia, 80126 Napoli, Italy; fusco@matna1.dma.unina.it.
Paolo Marcellini
Affiliation:
Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67 A, 50134 Firenze, Italy; marcell@udini.math.unifi.it.
Get access

Abstract

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.

Keywords

Nonconvex variational problemsuniform convexityregularityimplicit differential equations.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 69 - 95
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E. and Fusco, N., Regularity for minimizers of nonquadratic functionals: The case 1<p<2. J. Math. Anal. Appl. 140 (1989) 115-135. CrossRef
L. Ambrosio, N. Fusco and D. Pallara, Special functions of bounded variation and free discontinuity problems. Oxford University Press (2000).
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 15-52. CrossRef
Ball, J.M. and James, R.D., 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. CrossRef
Celada, P. and Perrotta, S., Minimizing non convex, multiple integrals: A density result. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 721-741. CrossRef
Cellina, A., On minima of a functional of the gradient: Necessary conditions. Nonlinear Anal. 20 (1993) 337-341. CrossRef
Cellina, A., On minima of a functional of the gradient: Sufficient conditions. Nonlinear Anal. 20 (1993) 343-347. CrossRef
Dacorogna, B. and Marcellini, P., Existence of minimizers for non quasiconvex integrals. Arch. Rational Mech. Anal. 131 (1995) 359-399. CrossRef
Dacorogna, B. and Marcellini, P., 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.
Dacorogna, B. and Marcellini, P., 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.
Dacorogna, B. and Marcellini, P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math. 178 (1997) 1-37. CrossRef
B. Dacorogna and P. Marcellini, Implicit partial differential equations. Birkhäuser, Boston (1999).
Dacorogna, B. and Marcellini, P., Attainment of minima and implicit partial differential equations. Ricerche Mat. 48 (1999) 311-346.
De Blasi, F.S. and Pianigiani, G., On the Dirichlet problem for first order partial differential equations. A Baire category approach. NoDEA Nonlinear Differential Equations Appl. 6 (1999) 13-34. CrossRef
Dolzmann, G., Kirchheim, B., Müller, S. and Sverák, V., The two-well problem in three dimensions. Calc. Var. Partial Differential Equations 10 (2000) 21-40. CrossRef
L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986) 227-252.
Evans, L.C. and Gariepy, R.F., Blowup, compactness and partial regularity in the calculus of variations. Indiana Univ. Math. J. 36 (1987) 361-371. CrossRef
Fonseca, I. and Francfort, G., 3D-2D asymptotic analysis of an optimal design problem for thin films. J. Reine Angew. Math. 505 (1998) 173-202.
Fonseca, I. and Fusco, N., Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997) 463-499.
Fonseca, I. and Leoni, G., Bulk and contact energies: Nucleation and relaxation. SIAM J. Math. Anal. 30 (1998) 190-219. CrossRef
Friesecke, G., 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. CrossRef
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. CrossRef
Mascolo, E. and Schianchi, R., Existence theorems for nonconvex problems. J. Math. Pures Appl. 62 (1983) 349-359.
Mascolo, E. and Schianchi, R., Nonconvex problems in the calculus of variations. Nonlinear Anal. 9 (1985) 371-379. CrossRef
Mascolo, E. and Schianchi, R., Existence theorems in the calculus of variations. J. Differential Equations 67 (1987) 185-198. CrossRef
S. Müller and V. Sverák, Attainment results for the two-well problem by convex integration, edited by J. Jost. International Press (1996) 239-251.
Raymond, J.P., Existence of minimizers for vector problems without quasiconvexity conditions. Nonlinear Anal. 18 (1992) 815-828. CrossRef
Sychev, M.A., Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 611-631.
Zagatti, S., Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000) 384-399. CrossRef
W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York, Grad. Texts in Math. (1989).