An existence result for a nonconvex variational problem via regularity

 Access to full text

 Full (PDF)

1. [1] E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case $1lt;plt;2$. J. Math. Anal. Appl. 140 (1989) 115-135. Zbl0686.49004MR997847
2. [2] L. Ambrosio, N. Fusco and D. Pallara, Special functions of bounded variation and free discontinuity problems. Oxford University Press (2000). Zbl0957.49001MR1857292
3. [3] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 15-52. Zbl0629.49020MR906132
4. [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. Zbl0758.73009
5. [5] P. Celada and S. Perrotta, Minimizing non convex, multiple integrals: A density result. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 721-741. Zbl0963.49001MR1776673
6. [6] A. Cellina, On minima of a functional of the gradient: Necessary conditions. Nonlinear Anal. 20 (1993) 337-341. Zbl0784.49021MR1206422
7. [7] A. Cellina, On minima of a functional of the gradient: Sufficient conditions. Nonlinear Anal. 20 (1993) 343-347. Zbl0784.49022MR1206423
8. [8] B. Dacorogna and P. Marcellini, Existence of minimizers for non quasiconvex integrals. Arch. Rational Mech. Anal. 131 (1995) 359-399. Zbl0837.49002MR1354700
9. [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. Zbl0846.35028
10. [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. Zbl0860.35020
11. [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. Zbl0901.49027
12. [12] B. Dacorogna and P. Marcellini, Implicit partial differential equations. Birkhäuser, Boston (1999). Zbl0938.35002MR1702252
13. [13] B. Dacorogna and P. Marcellini, Attainment of minima and implicit partial differential equations. Ricerche Mat. 48 (1999) 311-346. Zbl0939.49013MR1765691
14. [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. Zbl0922.35039MR1674778
15. [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. Zbl0956.74039MR1803972
16. [16] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986) 227-252. Zbl0627.49006MR853966
17. [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. Zbl0626.49007MR891780
18. [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. Zbl0917.73052MR1662252
19. [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. Zbl0899.49018MR1612389
20. [20] I. Fonseca and G. Leoni, Bulk and contact energies: Nucleation and relaxation. SIAM J. Math. Anal. 30 (1998) 190-219. Zbl0924.49010MR1656999
21. [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. Zbl0809.49017MR1286914
22. [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. Zbl0505.49009MR713812
23. [23] E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems. J. Math. Pures Appl. 62 (1983) 349-359. Zbl0522.49001MR718948
24. [24] E. Mascolo and R. Schianchi, Nonconvex problems in the calculus of variations. Nonlinear Anal. 9 (1985) 371-379. MR783584
25. [25] E. Mascolo and R. Schianchi, Existence theorems in the calculus of variations. J. Differential Equations 67 (1987) 185-198. MR879692
26. [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. Zbl0930.35038MR1449410
27. [27] J.P. Raymond, Existence of minimizers for vector problems without quasiconvexity conditions. Nonlinear Anal. 18 (1992) 815-828. Zbl0762.49001MR1162475
28. [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. Zbl0971.49004MR1769245
29. [29] S. Zagatti, Minimization of functionals of the gradient by Baire’s theorem. SIAM J. Control Optim. 38 (2000) 384-399. Zbl0948.49004
30. [30] W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York, Grad. Texts in Math. (1989). Zbl0692.46022MR1014685

1. Michela Eleuteri, Regularity results for a class of obstacle problems
2. Pietro Celada, Giovanni Cupini, Marcello Guidorzi, Existence and regularity of minimizers of nonconvex integrals with growth
3. Bruno De Maria, A regularity result for a convex functional and bounds for the singular set
4. Michela Eleuteri, Hölder continuity results for a class of functionals with non-standard growth
5. Giuseppe Mingione, Regularity of minima: an invitation to the Dark Side of the Calculus of Variations